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Semiconductors and p-n Junction | Intrinsic, Extrinsic & Working

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 “Types of Semiconductors and P-N Junction Explained with Energy Band Diagrams | Intrinsic, Extrinsic, n-Type & p-Type”

Author:

Prof. Kali Chandrakant

M.Sc., M.Ed., D.C.S.

50+ Years of experience in Physics teaching

 1.Introduction:

  Have you ever wondered what makes your smart-phone, laptop, or even your car’s smart features possible? The answer lies in semiconductors. These materials are the backbone of modern electronics, sitting perfectly between conductors (like copper) and insulators (like glass).

     In this guide, we’ll break down the different types of semiconductors, how “doping” changes their behavior, and the role of the Fermi level in making technology work.

   Also, this document provides a comprehensive overview of the P-N junction, a fundamental building block of modern semiconductor devices. It covers the formation of the junction, its working principle under different biasing conditions, the resulting voltage-current (V-I) characteristics, and a range of important applications. Understanding the P-N junction is crucial for anyone studying or working with electronics and semiconductor technology.

 2.What Are Semiconductors?

      A semiconductor is a material whose electrical conductivity lies between that of a conductor (like copper) and an insulator (like glass).

The most common semiconductor materials are:

  • Silicon (Si) – Energy gap: 1.1 eV
  • Germanium (Ge) – Energy gap: 0.72 eV

Semiconductors are classified into two main types:

  1. Intrinsic Semiconductors
  2. Extrinsic Semiconductors

Intrinsic & Extrinsic Semiconductors

 3. Intrinsic Semiconductors: The Pure Form:

    An intrinsic semiconductor is a substance in its chemically pure state, without any intentional impurities.

  • Structure: They are tetravalent, meaning they have four valence electrons that form covalent bonds with neighboring atoms. (Fig. A)

Fig.AIntrinsic semiconductor

  • At Absolute Zero (0 K): These materials act as perfect insulators because all electrons are locked in covalent bonds.
  • At Room Temperature: Thermal energy causes a few electrons to break free, leaving behind a “hole” (a positive charge carrier). In intrinsic semiconductors, the number of free electrons (n) is always equal to the number of holes (p).(Fig. A)
  • Energy Gaps:
  • Germanium: 0.72 eV 
  • Silicon: 1.1 eV

 4. Extrinsic Semiconductors: Power Through Impurity:

    Pure semiconductors aren’t very efficient at conducting electricity. To fix this, scientists use a process called doping—the intentional addition of specific impurities to increase the number of charge carriers.

Extrinsic Semiconductor = Intrinsic Semiconductor + Impurities

There are two primary types of extrinsic semiconductors: n-type and p-type.

 4.1. n-type (Negative Type Carriers):

        An n-type semiconductor is created by adding penta-valent impurities (atoms with 5 valence electrons) like Phosphorus (P), Arsenic (As), or Antimony (Sb) to a pure crystal.

     How it works: A penta-valent atom forms four covalent bonds with surrounding silicon atoms. (Fig. B)

  • The fifth electron is loosely bound.
  • Even a small amount of thermal energy frees this electron.

Fig.Bn type semiconductor

  • Majority Carrier: Electrons (negative charge).
  • Minority Carrier:
  • Charge: Despite having “extra” electrons, the material remains electrically neutral because the total number of protons in the nuclei equals the total number of electrons.

 4.2. p-type (Positive Type Carriers):

        A p-type semiconductor is created by adding trivalent impurities (atoms with 3 valence electrons) like Boron (B), Aluminum (Al), or Gallium (Ga).

How it works:

  • A trivalent atom forms three covalent bonds.
  • One bond remains incomplete. (Fig. C)
  • This creates a hole.

The impurity atom accepts an electron from a neighboring atom, creating hole conduction.

Fig.Cp type semiconductor

  • Majority Carrier: Holes (positive charge).
  • Minority Carrier:
  • Charge: Like n-type, p-type materials are electrically neutral.

 5. Understanding the Fermi Level (EF):

    “The Fermi Level is a theoretical energy level that represents the highest energy state an electron can occupy at absolute zero.”

Think of it as the “water level” of electrons in a material.

 5.1. Where does the Fermi Level sit?

Fig.DFermi level

 5.1.1. Fermi Level in Intrinsic Semiconductor:

  • Number of electrons = Number of holes
  • Equal probability of occupation in both bands
  • Fermi level lies exactly in the middle of the band gap. (Fig. D)

 5.1.2. Fermi Level in n-Type Semiconductor:

  • More electrons in conduction band
  • Higher probability of electron occupation
  • Fermi level shifts closer to the conduction band. (Fig. D)
  • Lies above the donor level

 5.1.3. Fermi Level in p-Type Semiconductor:

  • More holes in valence band
  • Higher probability of hole occupation
  • Fermi level shifts closer to the valence band. (Fig. D)

 

Semiconductor Type Fermi Level Position Reason
Intrinsic Exactly in the middle of the band gap Equal number of electrons and holes.
n-type Close to the Conduction Band High concentration of electrons (donors).
p-type Close to the Valence Band High concentration of holes (acceptors).

 6. Formation of a P-N Junction:

    “A P-N junction is formed when a p-type semiconductor material is joined with an n-type semiconductor material, creating an interface between them.”

 Fig.Ep-n juction Un-biased

      When a p-type and an n-type semiconductor are brought together, the concentration gradient of charge carriers causes diffusion. Electrons from the n-side diffuse into the p-side, and holes from the p-side diffuse into the n-side. (Fig. E)

 6.1. Working Principle:

   The diffusion of charge carriers across the junction leads to the following key phenomena:

 6.1.1. Depletion Region:

      As electrons diffuse from the n-side to the p-side, they recombine with holes near the junction. Similarly, holes diffusing from the p-side to the n-side recombine with electrons. This recombination depletes the region near the junction of free charge carriers (electrons and holes), creating a region called the depletion region or space charge region. The depletion region is essentially devoid of mobile charge carriers and acts as an insulator.

 6.1.2. Barrier Potential (Built-in Potential):

       The diffusion of charge carriers and the formation of the depletion region create an electric field across the junction. This electric field opposes further diffusion of charge carriers. The potential difference associated with this electric field is called the barrier potential or built-in potential (Vbi)The barrier potential depends on the doping concentrations of the p-type and n-type materials and the temperature.

  6.2. Biasing the P-N Junction:

    The behavior of the P-N junction is significantly affected by the external voltage applied across it. There are two primary biasing conditions:

 6.2.1. Forward Bias:

Fig.Fp-n juction forward bias

In forward bias, the positive terminal of the voltage source is connected to the p-side, and the negative terminal is connected to the n-side. This applied voltage opposes the barrier potential, effectively reducing the width of the depletion region. As the forward voltage increases, the barrier potential decreases, allowing more majority carriers to cross the junction. When the forward voltage exceeds the barrier potential, a large current flows through the junction.(Fig. F)

 6.2.2. Reverse Bias:

Fig.Gp-n juction reverse bias

In reverse bias, the positive terminal of the voltage source is connected to the n-side, and the negative terminal is connected to the p-side. This applied voltage reinforces the barrier potential, widening the depletion region. The increased depletion region further reduces the number of majority carriers that can cross the junction. Consequently, only a small reverse saturation current (Is) flows due to the minority carriers. (Fig. G)

 6.3. V-I Characteristics:

   The voltage-current (V-I) characteristics of a P-N junction diode describe the relationship between the voltage applied across the diode and the resulting current flowing through it. (Fig. H)

Fig.HV-I Characteristics curve

6.3.1. Forward Bias Region:

    In the forward bias region, the current increases exponentially with the applied voltage. The current is described by the diode equation:

 6.3.2. Reverse Bias Region:

    In the reverse bias region, small reverse saturation current (Is) flows, which is relatively independent of the applied voltage. However, if the reverse voltage exceeds a certain breakdown voltage (Vbr), a large reverse current flows, potentially damaging the diode.

6.3.3. Breakdown Region:

    When the reverse voltage applied to the diode exceeds the breakdown voltage, a large current flows in the reverse direction. This can occur due to two mechanisms:

 6.3.4. Avalanche Breakdown:

 High electric field accelerates minority carriers, which collide with other atoms, generating more electron-hole pairs. This process repeats, leading to a large current.

 6.3.5. Zener Breakdown:

     Occurs in heavily doped diodes. The high electric field directly breaks covalent bonds, generating a large number of carriers.

 6.4. Applications of P-N Junctions:

   P-N junctions are the fundamental building blocks of many semiconductor devices, including:

  • Diodes: Diodes are used for rectification (converting AC to DC), signal detection, switching, and voltage regulation.
  • Transistors: Bipolar junction transistors (BJTs) and field-effect transistors (FETs) utilize P-N junctions to amplify or switch electronic signals.
  • Solar Cells: Solar cells convert light energy into electrical energy using the photovoltaic effect, which relies on the properties of P-N junctions.
  • Light-Emitting Diodes (LEDs): LEDs emit light when electrons and holes recombine in the P-N junction under forward bias.
  • Photodiodes: Photodiodes detect light by generating a current when photons strike the P-N junction.
  • Varactor Diodes: Varactor diodes (also known as varicaps) are used as voltage-controlled capacitors, where the capacitance of the P-N junction varies with the applied reverse voltage.
  • Integrated Circuits (ICs): P-N junctions are essential components in the fabrication of integrated circuits, enabling the creation of complex electronic systems on a single chip.

 7. Conclusion:

Quick Comparison Table

Feature Intrinsic n-Type p-Type
Purity Pure Doped Doped
Majority Carrier Equal electrons & holes Electrons Holes
Minority Carrier Equal Holes Electrons
Conductivity Low High High
Fermi Level Middle of band gap Near conduction band Near valence band
  • Doping is the secret sauce that makes semiconductors useful.
  • n-type uses donor impurities to provide extra electrons.
  • p-type uses acceptor impurities to create holes.
  • The Fermi Level shifts depending on the type of doping, dictating how the material will behave in a circuit.

   The P-N junction is a crucial component in modern electronics. Its unique properties, stemming from the formation of the depletion region and the behavior under different biasing conditions, make it indispensable for a wide range of applications. Understanding the principles of P-N junction operation is essential for anyone working in the field of electronics and semiconductor technology.

    Understanding these fundamentals is the first step toward mastering electronics and solid-state physics.

 

Conductors, Insulators and Semiconductors | Classification and Examples

Posted on by Physics Prana

“Understanding Materials: A Guide to Conductors, Insulators, and Semiconductors”

Author:

Prof. Kali Chandrakant

M.Sc., M.Ed., D.C.S.

50+ Years of experience in Physics teaching

 1.Introduction:

      Have you ever wondered why copper is used in electrical wires while glass is used as an insulator? The answer lies in how materials conduct electricity. Based on their electrical conductivity, materials are classified into three main categories: Conductors, Insulators, and Semiconductors.

     In this article, we will explore these materials through the lens of Band Theory. We will explore their distinct properties, atomic structures, and applications, with a focus on understanding how these materials behave under the influence of an electric field.

 2.Band Theory: 

2.1. Main points:

  1. The bonding of atoms, due to the sharing of electrons, is called covalent bonding.
  2. In the crystal, closely spaced energy levels form a band called as the energy band.  Each orbit has a separate energy band.
  3. A band of energy levels associated with valence electrons and uppermost filled band is calledvalence band. Electrons from other bands cannot be removed but electrons from valence band can be removed by supplying a little energy.
  4. The empty band above valence band lowest unfilled band is conduction band.
  5. The valence band and conduction bandare separated by a gap called forbidden energy gap.

 2.2.Band structure:

   a) The electrons in an isolated atom occupy discrete energy levels. When atoms are close to each other, these electrons can use the energy levels of their neighbors.

   b) When the atoms are all regularly arranged is called the crystal lattice of a solid, the energy levels become grouped together in a band. This is a continuous range of allowed energies rather than a single level. There will also be groups of energies that are not allowed, is known as a band gap Eg.

  c)  Similar to the energy levels of an individual atom, the electrons will fill the lower bands first.

  d)  The Fermi levelgives a rough idea of which levels electrons will generally fill up to that level.  

Band structure of conductor,semiconductor,insulator

Fig. A Band diagram

3. Conductors:

 3.1. Definition and Properties:

     Conductors are materials that allow electric current to flow easily through them. They possess a large number of free electrons, which are electrons that are not bound to individual atoms and can move freely within the material.

  • High Electrical Conductivity: This is the defining characteristic of conductors. They offer very little resistance to the flow of electric current.
  • Low Resistivity: Resistivity is the inverse of conductivity. Conductors have very low resistivity values.
  • Metallic Bonding: Most conductors are metals, characterized by metallic bonding where electrons are delocalized and shared among many atoms.
  • Temperature Dependence: The conductivity of most conductors decreases with increasing temperature. This is because the increased thermal vibration of atoms hinders the movement of free electrons.

 3.2. Atomic Structure:

      Conductors have a partially filled valence band or overlapping valence and conduction bands. This means that electrons can easily move into the conduction band with minimal energy input. In simpler terms, there is no energy gap, or a very small one, between the valence and conduction bands. (Fig. A)

 3.3. Examples:

  • Copper (Cu): Widely used in electrical wiring due to its excellent conductivity and relatively low cost.
  • Aluminum (Al): Lighter than copper and also a good conductor used in power transmission lines.
  • Silver (Ag): The best conductor of electricity, but its high cost limits its use to specialized applications.
  • Gold (Au): Highly resistant to corrosion, making it suitable for use in electronic connectors and other critical applications.

 3.4. Applications:

     Conductors are essential components in virtually all electrical and electronic devices.

  They are used in:

  • Electrical Wiring: To carry electricity from power sources to appliances and other devices.
  • Electronic Circuits: To connect various components and allow current to flow through the circuit.
  • Power Transmission Lines: To transmit electricity over long distances.
  • Heat Sinks: Some conductors, like aluminum, are also good thermal conductors and are used to dissipate heat from electronic components.

 4. Insulators:

 4.1. Definition and Properties:

    Insulators are materials that resist the flow of electric current. They have very few free electrons and a large energy gap between the valence and conduction bands.

  • High Electrical Resistance: Insulators offer very high resistance to the flow of electric current.
  • High Resistivity: Insulators have very high resistivity values.
  • Covalent Bonding: Many insulators are characterized by covalent bonding, where electrons are shared between atoms in a way that restricts their movement.
  • Temperature Dependence: The conductivity of most insulators increases slightly with increasing temperature, but this increase is usually negligible.

 4.2. Atomic Structure:

      Insulators have a large energy gap between the valence and conduction bands. This means that a significant amount of energy is required to move electrons from the valence band to the conduction band, making it difficult for current to flow. (Fig. A)

 4.3. Examples:

  • Rubber: Used to insulate electrical wires and cables.
  • Glass: Used in insulators for power lines and in electronic components.
  • Plastic: Used in a wide variety of applications, including insulation for wires, cables, and electronic components.
  • Ceramics: Used in high-voltage insulators and other applications where high temperature resistance is required.
  • Wood: Used in some low-voltage applications, but its insulating properties can vary depending on moisture content.

 4.4. Applications:

    Insulators are used to prevent the flow of electric current in unwanted directions and to protect people from electric shock.

 They are used in:

  • Electrical Wiring and Cables: To insulate conductors and prevent short circuits.
  • Power Lines: To insulate the conductors from the supporting structures and prevent current leakage.
  • Electronic Components: To isolate different parts of a circuit and prevent interference.
  • Protective Gear: Used in gloves, boots, and other equipment to protect electricians and other workers from electric shock.

5. Semiconductors:

5.1. Definition and Properties:

     Semiconductors are materials that have electrical conductivity between that of conductors and insulators. Their conductivity can be controlled by factors such as temperature, light, and the presence of impurities.

  • Intermediate Conductivity: Semiconductors have conductivity values between those of conductors and insulators.
  • Controllable Conductivity: The conductivity of semiconductors can be significantly altered by doping (adding impurities) or by applying an electric field.
  • Temperature Dependence: The conductivity of semiconductors generally increases with increasing temperature.
  • Energy Gap: Semiconductors have a smaller energy gap than insulators, but a larger energy gap than conductors.

5.2. Atomic Structure:

    Semiconductors have an energy gap between the valence and conduction bands that is small enough to allow some electrons to move into the conduction band at room temperature. The number of electrons in the conduction band can be increased by doping the semiconductor with impurities.     (Fig. A)

 5.3. Examples:

  • Silicon (Si): The most widely used semiconductor material in the electronics industry.
  • Germanium (Ge): Used in some specialized applications, but less common than silicon.
  • Gallium Arsenide (GaAs): Used in high-speed electronic devices and optoelectronic devices.

 5.4. Doping:

   Doping is the process of adding impurities to a semiconductor to increase its conductivity.

There are two types of doping:

  • N-type Doping: Adding impurities with more valence electrons than the semiconductor material (e.g., adding phosphorus to silicon). This creates an excess of free electrons, increasing conductivity.
  • P-type Doping: Adding impurities with fewer valence electrons than the semiconductor material (e.g., adding boron to silicon). This creates “holes” (electron vacancies) that can move through the material, effectively carrying positive charge and increasing conductivity.

 5.5. Applications:

Semiconductors are the foundation of modern electronics. They are used in:

  • Transistors: Used to amplify and switch electronic signals.
  • Diodes: Used to allow current to flow in only one direction.
  • Integrated Circuits (ICs): Complex circuits containing millions or billions of transistors and other components on a single chip.
  • Solar Cells: Used to convert sunlight into electricity.
  • LEDs (Light-Emitting Diodes): Used to emit light when current flows through them.

  6. Conclusion:

    Conductors, insulators, and semiconductors are essential materials with distinct electrical properties that make them suitable for a wide range of applications. Understanding their properties and behavior is crucial for designing and developing electrical and electronic devices. The ability to control the conductivity of semiconductors through doping has revolutionized the electronics industry, leading to the development of countless innovative technologies.

 

How to Identify Whether a Photograph Is 2D or 3D | Physics of Imaging

Posted on by Physics Prana

How to Recognize Whether a Photograph Is 2D or 3D?

Author:

Prof. Kali Chandrakant

M.Sc., M.Ed., D.C.S.

50+ Years of experience in Physics teaching

 1. Introduction:

    Photography is an integral part of modern life. With the widespread use of smart phones, most people’s memories are filled with images, especially selfies. Digital platforms like Google provide a vast repository of visual information. Since the invention of the laser in 1960, methods for storing and representing visual information have advanced significantly.

    Today, we commonly encounter two types of photographic representations: two-dimensional (2D) and three-dimensional (3D) images. This article explains how to distinguish between a 2D photograph and a 3D photograph using principles of optics and human vision.

 2. Human Vision and Depth Perception:

Vision  Fig. A              Structure of eye  Fig. B

Fig. A: Light reflecting off an object toward the eye

   To understand photography, we must first understand the human eye. The process of “seeing” is a complex biological sequence: (Fig. B)

  1. Light Entry: Light reflects off an object and enters the eye through the cornea and pupil.
  2. Focusing: The eye’s lens refracts the light, focusing it onto the retina at the back of the eye.
  3. Signal Conversion: Photoreceptor cells on the retina convert light into electrical signals.
  4. Processing: The optic nerve carries these signals to the brain, which interprets them as the images we see.

    Although the images formed on the retina are fundamentally two-dimensional, the brain interprets them as three-dimensional by integrating information from both eyes and using depth cues such as stereopsis and perspective.

  Stereopsis refers to the brain’s ability to fuse slightly different views from the left and right eyes into a perception of depth and distance—a key aspect of 3D vision.

 3. Two-Dimensional (2D) Photography:

  3.1. Conventional Photography:

    Traditional photography—the kind we use every day—is limited. It only captures a fraction of the information available in light waves.

    A two-dimensional image exists only in length and width; it lacks depth. If a flat image is viewed edge-on, it appears as a line.(Fig. C)

Fig.C2 D image

Fig. C: Example of a two-dimensional representation

 Photography, in its conventional form, captures 2D images of three-dimensional objects by recording the intensity of light.(Fig. D)  Light from a scene strikes an image sensor or photographic film, which then records the brightness distribution.

Fig.D Camera

Fig. D: Formation of a photographic image on film or sensor

    In conventional photography, the phase information of light waves (which encodes depth) is lost. As a result, features at different distances from the camera (such as the nose and eyes in a portrait) appear to lie on the same plane.

  3.2. Key Features of 2D Photography:

  1. Only the intensity of light is recorded, without phase information.
  2. Normal light sources such as sunlight or artificial lighting suffice.
  3. A lens is required to focus the image onto a recording medium.
  4. Damage to the negative or digital image can irreversibly lose information.
  5. A single photograph cannot simultaneously record multiple distinct views.
  6. The information storage capacity is limited.
  7. Superimposing many images together is not feasible in a single exposure.
  8. Traditional photography produces a negative on film.

 4. Three-Dimensional (3D) Photography: Holography:

  4.1. Holography:

    A true three-dimensional image contains depth (the Z dimension) in addition to height and width. This allows visualization from different angles.

Fig.EHologram

Fig. E  A commercial hologram on product packaging

   One method to create a 3D image is to capture multiple views of an object from slightly different angles and combine them so that each eye sees a distinct perspective. This principle underlies stereoscopic photography and 3D movies.

  Fig. F            Hologram

Fig. F: A three-dimensional holographic image

 4.2. Dennis Gabor:

Fig.GDenis Gabor

Fig. G: Dennis Gabor, pioneer of holography

     In 1948, Dennis Gabor developed a technique to record both the amplitude and phase of light waves reflected from an object, enabling true 3D imaging. This process is known as holography. A hologram captures the full optical information of the object via interference and diffraction patterns formed by coherent light, such as from a laser.

  4.3. Key Features of 3D Photography (Holography):

  1. Both intensity and phase information of light are recorded.
  2. A coherent, monochromatic light source (usually a laser) is required.
  3. Light scattered directly from the object reaches the recording medium.
  4. Even a fragment of a hologram can reconstruct the entire 3D image.
  5. Multiple images can be recorded on a single hologram.
  6. The information storage capacity is high.
  7. Multilayered and superimposed images are possible.
  8. Holograms typically produce a positive reconstructed pattern.

 5. Recognizing Whether a Photograph Is 2D or 3D:

    Consider three photographs of a building taken from different directions:

 

Front view Left view Right view

Fig. H: Front view of the building

 Fig. I: Left-side view of the building

 

 Fig. J: Right-side view of the building

 

  Each photograph captures different details depending on the angle from which it was taken. However, once a 2D photograph is taken, changing the observer’s viewing position does not reveal additional information—the recorded scene remains unchanged.

   In contrast, a true 3D representation (such as a hologram) changes perceptually with the angle of view, revealing additional spatial information about the object as the observer’s perspective changes.

 6. Practical Identification Rule:

    2D Photograph:

   The recorded view does not change with a shift in viewing angle.

    3D Image (Hologram):

    The observed image changes or reveals additional spatial information when viewed from different angles.

   Thus, a simple method to determine whether an image is 2D or 3D is to observe the scene from varying angles: unchanging information indicates a 2D photograph, whereas changing information indicates a 3D image.

Feature 2D Photograph (Figs C, D, ) 3D Hologram (Figs E, F, )
Recorded Data Intensity only Intensity + Phase
Light Source Ambient/Flash Coherent Laser
Perspective Static / Flat Changes with viewing angle
Visual Depth Optical Illusion Physical Information
Technology Standard Camera Laser / Holography

  7. Conclusion:

     The transition from 2D to 3D is the transition from a “partial record” to a “total record” of light. Conventional photography captures two-dimensional images by recording only the intensity of light, resulting in flat representations of three-dimensional scenes.

     Holography, on the other hand, records both intensity and phase, enabling the reconstruction of full 3D images that reveal depth and parallax. Changing the viewing angle provides a straightforward way to distinguish between 2D photographs and 3D holographic images.

8. Video Support:

Please observe the 3d video by 360° video recording

 

LASER working principle, Types and Applications explained

Posted on by Physics Prana

Part 2: “How Laser work?  Stimulated emission, Types, and mind-blowing applications”

Author:

Prof. Kali Chandrakant

M.Sc., M.Ed., D.C.S.

50+ Years of experience in Physics teaching

 1. Introduction:

     Welcome back! In Part 1, we unpacked LASER’s origins and basics like population inversion. Now, see photons dance to create coherent beams—and transform industries.

 2. Nature of Radiation:

    “The energy which is propagating in the form the electromagnetic wave traveling at speed of 3×108 m/s is called radiation.”

Based on range of frequency or wave length the radiation can be classified as: radio waves, micro waves, infrared, visible light, ultraviolet, X-rays, γ-rays, and cosmic rays.

Visible light lies between 4000 Å – 7000 Å.

Light shows dual nature:

  • Wave
  • Particle (Photon)

Each photon carries energy:    E = hν
 3. The effect of incident photon on the collection of atoms:

   3.1. Daily life exaple:         

To understand the effect of an incident photon on a collection of atoms, let us consider a common experience. Suppose there is a mango tree full of mangoes. To detach a mango from the tree, a stone is thrown toward it. If the kinetic energy of the stone overcomes the binding energy of the mango, the mango will fall down.

   Different outcomes are possible when the stone is thrown toward the mango:

  1. The stone does not hit the mango.
  2. The stone hits the mango, but the mango does not detach due to insufficient supplied energy.
  3. The stone hits the mango and detaches it when the kinetic energy of the stone matches or exceeds the binding energy of the mango.

   In view of this example, let us now consider the effect of an incident photon on a collection of atoms.

 3.2. Photon passes away:

Fig F   Let E1​ represent the energy of the ground state with population N1, and ​ E2 represent the energy of the excited state with population N2 (N1>N2).  If a photon of energy hν≠E2−E1 is incident on a collection of atoms, it passes through the collection without interaction. (Fig F).

  3.3. Stimulated Absorption:

   “The process in which the atom absorbs photon energy and get excited is called stimulated absorption or simply absorption”

Fig G

 

   A photon of energy hν=E2−E1 is incident on a collection of atoms with populations N1>N2 . There is a maximum probability that the photon will interact with atoms in the ground state, because the population of the ground state is higher. The photon energy matches the energy difference between the atomic levels. Hence, when the photon collides with an atom in the ground state, it is completely absorbed by that atom. As a result, the atom gets excited and shifts to the energy level E2 (Fig. G).

  3.3.1. The conditions required for stimulated absorption:

a.The population of lower state (N1) must be greater than the population of excited state (N2) i.e. N1 ˃N2.

 b.The energy of the incident photon must match the energy difference between the atomic levels; that is     hν=E2-E1.

    This is similar to absorbing kinetic energy of stone by stem in our mango tree example.

                         Atom + hν = excited atom

 The rate of absorption transition Rab is the number of atoms per unit volume per second which are shifted from lower level to higher level. Rab is proportional to the population N1 of lower state and the energy density of photons ρ(ν). 

Rab = B12  ρ(ν) N1

  Where, B12 is known as the Einstein coefficient for stimulated absorption. B12 represents the probability of stimulated transition from state 1 to state 2.

 3.4. Spontaneous Emission:

   “The emission of photon by an excited atom spontaneously is called spontaneous emission”

Fig H After photon absorption, the atom in the excited state has a lifetime of excitation of the order of 10-8 second. Therefore, the time duration for which the atom remains in the excited state E2 is approximately 10-8 second. The atom then spontaneously returns to the ground state E1 . During this process, the electron of the atom jumps from energy level E2​ to E1​ by spontaneously emitting a photon of energy hν = E2 –E1 spontaneously.

(Fig. H) 


  Spontaneous emission is random in nature and is a completely disordered and uncontrolled process. The light emitted by usual sources such as the Sun, stars, electric bulbs, fire, etc., is due to spontaneous emission. Hence, the light from these sources is incoherent and un-polarized. Spontaneous emission is similar to people moving randomly in a bazaar.          

              Excited atom= atom + hν

The rate of spontaneous transition Rsp is the number of atoms per unit volume per second which are emitted from higher level to lower level. Rsp is proportional to the population N2 of excited state only.

Rsp= A21  N2

Where, A21 is known as the Einstein coefficient for spontaneous emission. A21 represents the probability of spontaneous transition from state 2 to state 1.

  3.5. Stimulated Emission:

     “The emission of a photon by an excited atom due to the stimulus (or induction) provided by an incident photon of matching energy is called stimulated emission.”

   Now consider the situation in the collection of atoms where in the population inversion (N2 ˃ N2) is brought between meta-stable level (E2) and ground level (E1).

Fig I The photon of energy hν = E2 – E1  is incident on the collection of atoms. There is a maximum probability that the photon will interact with atoms in the meta-stable level, since the population of this level is higher. The energy of the incident photon is in resonance with the excess energy possessed by atoms in level E2​. This resonance triggers the excited atom to emit a photon.

Due to the stimulus provided by the incident photon, the atom jumps from energy level E2​ to E1​ by emitting a photon of energy      hν = E2 – E1.

 (Fig I)

     The bosonic nature of photons ensures that the incident photon and the induced photon occupy the same quantum state. In other words, the induced photon finds itself in the same state as the incident photon. As a result, the stimulated photon has the same frequency, direction, phase, and polarization as the incident photon.

     In this process, the number of incident photons is one, while the number of output photons is two. Thus, stimulated emission leads to amplification of radiation. Since these two photons have the same phase, they emerge together as coherent radiation. The stimulated emission process is therefore an ordered and controlled process.

    This process was theoretically discovered by Albert Einstein in 1917 while re-deriving Planck’s law of radiation using the concept of probability coefficients, known as Einstein coefficients, for absorption, spontaneous emission, and stimulated emission. Stimulated emission is analogous to a military march, where motion is highly ordered and synchronized.

    Stimulated emission is the basic principle of laser action, and Albert Einstein laid the foundation for the invention of the laser.

The conditions required for stimulated emissions are:

  1. There should be population inversion (N2 ˃ N1)
  2. To bring the population inversion, the upper level E2,must be meta-stable state.
  3. The incident photon must have energy hν = E2-E1which brings resonance.  

                                Excited atom + hν = Atom + 2hν

 The rate of stimulated emission Rst is the number of atoms per unit volume per second which are shifted from excited level to lower level. Rst is proportional to the population N2 of excited state and the energy density of photons ρ(ν). 

Rst = B21  ρ(ν) N2

    Where, B21 is known as the Einstein coefficient for stimulated emission. B21 represents the probability of stimulated emission from state 2 to state 1.

 4. Pumping schemes:

         Laser action takes place through the process of stimulated emission. The energy levels involved in laser action are:

a.Ground level
b. Pumping level
c. Upper lasing level (usually a meta-stable state)
d. Lower lasing level (usually the ground state)

   However, in some cases, the same energy level may serve more than one purpose. Accordingly, there are three pumping schemes:

  1. Two-level system:
    The ground level also acts as the lower lasing level, whereas the pumping level and the upper lasing level together form the upper level.
  2. Three-level system:
    Atoms are pumped to a higher excited level, from which they quickly drop to a meta-stable state. Laser action then takes place between this meta-stable level and the ground level.
  3. Four-level system:
    In this system, all four energy states mentioned above are distinct and separate.

 5. Active medium and active centers:
    The material used in a laser is called the active medium, and the atoms or ions responsible for the actual laser action are called active centers.

          The active medium may be solid, liquid, or gaseous. In the case of a ruby laser, ruby is the active medium and the Cr+3 ions are the active centers.

6. Resonator:

Fig J The resonator required for laser action consists of an active medium in the form of a long cylinder. One end of this cylinder is perfectly polished to serve as a perfect reflector, while the other end is partially polished to act as a partial reflector. (Fig J)

In this case, the term partial reflector has a different meaning. The reflector reflects light of intensity below a certain threshold value. If the intensity of the light exceeds this threshold, the light is transmitted completely.

   7. Characteristics of a Laser:

    A laser possesses all the properties of light such as reflection, refraction, diffraction, and interference. However, it also has some special properties. A laser is
a. highly intense,
b. nearly perfectly monochromatic,
c. highly coherent,
d. highly directional, and
e. polarized light.

 7.1. Highly Intense light:
    Laser light is highly intense or bright because the energy of the laser source is concentrated in a single direction. A large number of photons travel in the same direction, whereas in an ordinary light source the photons are distributed in all possible directions.

 7.2. Perfectly Monochromatic light:
     Laser light is highly monochromatic because the resonant cavity allows only a selected wavelength of light to participate in laser action.

                              E2 – E1 = hν = hc / λ

     In an ordinary light source, photons are produced due to transitions between different energy levels, resulting in radiation of different wavelengths.

 7.3. Perfectly Coherent light:
     Due to the process of stimulated emission, laser light is perfectly coherent. That is, all the waves travel in the same phase because of the selective nature of stimulated emission. Laser light exhibits both spatial coherence (coherence with respect to space) and temporal coherence (coherence with respect to time). Hence, all the emitted photons are in the same phase.

 7.4. Highly Directional light:
      A laser beam is emitted in the form of an almost parallel beam and travels in a specific direction through a very small cross-sectional area. The divergence of laser light is very small. Therefore, a laser beam can travel long distances without significant spreading.

  7.5. Plane Polarized light:
      Laser light is plane-polarized. All the waves of the laser vibrate in only one plane of electric field oscillation. The stimulated emission process and the resonant cavity are responsible for the polarization of laser light.

 8. Types of Lasers:

    Lasers are commonly classified based on the type of active medium used. There are four main types of lasers:

  1. Solid-state lasers

 2. Gas lasers

 3. Liquid dye lasers

 4. Semiconductor diode lasers

Some lasers operate in continuous wave (CW) mode, while others operate in pulsed mode.

8.1. Solid-State Laser:

    In a solid-state laser, the active medium is a solid material containing active centers. The first historically successful laser was the ruby laser, built by Theodore Maiman in 1960. It is a pulsed laser based on a three-level pumping scheme and operates using optical pumping. The ruby laser emits red light with a wavelength of 6943 Å.

      Another popular solid-state laser is the Nd:YAG laser, developed in 1964. It operates on a four-level pumping scheme using optical pumping and emits radiation in the infrared region.

8.2. Gas Laser:

      In 1961, Ali Javan, Bennett, and Herriott developed the first helium–neon (He–Ne) gas laser. It operates on a four-level pumping scheme using electrical discharge. It is a continuous-wave laser with a wavelength of 6328 Å.

       The carbon dioxide (CO₂) laser is one of the most powerful lasers used in industry. It also operates on a four-level pumping scheme and emits radiation in the infrared region.

8.3. Liquid Dye Laser:

      Dye lasers are liquid lasers that use organic dye solutions as the active medium and operate using optical pumping.

8.4. Semiconductor Diode Laser:

      A semiconductor laser is a forward-biased p–n junction diode. The first semiconductor laser was developed by Hall and Nathan in 1962 using Ga-As as the active medium. These lasers are small in size and have high efficiency.

 9. Applications of Lasers:

    Lasers find applications in a wide variety of fields such as fundamental sciences, electronics, civil and mechanical engineering, medicine, and industry. The number of applications of lasers is continuously increasing.

  1. Holography is a technique for producing three-dimensional (complete) images of objects or scenes and is made possible using lasers.
  2. In the medical field, lasers are used for bloodless and painless surgeries, retinal welding, destruction of malignant tumors, dentistry, and ophthalmology.
  3. The intense laser beam is useful for material processing, as its energy density can be controlled precisely. Lasers are used in industry for welding, cutting, soldering, drilling holes, and heat treatment of various materials.
  4. Lasers have greatly increased the information-carrying capacity of optical communication systems through the use of optical fibers, revolutionizing modern communication.
  5. Other applications of lasers include laser printers, optical computing and signal processing, playing video discs, reading bar codes, measuring distances to remote objects, laser scanning, missile guidance, and laser weapons.

10. Summary of LASER – Part 2 (Working, Types & Applications):

     Part 2 focuses on the working principle of LASER, based on the interaction between photons and atoms. It explains the three fundamental processes involved: stimulated absorption, spontaneous emission, and stimulated emission. Among these, stimulated emission is the most important, as it produces photons that are identical in phase, frequency, direction, and polarization, leading to light amplification.

        The concept of population inversion and the presence of a meta-stable state allow stimulated emission to dominate over absorption. An optical resonator, consisting of two mirrors, reflects photons back and forth through the active medium, causing repeated amplification and producing a powerful laser beam through the partially reflecting mirror.

   This part also discusses different pumping schemes, such as two-level, three-level, and four-level laser systems, along with the role of the active medium. Various types of lasers—solid-state, gas, liquid (dye), and semiconductor lasers—are described with examples like Ruby laser, He-Ne laser, CO₂ laser, and semiconductor lasers.

     Finally, Part 2 highlights the wide range of applications of LASER in medicine, industry, communication, science, defense, and everyday life. It concludes that LASER technology has become an indispensable tool in modern science and technology due to its unique properties and versatility.

 

LASER Basics: History, Einstein Theory & Population Inversion

Posted on by Physics Prana

 

“Basics of LASER: From Einstein’s prediction to magical light machines”

Author:

Prof. Kali Chandrakant

M.Sc., M.Ed., D.C.S.

50+ Years of experience in Physics teaching

 1. Introduction:

 1.1. Why LASER feels like magic:

Have you ever wondered how a single beam of light can cut hard metal, heal human eyes, read barcodes, or even power science-fiction weapons?
      That powerful beam is called LASER, an acronym for Light Amplification by Stimulated Emission of Radiation.

    Unlike ordinary light from the Sun or electric bulbs—which spreads in all directions—laser light is highly intense, narrow, and precise. Because of these unique properties, lasers have become one of the most revolutionary inventions of modern science.

 1.2. Ordinary light vs Laser light:

Ordinary light Laser light
 Incoherent Highly coherent
Divergent Highly directional
Multi-wavelength Nearly monochromatic
Weak intensity Extremely intense

Scientists always dreamed of producing bright, directional and coherent light—and lasers made this dream real.

  1.3. Einstein’s revolutionary idea:

In 1917, Albert Einstein predicted a process called stimulated emission, which later became the foundation of laser technology.
    However, it took 43 years of scientific effort before the first working laser was built.

 2. Historical development of LASER:

Important milestones in laser history:

  • 1939 – Fabrikant (USSR) proposed radiation amplification
  • 1954 –
    • Charles Townes & Arthur Schawlow (USA)
    • Basov & Prokhorov (USSR)
      Developed MASER (Microwave Amplification)
  • 1960 – Theodore Maiman constructed the first Ruby Laser
  • 1960 – Ali Javan developed the He–Ne gas laser
  • 1962 – Robert Hall developed the semiconductor laser
  • 1964 – Kumar Patel invented the CO₂ laser

Nobel Prize 1964 was awarded to Townes, Basov, and Prokhorov.

     These discoveries gave birth to photonics, merging light with modern technology.

 3. Basic atomic concepts behind LASER:

 Energy states of atoms:

Atoms exist in discrete energy levels:

 3.1. Ground State (E1) :

   Lowest energy, most stable

  • The electrons revolving in different orbits round the nucleus constitutes stable atom. Let us consider the sodium atom 11Na23.

Fig A

 

 The electron distribution is 1S2 , 2S2 , 2P6 , 3S1 , 3P0 (Fig. A). In this case the entire atom has minimum energy, and the atom is said to be in ground state, which is stable state.  This ground state energy can be represented by E1.

 In the ground state:

1. The atom is stable.

2. The electron distribution is normal.

3. The atom has minimum energy.

  3.2. Excited State (E2) :

Higher energy, unstable

Fig. B

 Consider one electron in 2S state, which is excited to 3P state by supplying energy. Along with the electron the atom also has excess energy and the atom is in excited state. This excited state of the atom is represented by E2.(Fig B)

In the excited state:

1. The atom is unstable.

2. The electron distribution is not normal.

3. The atom has excess energy.
  • In further discussion, the atom as whole is taken in to consideration, with two energy states: Ground state E1and Excited state E.  

At room temperature, most atoms remain in the ground state.

 3.3. Boltzmann distribution law:

  • The number atoms in any atomic state are governed by Boltzmann’s distribution law.
  • Let N1 be the number of atoms (or population) in ground state of energy E1, N2 be the number of atoms (or population) in excited state of energy E2, k be Boltzmann’s constant and T be the Kelvin temperature of the system.
  • Boltzmann’s distribution law is stated in mathematical form:

N2=N1e−(E2−E1)/kT

Fig. C

 This law indicates that the population of ground state is very large as compared to the population of excited state.(Fig C)

(N1 ˃˃ N2).

Hence, under normal conditions, excited atoms are very few.

 3.4. Why excited states don’t last long:

     Nature prefers stability. “Every system in nature is in stable condition when it possesses minimum energy”.
Ordinary excited states decay quickly with a lifetime of about:  10-8 s.

3.5. Meta-stable state: The game changer:

    Some excited states—called meta-stable states—have much longer lifetimes: 10-3    seconds.   

        “The energy level having longer duration of excitation life time (of the order

10-3 second) is called meta-stable level or state”.

This is 100,000 times longer, allowing excited atoms to accumulate.

Example: Cr³⁺ ions in Ruby laser

 3.6. Pumping: Supplying energy:

   “The energy required to take atoms from lower energy state to higher energy state is pumping energy”.

 

Fig. D

 Fig D represents the process of pumping by which atoms in lower energy level E1 is taken upper energy level E2 by proper pumping energy.

Optical, electrical, chemical, thermal etc. energies are used for pumping. The pumping process is one of important process for laser action.

To excite atoms, external energy is supplied through:

  • Optical pumping
  • Electrical discharge
  • Chemical reactions

This process is known as pumping, similar to filling water into an upper tank.

 3.7. Population inversion:

      “If the population of atoms in higher energy state is greater than the population of atoms in lower state then this situation is called Population Inversion. ( N2 ˃ N1 )”

Fig E Fig E shows that when number of atoms (N2) of upper state E2 is greater than that of the number of atoms (N1) in lower state E1.

The population inversion can be achieved by using pumping energy. By pumping large number of atoms to meta-stable state population inversion is brought. Laser operation requires this non-equilibrium condition of population inversion.

Laser action requires a special condition called population inversion,

     where:N2 > N1

This is opposite to natural conditions and is achieved using meta-stable states.

 3.8. Fermions vs Bosons:

  • Fermions (electrons, proton,quarks): There is only one particle to occupy the same quantum state. 
    • Half-integer spin
    • Follow Pauli’s exclusion principle
  • Bosons (photons, pions): There is no restriction on the number of them that occupy the same quantum state. 
    • Integer spin
    • Unlimited particles in the same state

 Photons being bosons makes laser amplification possible.

 4. Summary of LASER – Part 1 (Basics & Theory):

    LASER (Light Amplification by Stimulated Emission of Radiation) is a device that produces highly intense, coherent, monochromatic, and directional light. Unlike ordinary light sources such as bulbs or the Sun, laser light does not spread randomly and can be precisely controlled. The theoretical foundation of LASER was laid by Albert Einstein in 1917 through the concept of stimulated emission, although the first practical laser was developed much later in 1960.

      This part explains the historical development of LASER, beginning from Einstein’s prediction to the invention of MASER and the first ruby laser. It introduces essential atomic concepts such as ground state, excited state, and energy levels. Under normal conditions, most atoms remain in the ground state, as explained by the Boltzmann distribution.

      A key idea discussed is the meta-stable state, which has a much longer lifetime than ordinary excited states. This allows atoms to accumulate in higher energy levels. Using an external energy source called pumping, atoms are excited to these levels. When more atoms exist in the excited state than in the ground state, a condition called population inversion is achieved, which is essential for laser action.

Part 1 builds a strong conceptual foundation by explaining why ordinary light cannot become laser light and how special atomic conditions are created to enable laser operation.

  In Part 2, we will see how photons multiply like an avalanche and create a powerful laser beam.

Units and Measurements: Dimensions, Errors & SI Units Guide

Posted on by Physics Prana

“Units and Measurements: Dimensions, Errors and Significant figures explained”

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

  1. Introduction:

     Physics is a quantitative science that relies on the measurement of physical quantities. Every experiment in physics involves observing, measuring, and comparing quantities such as length, mass, time, and temperature with internationally accepted standards.

  1.1. Unit of a quantity:

     A measurement is always a comparison between an unknown quantity and a standard quantity of the same kind.

“The standard measure of any physical quantity is called the unit of that quantity.”

     For example, to measure the mass of a fruit, we compare it with standard mass units such as 1 kg or 500 g.

 2. Units and Measurements:

  2.1. Requirements of a good unit:

A good unit of measurement must satisfy the following conditions:

  1. Invariable: Its value should not change with time, place, or physical conditions.
  2. Reproducible: It should be possible to reproduce the unit accurately anywhere in the world.
  3. Universally accepted: The unit should be accepted internationally.
  4. Accessible: It should be easy to use for comparison.

  2.2. Classification of Physical quantities:

     Physical quantities studied in physics are broadly classified into two types:

  2.2.1. Fundamental physical quantities:

   “A fundamental quantity is a physical quantity that does not depend on any other quantity.”

    Examples include length, mass, time, and temperature.

   There are seven fundamental physical quantities, and the units used to measure them are called fundamental units.

  2.2.2. Derived physical quantities:

     “A derived quantity is a physical quantity that depends on one or more fundamental quantities.”

    Examples include velocity, acceleration, force, work, pressure, and density.

    The units used to measure derived quantities are known as derived units.

2.3. Systems of Units:

      “A collection of units for measuring physical quantities is called a system of units”.

   Common systems of units are:

    FPS system:

       Foot, Pound, Second

     CGS system:

        centimetre, gram, second

    MKS system:

         metre, kilogram, second

    SI system:

    International System of Units (Système International d’Unités)

   2.4. Seven SI Fundamental units:

Physical quantity

 SI Unit Symbol

Length

 metre m
Mass kilogram kg
Time second s
Electric current ampere A
Thermodynamic temperature kelvin K
Amount of substance mole mol
Luminous intensity candela cd

 2.5. Supplementary Units:

Although no longer classified separately in SI, the following are commonly used:

Quantity Unit Symbol
Plane angle radian rad
Solid angle steradian sr

 2.6. Conventions for the Use of SI Units:

  1. The full name of a unit always begins with a lowercase letter, even if it is named after a scientist (e.g., newton, joule).
  2. The symbol of a unit named after a person is written with a capital letter (e.g., N for newton, J for joule).
  3. Symbols of other units are written in lowercase letters (e.g., m for metre, s for second).
  4. Unit symbols are never written in plural form (e.g., 25 m, not 25 ms).
  5. No full stop or punctuation mark is used after a unit symbol.
  6. Every physical quantity should be represented using its proper unit symbol.

    ( e.g. mass of ball= 2 kg )

  3. Dimensions and Dimensional formula:

    3.1. Definition of Dimensions:

  “The dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain the unit of that quantity.”

   When any derived quantity is expressed with appropriate powers of symbols of the fundamental quantities, then such an expression is called dimensional formula.       

      Dimensions are expressed in square brackets using symbols for fundamental quantities:

 Mass → M

 Length → L

 Time → T

    A general dimensional formula is written as:

[ Mᵃ Lᵇ Tᶜ ]

   3.2. Example: Dimensions of Velocity:

  1. Formula:

   Velocity = Displacement / Time

  1. Symbolic form:

         v = L / T

  1. Dimensional formula:

       [ v ] = [ M⁰ L¹ T⁻¹ ]

  3.3. Dimensional formulae of some physical quantities:

Sr. No. Physical Quantity Formula SI Unit Dimensional Formula
1 Velocity Velocity=displacement/time m/s [ M⁰ L¹ T⁻¹ ]
2 Volume Volume=cube of length m3 [ M⁰ L3 T0 ]
3 Density Density=mass/volume kg/m3 [ M1 L-3 T0 ]
4 Acceleration Acceleration=velocity/time m/s2 [ M⁰ L¹ T⁻2 ]
5 Momentum Momentum=Mass x Velocity kg m/s [ M1 L¹ T⁻¹ ]
6 Force Force= mass x acceleration N [ M1 L¹ T⁻2 ]
7 Work or Energy Work =Force x Displacement J [ M1 L2 T⁻2 ]
8 Power Power=Work/time W [ M1 L2 T⁻3 ]
9 Torque τ =rFsinϴ Nm or J [ M1 L2 T–2 ]
10 Moment of Inertia M.I = mr2 kg m2 [ M1 L2 T0 ]
11 Frequency n =cycles/second Hz [ M⁰ L0 T⁻¹ ]
12 Wavelength λ = Length m [ M⁰ L¹ T0 ]

  3.4. Applications of Dimensional analysis:

    Dimensional analysis is useful for:

  1. Checking the correctness of physical equations.

     “An equation involving different physical quantities is dimensionally correct if the dimensions of every term on both sides of the equation are identical.”

   Consider first equation of motion :

     v = u + at

   Dimensions of [v] =[M0 L1 T-1]  =[u]

    [at] =[M0 L1 T-2]  [M0 L0 T1]  =[ M0 L1 T-1] 

   Hence, dimensions of   [L.H.S.] = [R.H.S.]

   Thus, this equation is dimensionally correct.

  1. Converting units from one system to another.
  2. Deriving relationships between physical quantities.

   3.5. Limitations of Dimensional analysis:

  1. It cannot determine numerical (dimensionless) constants.
  2. It cannot be applied to equations involving trigonometric, exponential, or logarithmic functions.
  3. It fails when the proportionality constant has dimensions.
  4. It cannot identify additional terms having the same dimensions.

 4. Errors in measurements:

      In experimental physics, the measured value of a quantity may differ from its true value due to uncertainties known as errors.

 4.1. Causes of incorrect results:

  1. Mistakes – Caused by human negligence; these can be avoided.
  2. Errors – Uncertainties inherent in measurement; these cannot be eliminated but can be minimized.

 4.2. Types of Errors:

  1. Instrumental Errors:

   Errors caused due to faulty construction or calibration of instruments (e.g., improperly graduated thermometer).

  1. Systematic Errors:

    Errors arise due to defective experimental setup (e.g., an ammeter not reading zero when no current flows).

  1. Personal Errors:

   Errors introduced by the observer, such as parallax error while reading a scale.

  1. Random Errors:

    Errors caused by unpredictable variations in experimental conditions like temperature, pressure, or voltage.

  4.3. Minimization of Errors:

 Errors can be minimized by:

  1. Measuring large magnitudes of quantities.
  2. Taking multiple readings and calculating their mean.
  3. Using instruments with the smallest possible least count.

 4.4. Estimation of Errors:

  Absolute Error:

     “The difference between the measured value and the mean value of a quantity is called absolute error.”

   If a₁, a₂, a₃, …, an are measured values, then the mean value is:

         am = (a₁ + a₂ + a₃ + … + an) / n

   Absolute error:

Δaᵢ = am − aᵢ

   Mean absolute error:

Δamean = ( |Δa₁| + |Δa₂| + … + |Δan| ) / n

    Relative error:

Relative error = Δamean / am

    Percentage error:

Percentage error = (Δamean / am) × 100%

  4.5. Propagation of errors:

     Propagation of errors (or propagation of uncertainty) is the mathematical process used to determine how the uncertainties in individual measurements affect the uncertainty of a final result calculated from those measurements.

        In simpler terms, if you measure two different things (like length and width) and each has a small mistake or uncertainty, the final value you calculate (like area) will also have an uncertainty that depends on the errors of the original two measurements.

    Addition or Subtraction:

If A = a ± b, then:

ΔA = Δa + Δb

   Multiplication or Division:

If P = ab or P = a / b, then:

ΔP / P = (Δa / a) + (Δb / b)

   Power rule:

If Q=an, then

ΔQ/Q=nΔa/a

  5. Significant figures:

     “The number of digits in a measured value that are known with certainty and plus one uncertain digit are called significant figures.”

 5.1.  Rules for Significant figures:

  1. All non-zero digits are significant.

                e.g. Let mass= 345.57 kg, then the significant figure is 5.

  1. Zeros between non-zero digits are significant.

                e.g.  Consider, length = 2.05 m, then the significant figure is 3.

  1. Zeros to the left of the first non-zero digit are not significant.

              e.g.  If, time = 0.00 624 s, then the significant figure is 3.

  1. Zeros to the right of a non-zero digit in the decimal part are significant.

         e.g. Let, volume = 38.23000 m3 , then the significant figure is 7.

 5.2. Significant figures in calculations:

     Addition or Subtraction:

    The result should retain the least number of decimal places among the quantities.

     Multiplication or Division:

    The result should contain the least number of significant figures among the quantities.

 5.3. Rounding off Significant figures:

  1. If the digit to be dropped is less than 5, the preceding digit remains unchanged.
  2. If the digit to be dropped is greater than 5, the preceding digit is increased by one.
  3. If the digit to be dropped is 5 followed by non-zero digits, the preceding digit is increased by one.
  4. If the digit to be dropped is exactly 5 or 5 followed by zeros, the preceding digit is rounded to the nearest even number.

 6. Conclusion:

    The concepts of units and measurements, dimensions, errors, and significant figures form the foundation of experimental physics. A clear understanding of these topics is essential for accurate measurements, reliable calculations, and meaningful interpretation of experimental results.

    Mastery of this unit not only strengthens conceptual clarity but also enhances problem-solving skills required for higher studies and competitive examinations.

 

“Dual nature of matter and energy: Wave–Particle duality explained with examples”

Posted on by Physics Prana

“Dual nature of matter and energy: Wave–Particle duality explained with examples”

 

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

1.1. Every day example:    

 Popular Hindi films such as Seeta Aur Geeta, Chaalbaaz, Judwaa, and Tanu Weds Manu are well known for portraying double roles played by a single actor or actress, often with contrasting personalities. Interestingly, a similar idea of “double roles” exists in nature itself.

 1.2. Modern Physics:    

The dual nature of matter and energy is one of the most important concepts in modern physics. It reveals that matter and energy do not behave exclusively as particles or waves; instead, they exhibit both behaviors depending on the physical situation and method of observation. This concept, known as wave–particle duality, marked a turning point in the development of quantum mechanics.

 !.3.  Classical Physics: 

 Classical physics successfully explained many natural phenomena using Newton’s laws and Maxwell’s wave theory of light. However, experiments such as black body radiation, the photoelectric effect, and the Compton effect exposed the limitations of classical theories. These discoveries led scientists to propose revolutionary ideas that transformed our understanding of light, matter, and energy.

    In this article, we explore the dual nature of matter and energy through historical developments, key experiments, and theoretical insights, providing a clear and structured explanation suitable for students and physics enthusiasts.

  2. Matter and Energy: A scientific perspective:

    Nature (the universe) is constituted by matter and energy. To understand how and why matter and energy behave in particular ways, scientists made systematic observations and experiments. These observations led to the formulation of theories. A theory was accepted only when its predictions agreed with experimental results; otherwise, it was modified or rejected.

   From the 17th to the 19th century, Newtonian mechanics, Maxwell’s electromagnetic wave theory, and thermodynamics successfully explained most physical phenomena. This framework is collectively known as classical physics. Within this framework, light was firmly established as a wave.

 3. Classical view of light:

     The properties of light such as reflection, refraction, interference, diffraction, and polarization were satisfactorily explained by classical wave theory. By the end of the 19th century, it was widely accepted that light was purely a wave phenomenon.

   However, new experimental results discovered at the beginning of the 20th century challenged this classical viewpoint.

   4. Failure of Classical Physics and birth of Quantum theory:

  4.1. Black body radiation:

     One of the earliest challenges to classical physics came from black body radiation. Classical wave theory failed to explain the observed spectrum of radiation emitted by a black body.

 Wien (1896) explained the spectrum only at shorter wavelengths.

 Rayleigh and Jeans (1900) explained it at longer wavelengths but predicted the ultraviolet catastrophe.

    In 1900, Max Planck proposed a revolutionary idea: energy is not emitted or absorbed continuously, but in discrete packets called quanta. This marked the birth of quantum theory and introduced the particle-like behavior of light.

 4.2. Experimental evidence for particle nature of light:

 4.2.1. Photoelectric effect:

     The photoelectric effect was first observed by Heinrich Hertz (1887) and later studied in detail by Philipp Lenard. Classical physics failed to explain this phenomenon.

    In 1905, Albert Einstein successfully explained the photoelectric effect using Planck’s quantum hypothesis, proposing that light consists of discrete energy packets. Einstein received the Nobel Prize in 1921 for this explanation.

 4.2.2. Compton Effect:

      In 1923, A. H. Compton discovered that X-rays scattered by electrons showed an increase in wavelength, known as the Compton effect. This provided further confirmation of the particle nature of light.

   4.2.3. Photon concept and atomic spectra:

   In 1926, Gilbert N. Lewis named the quantum of light a photon.

      In 1913, Niels Bohr successfully explained the line spectrum of the hydrogen atom using quantum ideas.

 5. Dual nature of light (Energy):

    Developments between 1900 and 1930 led to a fundamental question: Does light behave as a wave or as a particle?

   5.1.  Wave nature of light is observed in propagation phenomena such as reflection, refraction, interference, diffraction, and polarization.

  5.2.   Particle nature of light is observed in interaction phenomena such as black body radiation, the photoelectric effect, the Compton effect, and atomic spectra.

    5.3.  These two behaviors appear contradictory, yet experiments show that light does not exhibit both natures simultaneously in a single observation. Instead, the observed nature depends on the type of experiment being performed.

     This led to the conclusion that light possesses a dual nature—wave as well as particle. These descriptions are complementary, not mutually exclusive.

  6. Everyday-life analogy for Dual nature:

     6.1. A water tank connected to a channel:

    Consider a water tank connected to a channel. Water may be transferred from the tank using a bucket, one bucket at a time (discontinuous transfer). However, after a certain distance, the water in the channel appears to flow continuously.

    6.2. A rainfall:

    A similar effect is seen during rainfall: rain falls drop by drop, yet the flow on the ground appears continuous.

  6.3. A light from a bulb:

   Likewise, light is emitted from atomic sources in discrete packets (photons), yet we observe light from a bulb as a continuous beam. This analogy helps in understanding how discontinuous emission can produce apparently continuous effects.

 7. Extension of Duality to matter:

     In 1924, Louis de Broglie proposed that if radiation can show particle nature, then matter should also exhibit wave nature. This idea was experimentally verified by Davisson and Germer (1927) through electron diffraction experiments. De Broglie was awarded the Nobel Prize in 1929.

  7.1. De Broglie hypothesis:

     According to de Broglie:

 Any material particle in motion is associated with a wave.

The wavelength of this matter wave (or de Broglie wave) is given by:

λ= h/mv

where:

 ( h ) = Planck’s constant

 ( m ) = mass of the particle

 ( v ) = velocity of the particle

 8. Conceptual difficulties and their resolution:

    8.1. Matter waves present a conceptual challenge because:

   ” Particles are localized in space. Waves are spread out.”

     This difficulty was addressed by Werner Heisenberg in 1927 through the uncertainty principle, which introduced the concept of wave packets.

      Further mathematical formulation was provided by Erwin Schrödinger (1926) through the Schrödinger wave equation, forming the foundation of quantum mechanics.

 8.2. Complementarity principle:

   Niels Bohr proposed the principle of Complementarity, which states:

   ” The wave and particle descriptions of radiation and matter are complementary and together provide a complete understanding of physical reality.”

    According to this principle, wave and particle behaviors cannot be observed simultaneously in a single experiment, but both are essential for a complete description.

 9. Dual nature in microscopic and macroscopic worlds:

     In the macroscopic world, matter generally behaves like particles because the associated de Broglie wavelengths are extremely small.

      In the microscopic world, particles such as electrons moving at high speeds exhibit significant wave properties.

      Light, being electromagnetic radiation, commonly exhibits wave behavior in everyday phenomena, while its particle nature becomes evident in specific interaction experiments.

 10. Conclusion:

     The discovery of the dual nature of matter and energy fundamentally changed the way we understand the physical universe. Experiments have shown that light exhibits wave behavior in phenomena such as interference and diffraction, while particle behavior becomes evident in interactions like the photoelectric and Compton effects. Similarly, matter particles, especially at microscopic scales, demonstrate wave properties as predicted by de Broglie and confirmed experimentally.

     These findings led to the development of quantum mechanics, where wave and particle descriptions are treated as complementary aspects of reality, as stated in Bohr’s principle of complementarity.         While macroscopic objects appear purely particle-like, microscopic entities reveal their wave–particle duality under appropriate conditions.

       Thus, matter and energy truly play a double role in nature, behaving as both waves and particles. This dual behavior is not a contradiction but a deeper reflection of the fundamental laws governing the universe.

“Two secrets of circular motion explained with diagrams”

Posted on by Physics Prana

 

“Two secrets of circular motion explained with diagrams”

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

 Understanding motion in a circle:

    According to Newton’s laws of motion, a force is required to change the state of rest or the state of motion of a body. When a force acts on a body, it produces acceleration in the direction of that force. If we want to change only the direction of motion while keeping the speed constant, the applied force must continuously act perpendicular to the direction of motion.

    Circular motion is a perfect example of this idea. Although the speed of a particle remains constant, its direction keeps changing at every instant. Therefore, circular motion is always an accelerated motion.

2. First secret of Uniform circular motion (U.C.M.):

2.1. Definition of uniform circular motion:

  When a particle moves along the circumference of a circle with constant speed, its motion is called Uniform Circular Motion (U.C.M.).

2.2.Tangential velocity in circular motion:

  Role of tangential velocity:Direction of velocity:

Fig.A

    Consider a particle of mass m moving along a circular path of radius r with centre O (Fig. A). At a given point on the circle:

  • The particle has a tangential velocity  vector V (AP).
  • This velocity always acts along the tangent to the circular path.

If no other force acts on the particle, it will continue to move along this tangent in a straight line due to inertia. Hence, tangential velocity alone cannot produce circular motion.

    Something must continuously pull the particle inward.

2.3.Need for an inward force:

Why straight-line motion must change?

To keep the particle on a circular path, its direction must change continuously. For this purpose, an inward force acting toward the centre of the circle is required.

This force must always act perpendicular to the instantaneous velocity.

Fig.B

2.4. How circular motion is produced? (Diagram B)

Vector explanation using parallelogram law:

 Now imagine a stone tied to a string and rotated in a circle. (Fig. B)

  1. Initially, the stone is given a straight line velocity along the tangent (shown by arrow AB )
  2. The string applies a force towards the fixed point O, along the radius (AO ).
  3. This force is always perpendicular to the instantaneous velocity.

At that instant, two vectors act on the stone:

  • Tangential velocity  vector V.
  • Radial force towards the center vector F.

  According to the law of parallelogram of vectors, the resultant of these two vectors gives the new direction of motion (shown by AC ) in the parallelogram (OABC).

  At the next position D, a similar parallelogram (ODEF) is formed, and the resultant motion is along DF .

    Because this process repeats continuously at every instant, the stone never moves in a straight line. Instead, its direction keeps changing smoothly, and the stone follows a circular path.

  3. Conditions required maintaining circular motion:

  Essential conditions for circular motion:

From the above diagrams, we clearly see that two conditions are essential for uniform circular motion:

  1. Tangential (linear) velocity V, which keeps the particle moving
  2. An inward force perpendicular to the velocity, which keeps changing the direction

This inward force is called centripetal force, and its magnitude is:

    F = mV2 /r

 4. Second secret of circular motion:

 4.1. Centripetal force in nature:

    Centripetal force is not a separate kind of force. Instead, different physical forces supply it in different situations.

 such as:

  • Gravitational force, which keeps planets revolving around the Sun
  • Tension in a string, which keeps a stone in circular motion
  • Electrostatic force, which acts as centripetal force in classical atomic models (in quantum mechanics, electrons exist in orbitals rather than circular paths)

 4.2. Why circular motion is accelerated motion?

    Although the speed of a particle in uniform circular motion remains constant, its velocity changes continuously because its direction changes at every point.

    Since acceleration depends on the rate of change of velocity, circular motion must be treated as an accelerated motion.

   Therefore, the frame of reference attached to a rotating system becomes a non-inertial (accelerated) frame.

4.3. Centrifugal force – A pseudo force:

   In a rotating or accelerated frame of reference, Newton’s laws of motion cannot be applied directly. To make them valid, an apparent force called centrifugal force is introduced.

Centrifugal force:

  1. is a pseudo (fictitious) force,
  2. acts along the radius but away from the centre of the circle,
  3. has magnitude equal to the centripetal force

      F = mV2 /r

4.4. Why Centrifugal force is not a reaction force?

It is important to understand that centrifugal force is not the reaction of centripetal force.

This is because: 

  • Both forces act on the same particle.
  • Action and reaction forces always act on different bodies, according to Newton’s third law.

  5. Real life example: Washing machine dryer:

  The working of a washing machine dryer clearly demonstrates the effect of centrifugal force.

  • Wet clothes rotate rapidly with the drum
  • Water droplets also move in circular paths
  • No centripetal force is available to hold the water droplets
  • Due to centrifugal effect, water is thrown outward through the holes of the drum

Consequently, the clothes become dry.

6. Conclusion:Summary of the two secrets:

  The attached diagrams clearly reveal the two secrets of circular motion:

  1. Circular motion is produced by the continuous combination of tangential velocity and centripetal force.
  2. Centrifugal force appears only in a rotating frame as a pseudo force, equal in magnitude and opposite in direction to centripetal force.

Thus, a clear understanding of these principles removes common misconceptions and builds a strong conceptual foundation in mechanics.

  7. Video support:

     Students are encouraged to study the diagrams carefully and observe animations to strengthen their understanding of circular motion.

“The real concept of electron flow in a conductor” Clearing a fundamental misconception in electricity

Posted on by Physics Prana

“The real concept of electron flow in a conductor”

Clearing a fundamental misconception in electricity.

 

Author:
Prof. Kali Chandrakant
              M.Sc., M.Ed., D.C.S.
50+ Years of experience in Physics teaching.

1. Introduction: A common but serious misconception:

     Many students believe that electrons start moving from the negative terminal to the positive terminal only after a battery is connected to a conductor. This idea looks logical at first, but it is scientifically incorrect. Surprisingly, this misconception is found not only among school students, but also among senior secondary and first-year B.Sc. students.

The truth is simple but important:

   Electrons are always in a random motion inside a conductor, even when no battery is connected.

      To clearly understand this, we must patiently look at the microscopic picture of a conductor. Let us proceed step by step.

2. Atomic structure and conductors:

2.1. Atomic structure:

 First consider the atomic structure. Every conductor is made up of atoms.

An atom consists of:

  • Electrons (e⁻) – negatively charged
  • Protons (p⁺) – positively charged
  • Neutrons (n⁰) – electrically neutral

   At the centre of the atom is the nucleus, which contains protons and neutrons. Because of protons, the nucleus is positively charged. Electrons revolve around the nucleus in different energy levels (shells).

   This basic atomic structure plays a crucial role in understanding electrical conduction. Now let us discuss about sodium atom.

 2.2. Sodium atom: Source of free electrons:

Fig.A

   Let us consider sodium metal, which is a good conductor.

  • Atomic number of sodium = 11
  • Electronic configuration = 2, 8, 1
  • Number of valence electrons = 1

    The single electron in the outermost shell is weakly bound to the nucleus. This weak binding is the key reason why sodium can conduct electricity.

      Figure A represents the electronic structure of a sodium atom. Using this structure sodium atom, let us see what changes occurs when large number (of the order 1022 atoms/cm3) of atoms come together to form metal. 

2.3. Formation of a metal crystal:

   When a large number of sodium atoms come together, they arrange themselves in a regular and orderly pattern, forming a crystal lattice.

  This is shown in Fig. B:

  • The green spheres represent sodium ions arranged in a crystal lattice.
  • The small highlighted entities between them represent free electrons, forming what is called an electron gas or electron sea.

Fig.B

In the sodium metal actually following thing occurs as shown in Fig. C:

  • The grey circles represent localised positive sodium ions fixed at lattice points.
  • The red symbols represent free electrons, which are not attached to any single atom and move randomly in the space between ions.

Fig.C

As we move farther from the nucleus:

  • The attractive force on electrons decreases.
  • Hence, the outermost electron experiences the least attraction.

During the formation of the metal crystal:

  • The outer electrons become delocalised.
  • They are no longer bound to individual atoms,
  • Each sodium atom becomes a positively charged Na⁺ ion.
  • The valence electrons form a common pool or “sea” of free electrons.

This situation is clearly shown in Fig. B and Fig. C.

3. Free electrons and random motion:

 Inside the metal:

  • Positive sodium ions remain at fixed lattice positions (they only vibrate slightly)
  • Free electrons move in the space between these ions

These free electrons are in continuous random motion due to:

  • Their thermal energy.
  • Frequent collisions with vibrating lattice ions.

3.1. Important Observation:

  Even though electrons are moving continuously and randomly, there is no electric current when no battery is connected.

3.2. Why does current not flow?

Because:

   The number of electrons crossing any cross-section in one direction is exactly equal to the number crossing in the opposite direction.

This balanced situation is illustrated in Fig. D.

 4. Why is there no current without a battery?

Fig.D

Let us observe the motion of a single electron.

  • Suppose an electron is at point A at a certain instant
  • Due to random motion, after time t, it may reach point B
  • Each electron follows a different zig-zag and unpredictable path

Since motion is random in all directions:

  • The net flow of charge across any cross-section is zero

Hence we conclude:

The random motion of electrons alone does not produce electric current.

5. Effect of applying voltage to a conductor:

  Now let us connect a battery of voltage V across the conductor.

The battery performs two important functions:

  • It establishes an electric field inside the conductor
  • It supplies energy E = eV to each electron

Due to the electric field, each electron experiences a force:

F = -eE

As a result:

  • Electrons acquire a small net motion towards the positive terminal of the battery.

6. Superposition of motions: Random + Drift:

Fig.E

 6.1. It is very important to understand that:

  • Random motion does not stop.
  • A small directional motion is added.

Thus, each electron now has:

  1. Random thermal motion.
  2. Slow drift motion towards the positive terminal.

In Figure E:

  • Without voltage, an electron moves from A to B in time t.
  • With voltage applied, it moves from A to B₁ in the same time t.

The small shift:

BB₁ = S

This shift represents the effect of the electric field.

6.2. Drift velocity (vₑ):

Drift velocity is defined as:

vd = S/t

   Although electrons move randomly at very high speeds, their drift velocity is extremely small, typically of the order of mm per second.

  Yet, this small drift is sufficient to produce electric current.

7. How is electric current produced?

   Because of drift:

  • More electrons cross a given cross-section in one direction
  • Let n electrons cross extra in time t
  • Total charge flowing:Q = ne
  • Hence current:I = Q/t = ne/t

  Therefore:

 The electric current is produced due to the net drift of electrons, not due to their random motion.

8. Direction of current:                   

  In metal wires:

  • Electrons drift from the negative terminal to the positive terminal
  • Conventional current is defined from positive to negative

   This convention exists for historical reasons. Both descriptions are correct; one refers to negative charge flow, the other to an imagined positive charge flow.

9. Everyday analogy: Ganapati procession:

Consider a Ganapati immersion procession with Lezim players:

  • Players performing Lezim → Random motion
  • Team standing at one place → No procession
  • Team leader gives a forward command → Electric field
  • Players continue performing while moving forward → Random + drift motion

Similarly:

  • Electrons move randomly
  • Battery creates a directional force
  • Slow drift produces electric current

10. Final conclusion:

  • Free electrons exist in a conductor even without a battery
  • Their motion is random and produces no current
  • Applying voltage creates an electric field
  • Random motion becomes biased
  • Net drift of electrons produces electric current

The electric current is not the speed of electrons, but the result of their organised drift.

11. Suggested animation:

For visual understanding of electron flow, refer to the following video:

  1. YouTube: Electron Flow in a Conductor
    https://www.youtube.com/watch?v=KprFTxjQAoE

How to draw ray diagrams for a Convex Lens, Compound Microscope and Telescope? (Student-friendly guide)

Posted on by Physics Prana

 

How to draw ray diagrams for a Convex Lens, Compound Microscope and Telescope? (Student-friendly guide)

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

     If ray diagrams feel confusing or “too neat to be real”, this guide is for you. This guide helps you draw scientifically accurate ray-diagrams for image formation by a convex lens, and further for a compound microscope and a simple astronomical telescope — using a “smart diagram” technique that is examination friendly, easy to draw, and visually clear.

   It walks you through each diagram step by step, with simple rules you can remember in examinations, and you can always pause and copy along with the video lesson.

 2. Key concepts and definitions:

Before drawing, know these basic terms:

2.1.  Principal Axis: 

  A straight horizontal line through the optical centre of a symmetrical lens.

2.2.  Optical Center (O):

   The midpoint of the lens (on the principal axis), through which a ray passes undeviated.

 2.3. Focus (F): 

  The point on the principal axis where rays parallel to the principal axis converge after passing through a convex (converging) lens.

 2.4. Twice the Focus (2F) (or  Center of curvature): 

  A point on the principal axis at twice the distance from the optical center as the focus; often used to locate object or image for standard cases.

 2.5. Real / Virtual Image:

  • Real image: Where refracted rays actually converge; can be projected on a screen.
  • Virtual image: Where refracted rays only appear to diverge from; cannot be projected, appears behind lens or eyepiece.

 2.6. Erect / Inverted Image:  

  Orientation of image arrow relative to object arrow (upright vs upside-down).

 3.Understanding these is essential before drawing:

   General “Smart Drawing” Tips (Use for All Diagrams):

 3.1.  Use roughly half an A4 page for a single diagram — large enough for clarity yet compact enough for examination notebooks.

3.2.  Draw with a pencil, sharpened at both ends (for neat lines), and use a full-size ruler (≈ 30 cm) for straight rays.

 3.3. First draw a clean principal axis — a horizontal line across the middle. Mark the optical centre (O) roughly at the center.

3.4.  Mark the focus (F) and twice focus (2F) points on both sides of the lens symmetrically.

 3.5. Draw the lens (just as a vertical line or simple double-concave/convex curve on the axis) — this keeps the diagram schematic and clear.

3.6.  Draw the object as an arrow (straight vertical line with arrow-head) perpendicular to the principal axis — on the left side (for standard lens problems).

4. Convex Lens:

4.1. Rules for ray diagram:

  1.  Parallel ray to principal axis passes through focus on other side of convex lens.
  2. Ray passing through Optical center of convex lens goes un-deviated.
  3. Intersection of these rays gives position of Image.
  4. Taking object on left side of lens, position of Image can be found.

4.2. Convex lens ray diagram for image due to object kept beyond 2F1.

Goal: To locate the image formed by a convex lens when the object is placed beyond 2F.

Step 1: Draw line in the middle of half page.

Step 2:

  1. Select a point 2F1at 3 cm from margin. 
  2. Draw a curve of length touching 3 lines above and below of middle line, having 6 cm radius and 2F1as center.

Step 3:

  1. Select a point 2F2 on principal axis at 6 cm from one end of curve.
    2. Draw opposite curve of 6 cm radius from 2F2.

Step 4:

     Select points F1, F2 on principal axis from O on both sides at a distance equal to half distance between O and 2F1.

Step 5:

  1. Draw 1 cm AB arrow headed perpendicular line to Principal axis near

     margin as object.

  1. Draw parallel line to axis from B up to lens, make this line to pass from F2.
  2. Then draw line passing through O from B till it intersects line through F2.
  3. Draw perpendicular line A1B1 from intersection of above lines.

Step 6 :

  The final image due to convex lens, when object is kept beyond 2F1 and image is formed between 2F2 and F2   and it is inverted, diminished, and real.

 

  4.3. Image due to object kept between F1 and O:

Practice to draw image due to object kept between F1 and O.

  4.4. Repeat same procedure to draw images due to object kept:
 1. at 2F1
 2. Between   2F1 and F1
 3.  At  F1  
 4. Between F1 and O

Images due to Convex Lens:

  4.5. Summary table — Image formed by convex lens:

Object position Image position Image nature Size
Beyond 2F Between F and 2F Real, Inverted Smaller
At 2F At 2F Real, Inverted Same
Between F and 2F Beyond 2F Real, Inverted Magnified
At F At infinity Real Highly enlarged
Between F and O Same side Virtual, Erect Magnified

  5. Ray diagram for a Compound Microscope:

  A compound microscope uses two convex lenses — an objective (closer to object) and an eyepiece (closer to eye). The final image you see is virtual and highly magnified.

 5.1. To draw image due to Compound Microscope, follow the procedure used for convex lens:

Step 1:


 1. Compound microscope consists of objective and eye-piece.
 2.  Turn the full page to horizontal, draw line at   middle of horizontal page.
 3.   Draw objective of radius 6 cm.
 4.   Fix the points 2F1O, F1O, O, 2F2O, F2O.

  5. Draw object of 1 cm height between 2F1O, F1O and complete diagram

         for image A1B1 as per procedure .

Final image A1B1, due to objective of microscope, which is magnified, inverted, real.

Step 2:

Select point X at 3 cm from image A1B1 .

Step 3:

  1. Draw Eye-piece with radius 8cm taking  1F1Eat 8 cm from X.
    2. Fix points 2F1E, F1E, O, 2F2E, F2E.

Step 4 :

1 Draw final image A2B2 with eye-piece taking   A1B1 as object.
2 Final image is magnified, inverted, virtual.

  6. Ray diagram for a Simple Telescope (Astronomical Telescope):

   A basic telescope also uses two convex lenses

  1. an objective (large focal length)
  2. an eyepiece (short focal length) ,

arranged so that the final image is for a distant object (effectively at infinity).

 6.1. To draw image due to Telescope, follow the procedure used for convex lens.

Step 1:

  1. Telescope consists of Objective and Eye-piece.
  2. Turn the full page to horizontal, draw line at middle of horizontal page.
  3. Draw objective of radius 8 cm. Fix the points 2F1O, F1O, O, 2F2O, F2O.

Step 2 :

  1. Draw a line 1 passing through O with small inclination to horizontal.

      2 . Then draw parallel line 2 to this above up to lens.

  1. Make this line 2 to pass through F2Etill it intersects line 1 at B1.
  2. Draw first image A1B1due to objective between F2E and 2 F2E.

Step 3:

Fix point X at a distance of 2 cm from image A1B1 . 

Step 4:

  1. Fix point 2F1E at 6 cm from X.
  2. Draw Eye-piece at X with radius 6 cm.
  3. Fix points 2F1E, F1E, O, 2F2E, F2E.

Step 5:

  1. Draw final image A2B2with eye-piece, taking image A1B1

      as object for eye-piece .

  1. Final image is magnified, inverted, and virtual.

Thus using the given procedure ray diagrams can be easily drawn.

 7. Common mistakes to avoid:

  7.1.  Marking F and 2F asymmetric or too close/far — destroys accuracy.

 7.2. Drawing lens as thick or unrealistic, better to use a simple vertical line or gentle convex curve.

 7.3. Not using a straight ruler for rays → leads to messy/inaccurate diagrams.

 7.4. Drawing object-arrow too big or too small → distorts relative size of image.

7.5.  Forgetting to draw at least two rays from object — then image location will be ambiguous.

 8. Why This Method Works (and Helps in Examinations):

   8.1. The simplified but systematic approach avoids messy drawing and reduces errors.

  8.2.  Using standard notation (O, F, 2F) — students easily interpret and generalize for different cases.

  8.3.  Making diagrams on half-page makes them exam-friendly (fits within answer sheet, remains legible).

  8.4. The reuse of the same two-ray method for all object positions and optical devices helps memorize logically rather than by rote.

  9. Conclusion and final advice:

  Once you master the logic behind the principal axis, focus points, and ray-rules, you don’t need to memorize separately for every case. Instead, apply the same “draw-object → draw rays → locate image → label” method systematically

  With proper pencil technique, carefully marked F and 2F, and a good ruler — your diagrams will look clean, correct, and exam-ready.

10. Practice and perfection tips:

  • Redraw each setup at least twice for hand memory.
  • Use the same notation consistently (O, F, 2F, A, B, etc.).
  • When revising, mentally visualize where the image forms for each case — this sharpens conceptual clarity.

Happy drawing — and best of luck mastering optics!

11. Video help:

   How to draw the ray diagram for images formed by convex lens, Microscope and Telescope — Motivational Physics.
https://youtu.be/d8qB-EhQ5Ik?si=aSI7E8Ku-gj9s2pr

 

“How to Draw Circuit Diagrams in Physics (HSC Level) – Smart Techniques with Step-by-Step Guide”

Posted on by Physics Prana

 

“How to Draw Circuit Diagrams in Physics (HSC Level) – Smart Techniques with Step-by-Step Guide”

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

        Many students can solve numerical in electricity but hesitate when the question is, “Draw the circuit diagram for this experiment.” Circuit symbols, series–parallel connections and crowded practical setups make the diagram look difficult.  This post shows a simple, step-by-step method to understand each instrument first and then combine them into neat, examination ready circuit diagrams for your class 11–12 electricity experiments.

2. Why start from symbols?

     Before drawing a big circuit like Whetstone’s bridge or a potentiometer, it is essential to know the basic electrical components and their standard symbols used in textbooks and board examinations. Once these 10–12 symbols are familiar, any practical circuit becomes just a meaningful arrangement of these small bricks rather than a scary picture. You may introduce this section with a sentence such as:
“First learn the language of circuit diagrams – the symbols – and then learn to write sentences with them – the full experiment circuits.”

3. Essential components up to HSC level:

Here is a compact description:

 3.1.  Cell / Battery
Source of emf for the circuit, always drawn with + and – terminals clearly shown. A single cell has one long and one short line; a battery is a group of such cells in series.

 3.2. Ammeter (DC)
Measures current in the circuit. It is always connected in series with the branch whose current you want to measure and must have its + terminal towards the higher potential side.

 3.3. Voltmeter (DC)
Measures potential difference between two points. It is always connected in parallel across the component (resistor, wire segment, cell, etc.) and again has + and – terminals marked.

 3.4. Galvanometer
Sensitive device to detect small current. It is connected in series in a branch where you want to check “current or no current”, as in Whetstone’s bridge or potentiometer balancing.

 3.5. Key / Switch
Simple on–off device. Drawn in series with the battery so that the whole circuit can be made live or dead by opening/closing the key.

 3.6. Resistor

     Resistor provides fixed resistance; rheostat gives variable resistance to control current.

 3.7. Rheostat
A rheostat is always in series with the circuit (or the relevant branch) and is drawn with an arrow indicating the moving contact.

 3.8. Connecting Wire Wires join components.

 3.9. Metal Strip
Metal strips represent brass or copper strips on experiment boards, often drawn as straight segments with endpoints used to connect other parts of the circuit. ​

 3.10. Earth (Ground) Earth symbol indicates a connection to ground.

 3.11. Diode

     A diode symbol shows a unidirectional device, used when you discuss rectifiers or protection circuits. Your original Table A can be retained but polished into a clean table (Name – Symbol – Use – Series/Parallel – Remark) so that students can revise quickly before the practical exam

  3.12. AC generator

          An AC generator (alternator) is a device that converts mechanical energy into electrical energy in the form of alternating current using electromagnetic induction.

3.13.   Load

       The load is the part of the circuit that consumes electrical power, such as a lamp, resistor, motor or any appliance connected across the source.

  4.  Table  A   gives the list of  instruments involved up to H. S. C. level with name, electric symbol, use, series/ parallel connection and remark  about each instrument.

Table A

No. Instrument Electric Symbol Use Series/Parallel Remark
1 Cell Source Series + , – terminals
2 Battery

Source Series + , – terminals
3 Ammeter D.C. Current measurement Series + , – terminals
4 Voltmeter D.C. Voltage measurement Parallel + , – terminals
5 Galvanometer Current detection Series  
6 Key On-Off  switch Series  
7 Resistor To control Current Series/Parallel Fixed resistance
8 Rheostat To vary current Series Variable Resistance
9 Wire Circuit Connection Series/Parallel  
10 Metal strip Part of Instrument Series/Parallel  
11 Ground Earthing Series/Parallel  
12 Diode Unidirectional Device Series/Parallel  
13

AC generator

  

 converts mechanical energy into electrical energy Series/Parallel  
14

Load

 consumes electrical power Series/Parallel  

  5. Standard source block (Fig A):

Fig.A

 

5.1 Why Use a Standard Block?

  To avoid redrawing battery, key and rheostat every time in a different style, fix a standard “source block” for all electricity experiments.

5.2 How to draw it

  • Draw a neat rectangle or group of symbols containing: battery, key and rheostat in series.
  • Mark its two terminals as A and B. These are the points where the rest of the experimental circuit will be attached. Below this figure, note: “This set acts as the common electric source for all HSC level electricity experiments.”

Once students remember this block, each experiment reduces to: “Attach the suitable arrangement of resistances, wire and galvanometer between A and B.”

6. Wheatstone bridge – idea and circuit:

  6.1. Aim of the circuit

     In a Wheatstone bridge, four resistances are arranged in the form of a bridge to compare an unknown resistance with known ones using the condition of null deflection in a galvanometer.

  6.2. Logical layout

    Tell students to imagine a rectangle of four resistors: left arm R, right arm X (unknown), and the top and bottom arms made of known resistances or strips with a 1 m wire between A and B. The galvanometer forms the “bridge” between the midpoints of the two vertical arms.

  6.3. Step-by-step drawing:

Fig. B

      1.  Draw two L‑shaped metal strips at the left and right margins of the page, and a straight strip between them to represent the wooden board. Join the top ends of the two strips by a straight line; this will be the 1 m wire AB used for balancing.

 2. Mark its ends clearly as A and B. At the bottom of the left strip draw the known resistance R and at the bottom of the right strip draw the unknown resistance X in series between A and B, forming the four arm bridge.

 3. From the midpoint of the left strip draw a line to the galvanometer symbol, and from the other terminal of the galvanometer draw a line to a sliding contact (jockey) touching the wire at some point C on AB.

 4. Attach the standard source block (battery + key + rheostat) between A and B using your fixed symbol set, keeping polarity consistent.

 5. Label the segments AC = lᵣ and CB = lₓ if you want to connect the diagram visually with the formula

 6. X=R.lx/lr

 7. “This completes the neat circuit diagram of Wheatstone’s bridge used to determine an unknown resistance by the null‑deflection method

7. Potentiometer – idea and circuit:

  7.1. Aim of the circuit

    In a potentiometer, a long uniform wire is used to compare emf of two cells or to find internal resistance by balancing potential differences without drawing current from the cell being measured.

  7.2. Logical layout

     Visually, the potentiometer board is shown as a long set of wires divided into equal segments, with end terminals A and B. The main battery with rheostat is connected across A and B, and the secondary circuit (cell + galvanometer + jockey) is attached along the length to find the balancing length.

 7.3. Step-by-step drawing:

Fig. C

  1. In the middle of the page, draw four long, straight, parallel strips and join their ends so they look like one continuous multi segment wire. Mark the left end as A and the right end as B; this is the potentiometer wire, total length usually 4 or 10 m in the real apparatus.

  2. Between A and B, below the wire, place the standard source block: battery, key, and rheostat in series. Make sure the positive terminal of this battery is connected to A. On one side, draw a second small cell whose emf is to be compared or measured.

 3. Connect its positive terminal to A and its negative terminal to a galvanometer. From the other terminal of the galvanometer, draw a line to the jockey symbol touching the potentiometer wire at some point C, which represents the balancing point.

 4. Indicate the balancing length AC = L1(for first cell) and AC = L2(for second cell) if two cells are used, and write the emf relation below the diagram, for example

  5. E1/E2 =L1/L2

  6. Close this section with:
“This gives the standard circuit diagram of a potentiometer used for comparison of emfs or measurement of internal resistance in the class 12 practical.”

  8. Why this method works (The Psychology behind it)

This technique is effective because:

 ✔ You draw using shapes, not memory

L-shapes, rectangles, and straight lines create a clear visual structure.

 ✔ You add instruments only after the structure is ready

No confusion about placement.

 ✔ You always begin from A & B

Consistent orientation makes every diagram uniform.

 ✔ You understand the role of each component

Instead of copying, you logically place each part.

 ✔ Your final diagram looks clean, professional, and exam-ready

  9. Common mistakes students make (and how to avoid them)

  1. Drawing voltmeter in series (should be parallel)

  2. Ammeter drawn without observing polarity

  3. Galvanometer wrongly connected to battery

  4. Jockey drawn at the wrong position

  5.  Main supply connected across wrong terminals

  6. Potentiometer wire drawn too short or too long

   7. No separation between primary and secondary circuits

   8. Symbols drawn incorrectly (especially diode, rheostat)

  9. Follow the step wise technique and these errors disappear.

 10. Conclusion:

 1. Learning to draw circuit diagrams is not about memorizing textbook figures.

  2. It is about understanding the structure, symbols, and sequence.

 3. Using the “Smarter Techniques” method:

 4.  You begin with simple lines and shapes

 5. Add only the essential components

 6.  Complete the diagram with clean, logical connections

 7. With practice, you will be able to draw any electricity experiment diagram quickly and confidently.

  

11. Video support:

To see the actual drawing process in action, refer to the demonstration in the following video:

YouTube Video: How to Draw Circuit Diagrams for Electricity Experiments?

How to draw the ray diagram used to derive the distance between the virtual sources in the Bi-prism experiment?

Posted on by Physics Prana

How to draw the ray diagram used to derive the distance between the virtual sources in the Bi-prism experiment?

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

       The Bi-prism experiment is a fascinating way to study interference in optics. One of its most important tasks is finding the distance between the two virtual sources, S₁ and S₂. We calculate this distance using the conjugate foci method.

       In this blog, you’ll learn how to draw the ray diagram for this method using simple and smart construction techniques. These steps follow the same rules explained in my earlier post, “How to Draw Diagrams in Physics Using Smarter Techniques.”

Let’s begin.

 2.  Understanding the bi-prism experiment: Conjugate foci method:

     2.1.  The conjugate foci method plays a vital role in measuring the distance (d) between two virtual sources in the Bi-prism experiment. To determine this distance, a lens of focal length f is placed between the bi-prism and the eyepiece, with the eyepiece located at a distance of 4f from the slit.

     The lens is then moved between two key positions:

 L₁: Lens close to the bi-prism

 L₂: Lens close to the screen

    In both positions, we observe different separations between the images of S₁ and S₂. These measurements help us calculate the true source separation.

     2.2.  Two key lens positions in the bi-prism experiment:

  1. Position L1 (Lens close to bi-prism):

    In this position, when the lens is closer to the Bi-prism (as shown in Fig. A), magnified images of the two virtual sources (S1 and S2) are formed.

    The distance d1 between the magnified virtual sources is measured.

  1. Position L2 (Lens near screen):

    In the second position, when the lens is moved towards the screen (as shown in Fig. B), diminished images of S1 and S2 are observed.

    The distance d2 between the diminished virtual sources is measured.

    3. Mathematical derivation for determining the distance between virtual        sources:

    In the bi-prism experiment, the object distance (u) and image distance (v) for the lens are related to the distances measured between the virtual sources.

 For position L1 (Fig. A below):

       d1/d = v/u

  where d1 is the distance between magnified virtual sources.

 For Position L₂  (Fig. B below):

    d2/d = u/v

  Where, d2 is the distance between diminished virtual sources.

   By multiplying these two equations and rearranging them, we can find the distance d between the two virtual sources as

          d2 = d1. d2

This is how the conjugate foci method is used to calculate the distance between S1 and S2.

4. Smarter Techniques for Drawing Ray Diagram:

Fig. A Fig. A

Fig. B     Fig. B

  4.1. Prepare the grid and principal axis:

   Step 1: Draw the frame:

  •  On unruled A4, first draw one vertical reference line near the left margin.

   Step 2: From this line, draw 16 equally spaced light horizontal grid lines across the page; number them 1 to 16 in your mind for counting.

   Step 3: Fix the optical length:

  • From the top reference line, mark a point so that the useful diagram height is about 12 cm.

   Step 4: Within this height, keep all important points (sources, lens, images) restricted to a central 2 cm vertical strip so the construction remains neat and symmetrical.

   Step 5: Choose the principal axis:

  • Take the 6th horizontal grid line as the principal axis; mark it clearly.
  • Count and note that the 13th line will be used for the eyepiece/screen side of the construction, matching the arrangement used in the biprism experiment. Mark the virtual sources S1 and S2. Mark coherent virtual sources
  • On the left half of the principal axis region, choose a short central segment and draw two small ticks: two grid steps above the axis for S1 and two grid steps below for S2.
  • Label them S1 and S2, and note that the vertical gap between them represents the virtual source separation d (same in both Fig A and Fig B).
  •  Mark lens positions and image separations.

   Step 6: Draw lens positions L1 and L2 

  • For Fig A (lens nearer bi-prism), go to about the 4 cm mark from the source side and draw a vertical line crossing all grid lines; thicken the central part and shape it as a convex lens; this is position L1.7.For Fig B (lens nearer screen), go to about the 8 cm mark from the source side and similarly draw another convex lens; this is position L2.

 Step 7: Fix magnified separation d1 in Fig A:

  • In Fig A, from the principal axis on the eyepiece side, count 4 lines up and 4 lines down and put small points on the eyepiece line; label the upper point A1 and the lower point B1.
  • The distance A1 B1 along the eyepiece line represents d1, the magnified separation of the virtual sources when the lens is at L1.8. Fix diminished separation d2 in Fig B

   Step 8:  In Fig B, on the eyepiece side, go just one line above and one line below the axis and mark two points on the eyepiece line; label them A2 (above) and B2 (below).

  • The distance A2 B2 now represents d2, the diminished separation of virtual sources for lens position L2.

  4.2. Construct rays for Fig A (lens near bi-prism):

 Step 1: Mark optical center and draw circular arcs

  • In Fig A, choose a point on the axis inside the lens as the optical centre O.
  • With a compass, take a convenient radius (about 6 cm on your grid) and from O draw two light circular arcs that pass through A1 and B1; extend these arcs back towards S1 and S2 to guide your ray directions.9.Join sources to images through the lens
  • Draw straight lines S1O and OB1; also draw S2O and OA1. These show rays passing undeviated through the optical center.
  • From A1 and B1, draw lines back to meet the lens boundary, and then from those points continue the incident parts of the rays towards S1 or S2, adding arrowheads to show the direction from sources to images.

   Step 2: Show incident, refracted rays and focus 

  • For each pair, thicken the part of the ray before the lens as incident and the part after the lens as refracted, with arrows pointing towards the images A1 and B1.
  • Extend two refracted rays (one from S1, one from S2) until they intersect on the axis behind the lens; mark this intersection as the focal point of the convex lens in position L1, indicating that A1 and B1 are magnified real

4.3. Construct rays for Fig B (lens near screen):

 Step 1: Join images back to sources

  • In Fig B, mark the optical center O of lens in position L2.

 Step 2: Draw lines A2O and OS1, and B2O and OS2, representing rays passing through O; then from A2 and B2 construct incident and refracted parts exactly as done in Fig A but now giving diminished images on the eyepiece line. Complete parallel ray construction

  • From the image side intersections on the lens surface, draw rays backward parallel to the principal axis to meet S1 and S2, adding arrows to show travel from S1, S2 through the lens to A2, B2.
  • Extend suitable pairs of refracted rays to meet at the focal point on the other side of the lens, indicating that A2 and B2 are diminished real images when the lens is near the screen.

  4.4. Label formula and result on the sheet:

  Show conjugate foci relation:

  • Beside Fig A, write d1 / d = v / u; beside Fig B, write d2 / d = u / v, indicating that u and v are interchanged in the two positions.
  • Below both figures, write the final relation d = √(d1 .d2), stating that this gives the true separation d of the two virtual sources in the bi-prism experiment by conjugate foci method.

This completes the ray construction for the Bi-prism conjugate foci method.

  5. Conclusion:

   In this blog, we explored a clear and practical method for drawing the ray diagram used to calculate the distance between virtual sources in the Bi-prism experiment. By following these steps, you can easily construct neat and accurate diagrams that help you understand how object distance, image distance, and source separation relate to each other.

 6. Video support:

To learn the smarter drawing technique visually, watch the video below:

How to draw diagrams for modes of vibrating air column in a tube open at both ends? (Smarter Techniques):

Posted on by Physics Prana

How to draw diagrams for modes of vibrating air column in a tube open at both ends? (Smarter Techniques):

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

      Understanding the vibration of an air column is difficult for many students — especially when it comes to drawing the diagrams correctly. Most students try to copy the curves from memory without understanding why the shapes look the way they do.

     With a simple, smart technique, you can draw these diagrams neatly on your A4 sheet within seconds.

    In this blog, I will walk you through:

  • What happens when air vibrates in a tube open at both ends
  • Why all harmonics (1st, 2nd, 3rd, …) appear
  • How to draw the 1st, 2nd and 3rd harmonics quickly
  • A simple method from my “Smarter Diagram Techniques” series

Let’s begin.

  2. What happens in a tube open at both ends?

    Consider a tube of length L open at both ends and containing air at room temperature. When a tuning fork of suitable frequency is held near one end, longitudinal waves travel along the air column and are reflected at the ends. At certain frequencies, the incident and reflected waves superpose to form stationary (standing) waves in the air column.

    At an open end of a tube, the air is free to move, so the displacement is maximum and the pressure variation is minimum. Hence each open end is a displacement antinode (A) and a pressure node. Between two successive antinodes there must be at least one displacement node (N). Only those patterns which satisfy “A at both open ends and N–A–N… sequence inside” are possible modes of vibration.

Because both ends are open:

  • Each open end is always an antinode (maximum vibration)
  • A certain number of nodes (no vibration) form inside the tube

This simple rule controls every standing-wave pattern in an open–open tube.

 3.Why all harmonics occur in an open tube?

   In an open–open pipe, the allowed patterns must satisfy:

  • Antinode at the top
  • Antinode at the bottom

These conditions are satisfied by:

  • 1st harmonic (fundamental)
  • 2nd harmonic
  • 3rd harmonic
  • 4th harmonic…and so on

   That’s why all harmonics are present in a tube open at both ends.

  4.Modes of vibration of the air Column in a tube open at both ends:

  1st harmonic ↓ (fundamental)   

                ↓   2nd harmonic  

            ↓   3 rd harmonic  

   An air column of length L is formed in the tube. When a tuning fork excites it, the air column vibrates in different modes, as shown in the figure above .

  4.1.  First mode (Fundamental / First Harmonic):

  • Antinode (A) at top
  • Antinode (A) at bottom
  • One node (N) exactly at the centre
  • Half a wavelength fits inside the tube

     Let V be the velocity of sound in air. For an open tube, the allowed frequencies form a harmonic    series. Let n be the frequency of 1st harmonics and λ be the correspond wavelength.

L = λ/2

Velocity of sound:
V = nλ

Substituting:
n = V/2L

This is the lowest  (Fundamental) frequency of the vibrating air column.

  4.2. Second mode (Second Harmonic):

  • Antinodes at both open ends
  • Two internal nodes
  • One complete wavelength fits in the tube
  • Let n1 be the frequency of 2nd harmonics and λ1 be the correspond

L = λ1

Velocity of sound:
V = n1 λ 1

Substituting:
n1 =  2.V/2L

      This is 2nd harmonic frequency of the vibrating air column.

  4.3. Third mode (Third Harmonic):

  • Antinodes at both open ends
  • Three internal nodes
  • One and a half wavelengths fit inside
  • Let n2 be the frequency of 3rd harmonics and λ2 be the correspond

L = 3.λ2 /2

Velocity of sound:
V = n2 λ2

Substituting:
n2 =3.V/2L

  This is 2nd harmonic frequency of the vibrating air column.

 Therefore, the frequencies are:

n =V/2L, n1 = 2.V/2L, n2 = 3.V/2L ……..so on

Thus, a tube open at both ends vibrates with all harmonics.

  4.4. Examples

  • Flute (embouchure hole acts as an open end)
  • Recorder
  • Open metal or PVC pipe (both ends not covered)
  • Organ pipe without a stopper (open pipe)
  • Paper straw
  • Wind chimes (hollow tubes open at both ends)

  5. Smarter Technique: How to Draw the Diagrams Easily:

      Students often draw uneven curves or place nodes at the wrong positions.
Here is the simple method that never fails . (See the diagram given above)

✔ Step 1: Draw the Tube

  • Draw two long, perfectly parallel vertical lines of length 6 cm.
  • Keep both ends open.
  • Mark the length L on the left for the first harmonic diagram.

✔ Step 2: Mark Antinodes and Nodes

    Apply the rule for every mode:

  • Top → Antinode (A)
  • Bottom → Antinode (A)
  • Internal nodes (N) depend on the harmonic number.
  • For 1st harmonic N at 3 cm.
  • For 2 nd harmonic one at 1.5 cm second one at 4.5 cm.
  • For3 rd harmonic one at 1cm second one at 3 cm and third at 5 cm.

✔ Step 3: Add Loops According to Harmonic

Each loop represents half a wavelength.

  • 1st harmonic → 1 loop (½ λ)
  • 2nd harmonic → 2 loops (1 λ)
  • 3rd harmonic → 3 loops (1.5 λ)

    Mark A’s at the ends and N’s at equal distances inside.

✔ Step 4: Draw Smooth Curves

  • Connect A → N → A with a smooth curve.
  • Repeat for each loop.
  • Ensure symmetry on both sides of the tube
  • This gives you neat, clear, textbook-quality diagrams every single time.

5. Conclusion:

       Understanding the rules of nodes and antinodes makes drawing these diagrams extremely simple. Using the Smarter Diagram Technique, you can draw the 1st, 2nd, 3rd… harmonics of an open–open air column quickly and neatly — exactly as shown in the figure.

 6. Video support:

     Do you want to know ‘ How to Draw Diagrams for Modes of vibrating Air column in a tube closed  at One End?’ using given guidelines.

    Let us see from the following video for an actual smarter method of drawing a diagram.

How to draw ray diagram for bi-prism experiment (step-by-step | Smart techniques)

Posted on by Physics Prana

How to draw ray diagram for bi-prism experiment (step-by-step | Smart techniques):

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

     Drawing neat diagrams in physics is not just about marks—it helps you understand the experiment more clearly. Many students remember the formula λ=Xd/D  but feel nervous when the question is, “Draw and explain the Fresnel bi‑prism experiment.”

    The ray diagram looks crowded – slit bi‑prism, virtual sources, interference region, fringes, screen and eyepiece – all on one page. This post shows a simple, construction style method so that any class 11–12 student can draw the bi‑prism ray diagram neatly on A4 sheet in two to three minutes.

Let’s begin.

 2.What is a Bi-prism? (Simple Explanation):

      A Fresnel bi‑prism is a very thin glass prism whose vertex angle is almost 1790. So it behaves as if two small prisms are joined at the base.
  When monochromatic light from a narrow slit falls on the bi‑prism, refraction makes the slit appear as two virtual coherent sources. These two virtual sources overlap and produce interference fringes on a distant screen, just like Young’s double slit experiment, and the pattern is observed with a micrometer eyepiece.

 3. What exactly happens in the Bi-prism Experiment?

    Here’s what you need to imagine:

  • A slit S emits monochromatic light of wavelength λ.
  • When this light passes through the bi-prism, refraction occurs at the two inclined faces.
  • Due to this refraction, the slit behaves as if it has split into two virtual coherent sources, named S₁ and S₂.
  • These two virtual sources overlap and produce interference fringes on a screen or using an eyepiece on an optical bench.

   So even though you have only one slit, the bi-prism creates two coherent sources—this is the beautiful trick behind the experiment!

 4. What do we measure in the experiment?

To find the wavelength of the light, we measure:

✔ Fringe width (X)

Distance between two consecutive bright fringes.

✔ Distance between virtual sources (d)

Found using lens displacement or optical bench method.

✔ Distance between sources and screen (D)

Once these three quantities are known, wavelength is calculated using the formula:

  λ = (X × d) / D

This is the heart of the experiment.

 5. Why the ray diagram is important:

The ray diagram explains:

  • How the bi-prism creates two virtual sources
  • How rays emerge from these sources
  • Where the fringe pattern is formed
  • Why interference occurs

    A clean, properly scaled diagram makes the concept very easy to understand—this is where smart drawing techniques help.

6. Construction of the Bi-prism Ray Diagram (Step-by-   Step):

Follow these steps carefully to draw a clear, neat, and properly labeled ray diagram of the bi-prism experiment.

  6.1. Draw the Principal (Optical) Axis

   First, select 5 horizontal lines of ruled A4 page, at middle of half page. Label them as 1, 2,3,4,5 in margin region with pencil.

     Draw a long, straight horizontal line at line number 3 — this will serve as the principal (optical) axis around which the entire diagram is structured.

 6.2. Mark the Single Slit

   On the left side of the axis, draw a small dot to represent the single narrow slit S. Label it “S”. This slit is the source of monochromatic light (wavelength λ). Then mark two dots at line 1 and 5 exactly above and below of slit S. Mark them as S1 and S2 .

 6.3. Position and Draw the Bi-prism
    At 3 cm to the right of the slit S draw the vertical line covering slightly above of line 1 and slightly below line 5. Mark point on axis (line 3) at 0.5 cm left from vertical line. Join end points of vertical line to this point. This forms bi-prism.

 6.4. Draw Incident Rays from the Slit
   From the slit S, draw 4 straight rays directed toward the two faces of the bi-prism and touching at line 1,2,4,5 at inclined face of bi-prism. Put arrow head on these lines directing to right. These represent the light emerging from the slit heading toward the prism.

  6.5. Show Refraction through the Prism
  Join S1 and point of intersection of ray 2 and bi-prism face by dotted line and extend this line up to line 5. From the intersection line 5 and extended line draw a vertical big line forming the screen.

  Repeat this for slit S2 . Now draw horizontal lines from S1 and S2 up to screen. Label interference region as shown in the figure. Add arrow heads to all lines heading to the right.

 6.6. Label Distances
On your diagram, clearly mark and label:

    • d: the distance between S₁ and S₂ (i.e., separation of virtual sources)
    • D: the distance from the virtual sources (S₁ / S₂) to the screen
  • Label bi-prism.
  • Labelling measurement parameters directly links the diagram to the experimental formula λ = X d / D, reinforcing understanding.

  6.7. Final Understanding:

After completing the diagram, you have a full picture:

  • How one slit becomes two sources
  • How interference arises
  • How the experiment leads to measurement of λ

  By using this smart, stepwise method, your diagram will look neat, logical, and exam-ready.

 8. Formula Highlight:

λ = X d / D
This formula gives the wavelength of light used in the Bi-prism experiment.

  9. Conclusion:

    Drawing the Bi-prism ray diagram becomes very simple when you follow a structured technique. Understanding the purpose of each line—slit, prism, virtual sources, rays, screen—makes the experiment clear and easy to learn.

10. Video support:

    Do you want to know  “How to draw ray diagram for bi-prism experiment (step-by-step | Smart techniques)”,using given guidelines.

Let us see from the following video for an actual smarter method of drawing a diagram.

How to Draw the Ray Diagram for Refraction at a Plane Boundary Using Huygens’ Principle – Smarter Techniques

Posted on by Physics Prana

How to Draw the Ray Diagram for Refraction at a Plane Boundary Using Huygens’ Principle – Smarter Techniques

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1.Introduction:

      Accurate ray diagrams are essential for understanding optics, especially for students of Class 10 to 12. Many students try to copy the diagrams mechanically, but the real beauty of Physics lies in understanding how a diagram emerges from fundamental principles.

       In my earlier blog, How to Draw Diagrams in Physics – Smarter Techniques, I explained a set of general guidelines for drawing neat and conceptually strong diagrams on an A4 sheet.
In this post, let us use the same guidelines to understand how to draw the ray diagram for refraction at a plane boundary using Huygens’ Principle.

  2.Laws of refraction:

    When light travels from air into glass, it bends at the surface separating the two media. This bending is called refraction and is described by Snell’s law, which relates the angles of incidence and refraction to the speeds of light in the two media. Using Huygens’ Principle, this behaviour can be shown with a clear, step-by-step ray diagram that students can easily reproduce in examinations.

 This bending is governed by: 

    2.1. . Refraction happens when light enters a medium with a different speed.

   2.2. . First law of refraction:

Snell’s law: n=sin i /sin r=v1/v2

Where

  • i = angle of incidence
  • r = angle of refraction
  • n = refractive index of the second medium with respect to the first
  • v1 = velocity in rarer medium
  • v2 = velocity in denser medium 

    2.3. Second law of refraction:

  The incident ray, refracted ray and the normal to the boundary all lie in the same plane.

These laws explain what happens—but Huygens’ Principle explains why it happens.

   3. Huygens’ principle in simple words:

         “Every point on a wavefront acts as a secondary source and sends secondary wavelets in all directions.”

   Using this idea, we can construct a wavefront in the refracted medium and from that, obtain the refracted ray.

    Huygens’ Principle not only explains refraction—it gives us a practical method to draw the ray diagram much more accurately.

   4. Why does refraction occur? (Wavefront explanation):

  When light enters a denser or rarer medium:

  • The speed of light changes.
  • Therefore, the secondary wavelets in the second medium grow at a different speed.
  • As a result, the wavefront tilts.
  • A ray drawn perpendicular to this new wavefront becomes the refracted ray.

     This is the core idea behind drawing the diagram.

   5. Smarter technique: step-by-step method to draw the ray diagram:

   Use an A4 sheet and follow the neat-diagram rules explained in the earlier blog.
         Then, draw the refraction diagram using these steps:

  5.1. Draw the boundary and normal

     Draw a horizontal line XY to represent the boundary between air (top) and glass (bottom).
  At point A on XY, draw a vertical line AM; this is the normal to the surface where the ray will meet     the boundary.

  5.2. Draw the incident rays from a distant source

  • Place the ruler at point A so that its left edge passes through A and is slightly inclined (about 30°) to the XY line, with about 5 cm of the ruler above XY.
  • Now draw two parallel lines along the edges of the ruler up to the XY line. Label the point where the right-side line meets the XY line as C.

   5.3. Mark the angle of incidence at A

    • At A, mark the angle between the incident ray PA and the normal AM as ∠i.
    • Indicate this angle clearly with a small arc and the symbol i.

  5.4. Show an incident wavefront and its secondary source

  • Draw perpendicular to the line QC from A, mark a point B at foot of the perpendicular on QC.
  • Line segment AB indicates the part of the incident wavefront that has already reached the mirror at A, while the point near B is still in the incident medium.
  • When point A touches the boundary first, it starts sending secondary wavelets into the glass, while point B continues to advance in air toward the boundary.

  5.5. Mark distances in equal time
       Choose a small time interval t.

  • In time t, point B moves in air to C on the boundary, a distance v1t
  • In the same time t, the secondary wavelet from A travels into glass a distance v2t
  • From A, draw an arc inside the glass with radius v2t

  5.6. Draw the new refracted wavefront
      Join point C on the boundary to a point D on the arc so that CD just touches (is tangent to) the       arc from A.
      Line CD is the new refracted wavefront in the glass.

  5.7. Draw the refracted ray and identify angles
      From A, draw a line AR perpendicular to the refracted wavefront CD; this is the refracted ray in      glass.
   The angle between AR and the normal is the angle of refraction r.

  5.8. Connect the construction to Snell’s law
      In the geometry of triangle ACD:

  • AC corresponds to distance v1t in air.
  • AD corresponds to distance v2t in glass.
    From the similar triangles formed, the ratio of the sines of the angles equals the ratio of these distances, giving
    : n = sini/sinr = v1/v2

  5.9. Conclusion:

   Why this diagram is a “Smarter Technique”
     This construction lets students see both the ray and the wavefront picture in one diagram. It also      proves Snell’s law directly, instead of just stating it, and encourages neat, exam ready diagrams     that can be drawn accurately on an A4 sheet.

   6. Video support:

   Do you want to know “How to draw the Ray Diagram for Reflection at a plane      surface based on Huygens’ Principle?” using given guide lines?

  Let us see from following video for actual smarter method of drawing diagram. 

How to Draw the Ray Diagram for Reflection at a Plane Surface Using Huygens’ Principle– Smarter Techniques

Posted on by Physics Prana

How to Draw the Ray Diagram for Reflection at a Plane Surface Using Huygens’ Principle– Smarter Techniques

 

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

    Most students know the laws of reflection, but when asked to draw the ray diagram based on Huygens’ Principle, they often get confused.
Why?

     Because students often replicate the diagram mechanically rather than understanding the underlying wavefront method.

   In my series “How to Draw Diagrams in Physics — Smarter Techniques,” I focus on teaching students scientific, accurate, and exam-perfect ways to draw on an A4 sheet using simple tools like a scale, set-square, and compass.

In this article, you will learn:

  • What Huygens’ Principle says
  • How it explains reflection
  • And—most importantly—

How to draw the ray diagram step-by-step using a smarter technique?

Let’s begin.

2. Laws of Reflection:

  2.1. Angle of incidence (i) = Angle of reflection (r)

  2.2Incident ray, reflected ray, and normal lie in the same plane

These laws are not assumptions—they can be beautifully proved using wave fronts.

3. Huygens’ principle (Quick recap):

“Every point on a wave front acts as a secondary source and emits secondary wavelets in all directions.”

This simple idea helps us reconstruct the next position of a wave front.

4. How Huygens’ principle explains reflection:

 When a wave front strikes a plane mirror:

  • The point touching the mirror acts as a source of secondary wavelets
  • These wavelets reflect according to the rule: Angle of reflection = angle of incidence
  • By constructing the reflected wave front, we obtain the reflected ray

This is more scientific than simply “drawing rays.”

5. Why this is a “Smarter technique”?

  1. The diagram uses only straight lines, one arc, and simple perpendiculars, so students can reproduce it quickly and neatly in the exam.
  2. Each construction step has a clear physical meaning: incident wave front, secondary wavelet, reflected wave front, and rays perpendicular to wave fronts.
  3. From a single, clean diagram, both laws of reflection are obtained using Huygens ’ Principle, giving students both clarity in concept and confidence in drawing.

Here is a step-by-step construction matching your given diagram.

6. Step-by-step construction:

6.1. Draw the plane reflecting surface

  • Draw a horizontal line and mark it as plane reflecting surface XY.
  • Choose the point on it, near the left, and label them A.

  6.2.  Draw the incident rays from a distant source

  • Place the ruler at point A so that its left edge passes through A and is slightly inclined (about 30°) to the XY line, with about 5 cm of the ruler above XY.
  • Now draw two parallel lines along the edges of the ruler up to the XY line. Label the point where the right-side line meets the XY line as C.

   6.3. Draw the normal at points of incidence

  •     At point A, draw a vertical line upwards; label its upper part as M. This line is the normal to the surface at A.
  • At point C, draw another vertical line upwards; label its upper part as N. This line is the normal to the surface at C.

  6.4. Mark the angle of incidence at A

    • At A, mark the angle between the incident ray PA and the normal AM as ∠i.
    •   Indicate this angle clearly with a small arc and the symbol i.

  6.5. Show an incident wave front and its secondary source

  • Draw perpendicular to the line QC from A, mark a point B at foot of the perpendicular on QC.
  • Line segment AB indicates the part of the incident wave front that has already reached the mirror at A, while the point near B is still in the incident medium.

   6.6. Draw secondary wavelets from point A

  • Taking A as centre, draw an arc with radius AB cutting the vertical through N at a point; this arc represents the secondary wavelet that has travelled from A during the time the disturbance moves from B to C.
  • Indicate this arc smoothly from near M towards the region near C, as in your diagram.

  6.7. Construct the new (reflected) wave front

  • From point C on the surface, mark off on the vertical CN a distance equal to the radius used at A; label the point where the arc intersects the construction as D.
  • Join B to D by a straight line; this line BD represents the reflected wave front obtained as the common tangent to the secondary wavelets.

  6.8. Draw the reflected rays

  • Through point A, draw a straight line from left to right such that it is symmetric to the incident ray PA about the normal AM; this is the reflected ray, and you may extend it towards the right and label it AR with an arrow pointing away from the mirror.
  • Through point C, draw another reflected ray starting from C, making the same angle with the normal CN as the ray from B does with the vertical; mark this ray as CS with an arrow.

  6.9. Show equality of angles and explain

    1. Mark the angle between the reflected ray at A and the normal AM as ∠r
    2. Note that by construction the triangles formed using distances along the surface and verticals are congruent, so ∠i=∠r, verifying the law of reflection using Huygens’ Principle.

6.10.  Final labelling and neatness

  • Label all important points and lines: surface XY, normals AM and CN, incident rays PQ and reflected rays AR and CS, wave front segment AB, secondary wave front arc through D.
  • Thicken or darken the final incident and reflected rays and keep construction lines slightly lighter so students see clearly what they must finally reproduce in the examination.

7. Video support:

   Do you want to know “How to draw the ray diagram for reflection at a plane surface based on Huygens’ principle?” using given guide lines.

Let us see from following video for actual smarter method of drawing diagram. 

 

 

How to draw diagrams for modes of vibrating air column in a tube closed at one end – Smarter Techniques

Posted on by Physics Prana

How to draw diagrams for modes of vibrating air column in a tube closed at one end – Smarter Techniques

“Harmonics in Closed Tube Explained with Diagram”

 

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1.Introduction:

   Understanding the vibration of an air column can be confusing for many students—especially when it comes to drawing the diagrams. Most of the time, students memorize the shapes without really knowing why they look that way.

But with a smart technique, you can draw these diagrams neatly on your A4 sheet within seconds.

In this blog, I will walk you through:

  • What happens when air vibrates in a tube closed at one end
  • Why only odd harmonics appear
  • How to draw 1st, 3rd and 5th harmonics quickly
  • A simple method from my “Smarter Diagram Techniques” series

Let’s begin.

    2. What Happens in a Tube Closed at One End?

    Imagine a tube, one end closed, and the other open, filled with air. Strike a tuning fork and hold it nears the open end, then, the air column inside vibrating, creating “standing waves.” These vibrations happen in specific ways known as “modes.”

Because one end is closed:

  • The closed end is always a node (no vibration)
  • The open end is always an antinode (maximum vibration)

This simple rule controls the entire pattern.

    3. Why Only Odd Harmonics?

        In a closed pipe, all allowed patterns must satisfy:

  • Node at closed end
  • Antinode at open end

This happens only in the:

  • 1st harmonic (fundamental)
  • 3rd harmonic
  • 5th harmonic
  • And so on…

That’s why a closed tube has only odd harmonics.

      4. Modes of vibration of the air column in the tube closed at one end:

Fundamental frequency↓ |  3 rd  Harmonics frequency↓    ↓   5 th  Harmonics frequency

Fig. A

   An air column of length L is formed in a tube closed at one end. This air is made to vibrate by a tuning fork of frequency n. Then the air column will vibrate in different modes.

   4.1. First mode (Fundamental/First harmonic):

     In this mode, there’s a node (point of no vibration) at the closed end and an antinode (point of maximum vibration) at the open end. Only one-quarter of a wave fits inside the tube for this lowest frequency vibration.

    Let  V be the velocity of sound waves in air and n fundamental frequency with λ wavelength because of one node and one antinode only. The tube length L covers half wavelength.

Thus L = λ/4

The velocity of sound waves in air V = n λ

Putting the value of λ, we get

V = 4nL

Then the fundamental frequency n is given by

n= V /4L

  4.2. Higher Modes (Third and Fifth Harmonics)

      4.2.1. Third harmonic (First overtone):

    Here, three-quarters of a wave fit inside the tube, including two nodes (one at the closed end and one inside the tube) and two antinodes. Let n1 be third harmonic frequency with λ1 wavelength

Thus L = 3λ1/4

The velocity of sound waves in air V = n1 λ1

Putting the value of λ1, we get

V = 4nL/3

Then the third harmonic frequency n1 is given by

n1= 3V /4L

 

    4.2,2. Fifth Harmonic (Second Overtone):

    Now, one and one-quarter wavelengths fit inside. Draw three antinodes and two additional nodes between the ends.

   Let n2 be fifth harmonic frequency with λ2 wavelength

Thus L = 5λ2/4

The velocity of sound waves in air V = n2 λ2

Putting the value of λ2, we get

V = 5nL/3

Then the fifth harmonic frequency n2 is given by

n2= 5V /4L

Next frequencies are 7,9 …… times of V/4L.

Thus, an air column in a tube closed at one end vibrates only with odd harmonics.

 4.3. Examples

  • Bottle / glass bottle (blowing across the top acts like a closed–open tube)

  • Tube with one end capped (PVC pipe with one end plugged)

  • Didgeridoo

  • Clarinet (approximately behaves like a closed–open tube because of the reed end)

  • Organ pipe with a stopper (stopped pipe)

  5.  Smarter Technique: How to draw the diagrams easily:

     Students often draw wrong loops or uneven curves.
    Here is a simple method that never fails.

    See the figure Fig. A

✔ Step 1: Draw the Tube

  • Draw a long vertical rectangle.
  • Mark the closed end with a thick, dark line.
  • Leave the top open.

This sets the boundary conditions.

✔ Step 2: Mark Node and Antinode

Always apply the rule:

  • Bottom (closed) end → node
  • Top (open) end → antinode

Keep this fixed for every pattern.

✔ Step 3: Add Loops Depending on Harmonic

Here is the trick:

One loop is formed by two nodes with antinode between them. The half loop consists of one node and one antinode.

  • 1st harmonic → Half loop
  • 3rd harmonic → One and half loops
  • 5th harmonic → Two and half loops

Each loop represents half a wavelength.

✔️ Step 4: Draw the Wave Smoothly

  • Start from a node at the closed end.
  • End at an antinode at the open end.
  • Use smooth, equal curves for each loop.

This gives a neat, textbook-perfect diagram.

 6. Video support:

  Do you want to know ‘ How to Draw Diagrams for Modes of vibrating Air column in a tube closed at One End?‘ using given guidelines.

Let us see from the following video for an actual smarter method of drawing a diagram.

How to draw diagrams for Modes of vibrating Stretched String? – Smarter Techniques

Posted on by Physics Prana

How to draw diagrams for Modes of vibrating Stretched String? – Smarter Techniques:

 

 

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

      Physics students frequently encounter questions about “vibrating string modes” in exams and competitions. Drawing clear, accurate diagrams for these modes not only helps in scoring high marks, but also builds a deeper understanding of wave physics. This comprehensive guide explains the science, offers smarter diagramming techniques.

    When a string is fixed at both ends and stretched with a tension T, it vibrates in different patterns or modes when plucked. These patterns form the basis of harmonics in physics.

Let:

  • L = length of the string
  • m = mass per unit length
  • p = number of loops (segments) formed in the vibrating string
  • n = frequency of vibration

     Understanding the relationship between these elements makes it easy to draw accurate physics diagrams for vibrating strings.

2.What Are Vibrating String Modes?

     When you pluck or strike a string that is fixed at both ends, such as on a guitar or lab apparatus, the string vibrates in characteristic patterns known as vibrating string modes or harmonics. Each mode displays loops (antinodes separated by nodes points that remain stationary). Understanding and drawing these modes forms a core part of learning about waves, sound, and musical instruments in physics.

3.Harmonic Modes of a Stretched String:

When a stretched string vibrates, it can produce several harmonic frequencies:

3.1.  Fundamental Mode (1st Harmonic)

  This is the simplest vibration pattern.

  • The string has one loop (p = 1).
  • The frequency is known as the fundamental or first harmonic.     

3.2. Second Harmonic:

  • The string forms two loops (p = 2).
  • The frequency corresponds to the second harmonic.

3.3. Third Harmonic:

  • The string forms three loops (p = 3).
  • This gives the third harmonic, and so on…

In general:
When p = 1, 2, 3…, the string vibrates in the 1st, 2nd, 3rd… harmonics respectively.

A stretched string naturally vibrates with all possible harmonic frequencies, not just the fundamental.

4. Step-by-step: Smarter techniques to draw vibrating string modes:

 

←  Fundamental or 1 st Harmonic frequency.

 

 

←  2 nd Harmonics frequency.

 

←  3 rd Harmonics frequency.

Fig. A   

   Ready to draw diagrams with confidence? Here are step wise, smarter approaches to ensure accuracy and speed—perfect for examinations or teaching:

4.1. Draw the Baseline

     Use your ruler to create a perfectly straight line representing the string. Leave space at both ends to mark the fixed nodes.

4.2. Mark Nodes and Antinodes

    Use dots or tiny vertical ticks for nodes (points of no displacement). Place them at both ends, and for higher harmonics, divide the string into equal segments for additional nodes.

4.3. Sketch Loops (Antinodes)

   For the fundamental, draw one smooth arc peaking at the centre. For the second and third harmonics, sketch two or three arcs, each touching the baseline at the nodes.

4.4. Label Diagram Elements

     Mark “N” for node and “A” for antinode. Indicate the total length as “L.” This reduces examiner confusion and looks professional.

4.5. Use Consistent, Clear Symmetry

      Keep the loops and distances between nodes consistent and symmetrical. A clean diagram reflects proper understanding and care.

4.6. Caption Each Diagram

   Write a brief caption beneath each: “Fundamental (1st Harmonic),” “Second Harmonic (1st Overtone),” etc. This helps both for grading and for writing clarity.

5. Quick Reference Table: String Modes at a Glance:

Mode Nodes Antinodes Wavelength Frequency Diagram Structure
Fundamental (1st) 2 1 λ=2L N One big arc
Second Harmonic 3 2 λ=L 2n “M” shape (2 arcs)
Third Harmonic 4 3 λ=2L/3 3n “W” shape (3 arcs)

6. Common mistakes (and How to avoid them):

  Avoid these frequent pitfalls for stress-free, high-scoring answers:

  • Uneven node spacing: Always calculate divisions carefully.
  • Arcs not touching baseline only at nodes: Double-check with a ruler and light pencil sketch first.
  • Forgetting to label: Always label nodes, antinodes, and length.
  • Missing captions: Add short, clear captions for clarity and examiner friendliness.

   For teachers and tutors, encourage practice with real exam questions and cite schematic sources (including this guide!) for additional support.

7. How are vibrating string modes appear in real life?:

   Understanding these diagrams goes beyond exam scores. The same principles govern:

  • Musical instruments, like guitars and violins, where different vibration modes produce unique notes.
  • Lab experiments in physics to demonstrate resonance and frequency relationships.
  • Advanced concepts of wave mechanics and quantum physics.

     If you’re studying sound waves, learning to visualize these patterns builds intuition that lasts through university and beyond.

8. Video support:

   So do you want to know  ‘ How to draw Diagrams for Modes of vibrating Stretched String  ‘ using given guide lines?

Let us learn with the help of following video the actual smarter method of drawing the diagram.

How to draw a vector diagram for the motion of Simple pendulum as a S. H. M.? (Smarter Techniques)

Posted on by Physics Prana

How to draw a vector diagram for the motion of Simple pendulum as a Simple Harmonic Motion? (Smarter Techniques)

 

 

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1.Introduction:

Many students understand the theory of a simple pendulum, but when it comes to drawing the vector (force) diagram, confusion begins.

    Questions like “Which force acts where?”, “Why do we resolve mg?”, and “How does SHM appear in the diagram?” are very common.

    In this article, we will remove that confusion completely by using a smart, visual, exam-oriented method, based on the diagram shown above and explained step by step in the linked video.

2.What is a simple pendulum?

      A simple pendulum is a heavy particle tied at the free end of a long suspended string at the support.

But in practice, instead of a heavy particle, a heavy bob is attached.

The length (ℓ) of the pendulum is the distance between the point of support and the center of the bob.

When displaced slightly from its mean position and released, the bob oscillates to and fro.

For small angular displacements, the motion of the bob is Simple Harmonic Motion (S.H.M.).

3. Why the motion of a simple pendulum is linear simple harmonic motion.:

   As the bob swings, it oscillates about its mean position, and for small angles, its motion becomes Simple Harmonic Motion (S.H.M.).

 3.1. Let’s look at what happens when the pendulum is swinging:

Fig. A represents an oscillating simple pendulum.Fig. A

Consider bob of mass m is attached to a string of length L and is oscillating between A and C. Let the pendulum be at an angle Ө from vertical.

B is the mean (equilibrium) position

 At position A, the string makes an angle θ with the vertical

This is the position where we draw the vector diagram.

 

3.2. Forces acting on the bob:

At any instant (say at position A), two forces act on the bob:

  1. Weight (mg) acting vertically downward
  2. Tension (T) acting along the string towards the point of suspension

  The weight mg is resolved into two components: mgcosӨ as the ǁ component and mgsinӨ as the ⏊ component.

    mgcosӨ is balanced by tension in the string, and mgsinӨ acts as a restoring force to bring the bob to mean position B.

As Ө is small,

sinӨ = Ө = x/ ℓ.

Restoring force

F = -mgx / ℓ ,

∴  F  ⍺ – x

The restoring force is directly proportional to displacement and opposite to it.

This is Simple Harmonic Motion  of a simple pendulum.

Thus, the motion of a simple pendulum is linear simple harmonic motion.

4.Smarter way to draw the vector diagram:

This is the most important part for students and teachers.

 4. 1: Draw the pendulum geometry:

 Draw a horizontal support at the top

 Draw the string making an angle θ with the vertical

 Mark the bob at position A

 Show the mean position B vertically below the support

Keep the diagram large and clean (as shown in the Fig.A).

  4.2: Draw the weight vector (mg):

 From the center of the bob at A, draw an arrow vertically downward

 Label it clearly as mg

This is the actual force due to gravity.

  4.3: Resolve mg into two components (Smart trick):

Now comes the key idea.

Resolve mg into:

  4.3.1. Component along the string:

 Draw a component of mg along the string

 Label it as mg cosθ

 Mention: “Balanced by tension T”

This component does not cause motion.

  4.3.2. Component perpendicular to the string:

 Draw the second component perpendicular to the string

 Label it as mg sinθ

 Show its direction towards the mean position B

This is the restoring force.

  4.4: Mark the angle clearly:

 Mark the angle θ between the string and the vertical

 This is essential for exams and concept clarity.

 5. Why this diagram method is “Smarter”?

5.1.  No confusion between forces

 5.2. Easy to remember in exams

 5.3. Clear separation of balanced and restoring forces

 5.4.  Neat, scoring diagram on A4 answer sheets

This is exactly why students struggle less when they draw first and think later.

6. Video support:

   So do you want to know ‘How to draw a vector diagram for the motion of Simple pendulum as a Simple Harmonic Motion ‘ using the given guidelines?

Let us learn with the help of the following video, the actual smarter method of drawing the diagram.

How to Draw a Vector Diagram for a Vehicle Moving on a Curved Banked Road? (Smarter Techniques)

Posted on by Physics Prana

How to Draw a Vector Diagram for a Vehicle Moving on a Curved Banked Road? (Smarter Techniques)

 

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1. Introduction:

      Vector diagrams are one of the most important tools for understanding physics problems.
Many students often ask:

“How do I draw the vector diagram of a vehicle moving on a banked curved road?”

    Where to start? How to show the forces? Which vector should come first?

    In this blog, we will learn a simple, neat and smarter technique to draw this diagram—perfect for Class 10–12 students.

2. What is a banked road?

   Banking means making the road bed slightly inclined to the horizontal. This is done at curved roads for safe driving.

3.Forces Acting on a Vehicle on a Banked Curve:

Fig. A   Fig. A

    Fig. A represents a cross section of a banked road along with a vehicle on a curved road.

Consider a vehicle of mass m moving with velocity v on a curved road of radius r and banked at an angle of Ө.

Two major forces act on the vehicle:

(i) Weight (mg) acts vertically downward.

(ii) Normal Reaction (N) acts perpendicular to the inclined surface.

     The normal reaction by the roadbed on the vehicle N is resolved into NsinӨ and NcosӨ.

NsinӨ acts as a centripetal force  mv2 / r  on  the vehicle.

∴  NsinӨ = mv2 /r

And NcosӨ  is balanced by the weight of the vehicle,

NcosӨ = mg = weight of vehicle.

Using these equations we get:

v2  = r. g.tan Ө

This equation helps us find the safe velocity for the vehicle on a banked curved road.

4. How to draw the vector diagram? (Step-by-step):

(Smarter Techniques for Physics Diagrams)

Follow the steps below to draw a clean, examination-perfect vector diagram.

✔ Step 1: Draw the inclined plane:

  • On an A4 sheet, take a point A near left side, draw full horizontal line from A, and then draw a slanted line slightly making the angle θ to horizontal. .
  • This represents the banked road.

✔ Step 2: Mark the vehicle:

  • Draw a small rectangle or dot on the surface (slanted line)
  • Mark point O at the centre of rectangle.
  • This point represents the centre of gravity of the vehicle.

✔ Step 3: Draw the normal reaction (N):

  • From the same dot O, draw an arrow perpendicular to the inclined plane.
  • Label it N.

(Most students make mistakes here—normal reaction is always perpendicular to the surface.)

✔ Step 4: Resolve the normal reaction:

Draw two components of  N with arrow heads:

  • N cosθ → vertically upward
  • N sinθ → horizontally towards the centre of the circular path

Make both components as the sides of parallelogram and complete parallelogram.

✔ Step 5: Draw the weight (mg):

  • From the dot O, draw a vertical downward arrow equals to N cosθ line .
  • Label it mg.

✔ Step 6: Show the direction of Centripetal force:

  • The arrow headed N sinθ line represents centripetal force.
  • Label: Centripetal force = mv²/r

✔ Step 7: Neatness and final touches:

  • Show the angle θ clearly
  • Keep all arrows emerging from the same point
  • Align arrows straight and sharp
  • Recheck labels and arrow directions

Your vector diagram is ready!

5.  Video  Support:

   So do you want to know ‘How to Draw a Vector Diagram for a Vehicle Moving on a Curved Banked Road’ using given guidelines?

Let us learn with the help of the following video, the actual smarter method of drawing the diagram.

 

How to Draw Physics Diagrams Smartly ( Board examination of Class 10–12 Students)

Posted on by Physics Prana

How to Draw Physics Diagrams Smartly

( Board examination of Class 10–12 Students)

By Prof. Chandrakant Kali

Smarter Diagram Skills = Better Understanding + Better Marks

        Physics is a scoring subject—if your diagrams are neat, clear and accurate.
Most students learn diagrams from books, teachers, or online images. But during board examinations, you must draw them perfectly on an A4 answer sheet, and this is where many students struggle.

With over 50 years of teaching Physics, I’ve discovered that students can master diagrams easily by learning smart, structured techniques. These techniques not only improve presentation but also help you understand the concept better.

Let’s explore how to draw diagrams the smart way.

1. Why Smart Diagram Techniques Matter:

A diagram is not just a drawing—it’s a visual explanation. When you draw it correctly, you automatically understand the concept better.
Here’s how smarter diagram techniques help you:

A. Show the exact physical meaning clearly

A well-drawn diagram immediately tells you what’s happening in the concept. Smart diagrams emphasize the real concept—optics rays, field directions, etc.—not random lines.

B. Improve understanding of the topic

  Neat visuals deepen memory and reduce confusion.

C. Increase interest in Physics

  When diagrams look good, learning becomes enjoyable. They make learning physics more interesting, helping you visualize and remember principles

D. Save time in board examinations

Smart methods allow you to draw fast without sacrificing clarity.

E.  Build neatness, accuracy and confidence

You spend less time struggling and more time learning, because accuracy comes naturally.

    A clean diagram leaves a strong impression on the examiner and helps you score higher. Neat diagrams impress examiners—they show you understand the topic, not just reproduce it.

2. Before, You Start Drawing – Follow These Must-Know Points:

To create examination-quality diagrams on A4 paper, keep these basics in mind:

A. Use only half page of an A4-sized ruled sheet

    This helps maintain proportion, spacing and balance.

B. Use an HB pencil only

HB gives perfect line thickness—neither too dark nor too faint.

C. Keep your compass ready with a sharp pencil tip

This ensures smooth circles and arcs—important for lenses, mirrors, SHM, pendulum, electricity diagrams, etc.

 

 

D. Use 30 cm and 15 cm rulers as required

Choose the ruler based on diagram size.

  • Large diagrams → 30 cm ruler
  • Small and quick diagrams → 15 cm ruler.

 

 

 

 

 

3.The Smart Way to Learn Physics Through Diagrams:

When diagrams follow correct technique + concept logic, Physics becomes truly simple.

If you practice consistently:
✔ your speed increases
✔ your confidence grows
✔ your examination answers look professional
✔ your conceptual clarity improves

4. What’s coming next:

In the upcoming blog posts, I will share step-by-step smart techniques to draw diagrams like:

  • Ray diagrams (Lenses , Reflection, Refraction Biprism)
  • Simple pendulum
  • Vibration of string, Air column
  • SHM vector diagrams
  • Electric circuits
  • Graphs and force diagrams

Each one explained in a simple, practical and student-friendly way.

Stay connected—smart learning starts here!

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