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Electromagnetic Waves

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Chapter Notes

Overview
This chapter builds on the concepts of electricity and magnetism from earlier chapters to introduce electromagnetic waves. The central idea is that a time-varying electric field generates a magnetic field (displacement current), and together with Faraday's law, this leads to self-sustaining electromagnetic waves that travel at the speed of light. Students must understand that accelerated charges produce electromagnetic waves, and that light is an electromagnetic wave, unifying electricity, magnetism, and optics.
Key Concepts (22)
Displacement current
The current due to a time-varying electric field, given by id = ε0 (dΦE/dt), which acts as a source of magnetic field in the same way as conduction current.
e.g. When charging a capacitor, there is no conduction current between the plates, but a changing electric field exists, producing a displacement current that generates a magnetic field.
Conduction current
The current carried by conductors due to the flow of charges.
e.g. The current flowing through a wire connecting a battery to a resistor.
Ampere-Maxwell law
The generalised form of Ampere's circuital law: ∮ B·dl = μ0 (ic + ε0 dΦE/dt), stating that the total current (conduction + displacement) passing through a surface is the source of the magnetic field.
e.g. Used to find the magnetic field inside a charging capacitor where only displacement current exists.
Electromagnetic waves
Coupled time-varying electric and magnetic fields that propagate in space, perpendicular to each other and to the direction of propagation.
e.g. Light, radio waves, X-rays, and microwaves are all electromagnetic waves.
Maxwell's equations
A set of four equations (Gauss's law for electricity, Gauss's law for magnetism, Faraday's law, and Ampere-Maxwell law) that mathematically express all basic laws of electromagnetism.
e.g. These equations predict the existence of electromagnetic waves and that light is an electromagnetic wave.
Speed of electromagnetic waves in vacuum
The constant speed c = 1/√(μ0ε0) = 3 × 10^8 m/s, which is the speed of light.
e.g. All electromagnetic waves, from gamma rays to radio waves, travel at this speed in vacuum.
Electromagnetic spectrum
The classification of electromagnetic waves according to frequency or wavelength, ranging from gamma rays (shortest wavelength) to radio waves (longest wavelength).
e.g. Visible light is a small part of the spectrum between ultraviolet and infrared.
Accelerated charge
A charge that changes its velocity (speed or direction) and radiates electromagnetic waves.
e.g. An oscillating charge in an antenna produces radio waves.
Plane electromagnetic wave
An electromagnetic wave in which the electric and magnetic fields lie in a plane perpendicular to the direction of propagation and vary sinusoidally.
e.g. A wave propagating along the z-direction with Ex = E0 sin(kz - ωt) and By = B0 sin(kz - ωt).
Wave vector (k)
A vector whose magnitude is k = 2π/λ and whose direction is the direction of propagation of the wave.
e.g. For a wave travelling along the x-axis, k points along the x-direction.
Relation between E and B in an electromagnetic wave
The magnitudes of the electric and magnetic fields are related by E0 = c B0, and they are perpendicular to each other and to the direction of propagation.
e.g. If E0 = 120 N/C, then B0 = E0/c = 4 × 10^-7 T.
Hertz's experiment
The first experimental demonstration of electromagnetic waves (radio waves) by Heinrich Hertz in 1887, confirming Maxwell's prediction.
e.g. Hertz produced and detected electromagnetic waves of a few metres wavelength.
Jagdish Chandra Bose's experiment
Bose produced and observed electromagnetic waves of much shorter wavelength (25 mm to 5 mm) in 1894.
e.g. His work was confined to the laboratory but advanced the study of microwaves.
Marconi's experiment
Guglielmo Marconi succeeded in transmitting electromagnetic waves over distances of many kilometres, marking the beginning of communication using electromagnetic waves.
e.g. Marconi's work led to the development of radio communication.
Speed of electromagnetic waves in a medium
The velocity v = 1/√(με), where ε is the permittivity and μ is the permeability of the medium.
e.g. Light travels slower in glass than in vacuum because glass has higher ε and μ.
Radio waves
Electromagnetic waves produced by accelerated motion of charges in conducting wires, used in communication systems (frequency range 500 kHz to about 1000 MHz).
e.g. AM radio uses frequencies from 530 kHz to 1710 kHz.
Microwaves
Short-wavelength radio waves in the GHz range, produced by special vacuum tubes, used in radar and microwave ovens.
e.g. Microwave ovens use a frequency that matches the resonant frequency of water molecules to heat food.
Infrared waves
Electromagnetic waves produced by hot bodies and molecules, often called heat waves, used in physical therapy and remote controls.
e.g. Infrared lamps are used in physical therapy to heat muscles.
Visible rays
The part of the electromagnetic spectrum detected by the human eye, with wavelengths from about 400 nm to 700 nm.
e.g. Sunlight is composed of visible light, which allows us to see objects.
Ultraviolet rays
Electromagnetic waves with wavelengths from about 400 nm down to 0.6 nm, produced by very hot bodies and special lamps, harmful in large quantities.
e.g. The sun emits UV radiation, but most is absorbed by the ozone layer.
X-rays
Electromagnetic waves with wavelengths from about 10 nm down to 10^-4 nm, generated by bombarding a metal target with high-energy electrons, used in medical imaging.
e.g. X-rays are used to take images of bones in a hospital.
Gamma rays
High-frequency electromagnetic waves with wavelengths from about 10^-10 m to less than 10^-14 m, produced in nuclear reactions and by radioactive nuclei, used to destroy cancer cells.
e.g. Gamma rays are used in radiotherapy to treat cancer.
Formulas & Equations (13)
∮ B·dl = μ0 i(t)
Reads as: The line integral of B dot dl equals mu naught times the current i of t
Variables: B = magnetic field (T), dl = infinitesimal length element (m), μ0 = permeability of free space (4π × 10^-7 T m/A), i(t) = time-dependent conduction current (A)
Use when: Applying Ampere's circuital law to find magnetic field due to a steady or time-varying conduction current
⚠ Students forget that this original form is incomplete and fails for surfaces not pierced by conduction current, e.g., between capacitor plates.
id = ε0 (dΦE/dt)
Reads as: Displacement current i_d equals epsilon naught times the rate of change of electric flux
Variables: id = displacement current (A), ε0 = permittivity of free space (8.85 × 10^-12 F/m), ΦE = electric flux (N m^2/C), t = time (s)
Use when: Calculating the displacement current in a region where the electric field is changing with time, e.g., between capacitor plates
⚠ Students often write id = ε0 dE/dt, forgetting that it is the flux, not the field, that must be differentiated.
∮ B·dl = μ0 (ic + ε0 dΦE/dt)
Reads as: The line integral of B dot dl equals mu naught times the sum of conduction current and epsilon naught times the rate of change of electric flux
Variables: B = magnetic field (T), dl = infinitesimal length element (m), μ0 = permeability of free space (4π × 10^-7 T m/A), ic = conduction current (A), ε0 = permittivity of free space (8.85 × 10^-12 F/m), ΦE = electric flux (N m^2/C), t = time (s)
Use when: Using the generalised Ampere-Maxwell law to find magnetic field when both conduction and displacement currents are present
⚠ Students omit the displacement current term when solving for magnetic field inside a capacitor, leading to a zero field incorrectly.
c = 1/√(μ0ε0)
Reads as: c equals one over the square root of mu naught times epsilon naught
Variables: c = speed of light in vacuum (3 × 10^8 m/s), μ0 = permeability of free space (4π × 10^-7 T m/A), ε0 = permittivity of free space (8.85 × 10^-12 F/m)
Use when: Calculating the speed of electromagnetic waves in vacuum from fundamental constants
⚠ Students confuse μ0 and ε0 values or forget that this formula applies only in vacuum.
νλ = c
Reads as: nu times lambda equals c
Variables: ν = frequency (Hz), λ = wavelength (m), c = speed of light in vacuum (3 × 10^8 m/s)
Use when: Relating frequency and wavelength of any electromagnetic wave in vacuum
⚠ Students forget to convert units (e.g., MHz to Hz, cm to m) before substituting.
E0 = c B0
Reads as: E naught equals c times B naught
Variables: E0 = amplitude of electric field (V/m), c = speed of light (3 × 10^8 m/s), B0 = amplitude of magnetic field (T)
Use when: Finding the amplitude of the electric field from the magnetic field amplitude or vice versa in an electromagnetic wave
⚠ Students incorrectly use E = cB for instantaneous values instead of amplitudes, or forget that E and B are perpendicular.
v = 1/√(με)
Reads as: v equals one over the square root of mu times epsilon
Variables: v = speed of electromagnetic wave in a medium (m/s), μ = permeability of the medium (H/m), ε = permittivity of the medium (F/m)
Use when: Calculating the speed of light in a material medium given its electromagnetic properties
⚠ Students use μ0 and ε0 instead of the medium's μ and ε, or forget that v < c in a medium.
Ex = E0 sin(kz - ωt)
Reads as: E sub x equals E naught times sine of k z minus omega t
Variables: Ex = electric field component along x-axis (V/m), E0 = amplitude (V/m), k = wave number (rad/m), z = position along propagation direction (m), ω = angular frequency (rad/s), t = time (s)
Use when: Writing the expression for the electric field of a plane electromagnetic wave propagating along the z-direction
⚠ Students confuse the sign in the argument: for a wave travelling in +z direction, it is (kz - ωt); for -z direction, it is (kz + ωt).
By = B0 sin(kz - ωt)
Reads as: B sub y equals B naught times sine of k z minus omega t
Variables: By = magnetic field component along y-axis (T), B0 = amplitude (T), k = wave number (rad/m), z = position along propagation direction (m), ω = angular frequency (rad/s), t = time (s)
Use when: Writing the expression for the magnetic field of a plane electromagnetic wave propagating along the z-direction
⚠ Students forget that E and B are perpendicular and in phase, so both have the same sine argument.
k = 2π/λ
Reads as: k equals two pi over lambda
Variables: k = wave number (rad/m), λ = wavelength (m)
Use when: Finding the wave number from the wavelength
⚠ Students confuse k with the propagation vector or forget that λ must be in metres.
ω = 2πν
Reads as: omega equals two pi nu
Variables: ω = angular frequency (rad/s), ν = frequency (Hz)
Use when: Converting frequency to angular frequency
⚠ Students forget that ν is in Hz, not kHz or MHz, and must convert.
ω = ck
Reads as: omega equals c times k
Variables: ω = angular frequency (rad/s), c = speed of light (3 × 10^8 m/s), k = wave number (rad/m)
Use when: Relating angular frequency and wave number for electromagnetic waves in vacuum
⚠ Students apply this relation in a medium without adjusting for the speed v.
E = hν
Reads as: E equals h times nu
Variables: E = energy of a photon (J), h = Planck's constant (6.63 × 10^-34 J s), ν = frequency (Hz)
Use when: Calculating the energy of a photon of electromagnetic radiation
⚠ Students forget to convert energy from joules to eV (1 eV = 1.6 × 10^-19 J) when required.
How It Works
Displacement current in a charging capacitor
  1. A time-varying current i(t) flows in the circuit, charging the capacitor.
  2. Between the plates, there is no conduction current (ic = 0), but the electric field E changes with time as charge accumulates.
  3. The electric flux ΦE through a surface between the plates changes at a rate dΦE/dt.
  4. This changing flux gives rise to a displacement current id = ε0 dΦE/dt, which is equal in magnitude to the conduction current i(t) in the wire.
  5. The displacement current acts as a source of magnetic field, so the magnetic field at a point between the plates is the same as just outside the plates.
💡 The displacement current is not a flow of charge; it is a term added to Ampere's law to maintain consistency. It exists wherever the electric field changes with time.
Generation of electromagnetic waves by an oscillating charge
  1. An accelerated charge (e.g., an electron oscillating with frequency ν) produces a time-varying electric field.
  2. This time-varying electric field, according to the Ampere-Maxwell law, produces a time-varying magnetic field.
  3. The time-varying magnetic field, according to Faraday's law, produces a time-varying electric field.
  4. The electric and magnetic fields regenerate each other, propagating as a self-sustaining electromagnetic wave.
  5. The frequency of the wave equals the frequency of oscillation of the charge.
  6. The energy of the wave comes from the energy of the accelerating charge.
💡 Stationary charges and charges in uniform motion do not produce electromagnetic waves; only accelerated charges do.
Propagation of a plane electromagnetic wave
  1. The wave propagates along a direction (say z-axis).
  2. The electric field E oscillates sinusoidally along a perpendicular direction (say x-axis): Ex = E0 sin(kz - ωt).
  3. The magnetic field B oscillates sinusoidally along a direction perpendicular to both E and propagation (say y-axis): By = B0 sin(kz - ωt).
  4. E and B are in phase (both have the same sine argument).
  5. The magnitudes are related by E0 = c B0.
  6. The wave travels at speed c = 1/√(μ0ε0) in vacuum.
💡 E, B, and the direction of propagation are mutually perpendicular, forming a right-handed system (E × B gives the direction of propagation).
Hertz's experimental demonstration of electromagnetic waves
  1. Hertz used an oscillating circuit (a spark gap) to produce electromagnetic waves of radio frequency.
  2. The waves were transmitted through space.
  3. A receiving circuit (another spark gap) detected the waves by showing a spark across its gap.
  4. Hertz measured the wavelength and speed of the waves, confirming they travel at the speed of light.
  5. He also demonstrated reflection, refraction, and polarisation of these waves, establishing their identity with light.
💡 Hertz's experiment was the first direct verification of Maxwell's prediction of electromagnetic waves.
Interaction of electromagnetic waves with matter (absorption and heating)
  1. Electromagnetic waves interact with matter through their electric and magnetic fields.
  2. These fields set charges (electrons, atoms, molecules) into oscillation.
  3. The oscillation increases the internal energy of the substance, raising its temperature.
  4. Different wavelengths interact differently: infrared waves vibrate entire atoms/molecules (heat waves), UV and X-rays affect electrons and can cause ionisation.
  5. The ozone layer absorbs most UV radiation from the sun, protecting life on Earth.
💡 Infrared waves are often called heat waves because they are readily absorbed by water molecules, increasing thermal motion.
Exam Traps — Don't Make These Mistakes
⚠ Confusing displacement current with actual current
✗ Displacement current is the flow of charges between the capacitor plates.
→ Displacement current is not a flow of charge; it is a term ε0 dΦE/dt that accounts for the changing electric field. There is no charge flow between the plates.
✓ Displacement current is the current due to a time-varying electric field, given by id = ε0 dΦE/dt. It is not a conduction current but acts as a source of magnetic field.
⚠ Incorrect direction of B relative to E and propagation
✗ In an electromagnetic wave, E and B are parallel to each other and perpendicular to the direction of propagation.
→ E and B are perpendicular to each other, and both are perpendicular to the direction of propagation. They are not parallel.
✓ In an electromagnetic wave, E, B, and the direction of propagation are mutually perpendicular. For a wave along the z-axis, E is along x and B is along y.
⚠ Forgetting that E and B are in phase
✗ In an electromagnetic wave, the electric field and magnetic field are 90 degrees out of phase.
→ From Maxwell's equations, for a plane electromagnetic wave in vacuum, E and B are in phase (both have the same sine argument). They reach their maxima and minima at the same time.
✓ In a plane electromagnetic wave, the electric and magnetic fields are in phase, meaning E = E0 sin(kz - ωt) and B = B0 sin(kz - ωt).
⚠ Misapplying Ampere's law without displacement current
✗ The magnetic field between the plates of a charging capacitor is zero because there is no current between the plates.
→ Although there is no conduction current between the plates, there is a displacement current due to the changing electric field. The Ampere-Maxwell law includes this term, so the magnetic field is non-zero.
✓ Between the plates of a charging capacitor, there is a displacement current id = ε0 dΦE/dt, which produces a magnetic field. Using the Ampere-Maxwell law, the magnetic field at a point between the plates is the same as just outside the plates.
⚠ Confusing speed of light in vacuum and in a medium
✗ The speed of light in glass is the same as in vacuum, i.e., 3 × 10^8 m/s.
→ The speed of light in a medium is v = 1/√(με), which is less than c because ε > ε0 and μ ≈ μ0 for most dielectrics.
✓ The speed of light in a medium is v = 1/√(με), which is less than the speed in vacuum c = 1/√(μ0ε0). For example, in glass, v ≈ 2 × 10^8 m/s.
⚠ Incorrectly writing the wave equation for propagation in the negative direction
✗ For a wave propagating along the negative z-direction, the electric field is Ex = E0 sin(kz - ωt).
→ For propagation in the negative z-direction, the argument should be (kz + ωt) because as t increases, the wave moves to smaller z.
✓ For a wave propagating along the negative z-direction, Ex = E0 sin(kz + ωt) and By = B0 sin(kz + ωt).
Compare & Contrast
Nature of current
Conduction current
Displacement current
Conduction current is due to the flow of charges in a conductor, while displacement current is due to a time-varying electric field and exists even in vacuum.
Speed of propagation
Electromagnetic wave in vacuum
Electromagnetic wave in a medium
In vacuum, the speed is c = 1/√(μ0ε0) = 3 × 10^8 m/s, while in a medium, the speed is v = 1/√(με), which is always less than c because ε > ε0 and μ ≥ μ0.
Exam Focus
Marks pattern: 1 MCQ (1 mark) on displacement current or properties of EM waves, 1 SA-I (2 marks) on E0 = cB0 or wave equation, 1 SA-II (3 marks) on Ampere-Maxwell law or numerical problem, 1 LA (5 marks) on electromagnetic spectrum or derivation of speed of light
Most asked: Displacement current and its necessity (Ampere-Maxwell law) · Properties of electromagnetic waves (E ⊥ B ⊥ direction of propagation, E0 = cB0) · Speed of electromagnetic waves in vacuum and medium · Electromagnetic spectrum and applications of different regions · Relation between frequency, wavelength, and speed (νλ = c)
Don't skip: Numerical problems involving E0 = cB0 and νλ = c; Derivation or explanation of displacement current and its role in Ampere-Maxwell law; Questions on the electromagnetic spectrum: naming regions, sources, and applications
One-Line Revision (read aloud before the exam)
  1. Displacement current is given by id = ε0 dΦE/dt and is necessary to maintain consistency in Ampere's law.
  2. The Ampere-Maxwell law is ∮ B·dl = μ0 (ic + ε0 dΦE/dt).
  3. Maxwell's equations consist of Gauss's law for electricity, Gauss's law for magnetism, Faraday's law, and the Ampere-Maxwell law.
  4. Electromagnetic waves are produced by accelerated charges.
  5. In an electromagnetic wave, E, B, and the direction of propagation are mutually perpendicular.
  6. The speed of electromagnetic waves in vacuum is c = 1/√(μ0ε0) = 3 × 10^8 m/s.
  7. The relation between electric and magnetic field amplitudes is E0 = c B0.
  8. The wave equation for a plane electromagnetic wave is Ex = E0 sin(kz - ωt) and By = B0 sin(kz - ωt).
  9. The frequency ν and wavelength λ of an electromagnetic wave are related by νλ = c.
  10. The electromagnetic spectrum includes gamma rays, X-rays, ultraviolet, visible, infrared, microwaves, and radio waves.
  11. Infrared waves are often called heat waves because they increase thermal motion in materials.
  12. The ozone layer absorbs most ultraviolet radiation from the sun.
  13. X-rays are used in medical imaging and gamma rays in cancer treatment.
  14. Microwaves are used in radar and microwave ovens, matching the resonant frequency of water molecules.
  15. The speed of light in a medium is v = 1/√(με), which is less than c.

MCQs — 10 Questions with Answers

Q1. A parallel plate capacitor is being charged by a current i. The electric flux through the capacitor plates is Φ_E. What is the displacement current between the plates?

  • A. i_d = ε₀(dΦ_E/dt) ✓
  • B. i_d = μ₀(dΦ_E/dt)
  • C. i_d = (1/ε₀)(dΦ_E/dt)
  • D. i_d = i (always equal to conduction current)

Answer: A — By definition, displacement current is i_d = ε₀(dΦ_E/dt); it equals conduction current only when the changing flux rate matches.

Q2. In a charging capacitor, when a circular loop is drawn around the connecting wire outside the capacitor and another identical loop is drawn between the plates with the same boundary, Ampere's law initially seems to give different values of B. Why?

  • A. Because conduction current is zero between the plates, but displacement current is not ✓
  • B. Because the magnetic field is not actually the same at both locations
  • C. Because Ampere's law is invalid for time-varying current
  • D. Because the capacitor plates create a magnetic field that cancels B outside

Answer: A — Ampere's law fails without displacement current because the wire loop encloses conduction current i, but the loop between plates encloses no conduction current—only the changing electric flux (displacement current ε₀dΦ_E/dt) fixes this.

Q3. A parallel plate capacitor with plate area A = 0.1 m² is charged such that the electric field between the plates increases at a rate dE/dt = 10⁵ V/(m·s). Find the displacement current. (ε₀ = 8.85 × 10⁻¹² F/m)

  • A. 0.0885 A
  • B. 0.00885 A ✓
  • C. 8.85 × 10⁻⁶ A
  • D. 8.85 × 10⁻³ A

Answer: B — i_d = ε₀A(dE/dt) = 8.85 × 10⁻¹² × 0.1 × 10⁵ = 8.85 × 10⁻⁹ × 10⁵ = 8.85 × 10⁻⁴ × 10 = 0.00885 A.

Q4. Which statement about electromagnetic waves is NOT correct?

  • A. EM waves require a medium to propagate ✓
  • B. The electric and magnetic fields are perpendicular to each other
  • C. All EM waves travel at speed c = 3 × 10⁸ m/s in vacuum
  • D. A changing magnetic field can create an electric field

Answer: A — EM waves do NOT require a medium; they propagate through vacuum at constant speed c, unlike sound waves which need a medium.

Q5. In an electromagnetic wave, the electric field amplitude is 100 V/m. What is the magnetic field amplitude? (c = 3 × 10⁸ m/s, μ₀ = 4π × 10⁻⁷ T·m/A)

  • A. 3.33 × 10⁻⁷ T ✓
  • B. 3.33 × 10⁻⁶ T
  • C. 3.33 × 10⁻⁸ T
  • D. 3.33 × 10⁻⁹ T

Answer: A — For EM waves, E/B = c, so B = E/c = 100/(3 × 10⁸) = 3.33 × 10⁻⁷ T.

Q6. Consider the following statements: (I) Displacement current is a real flow of charge. (II) Displacement current produces a magnetic field just like conduction current does. Which is/are correct?

  • A. Both (I) and (II) are correct
  • B. Only (I) is correct
  • C. Only (II) is correct ✓
  • D. Neither (I) nor (II) is correct

Answer: C — Displacement current is NOT a real flow of charge but a changing electric field; however, it DOES produce a magnetic field, just as conduction current does.

Q7. The electromagnetic spectrum includes (in order of increasing frequency): radio, microwave, infrared, visible, ultraviolet, X-ray, gamma ray. Which type of EM wave is used for sterilization in hospitals?

  • A. Microwave
  • B. Ultraviolet ✓
  • C. X-ray
  • D. Gamma ray

Answer: B — Ultraviolet (UV) radiation has sufficient energy to damage DNA in microorganisms, making it ideal for sterilization; gamma rays are also used but UV is the primary choice in hospitals.

Q8. A charged parallel plate capacitor is disconnected from a battery. The electric field between the plates remains constant. What is the displacement current at this instant?

  • A. Zero, because the field is constant ✓
  • B. Zero, because no current flows in the external circuit
  • C. Equal to the conduction current that was flowing before disconnection
  • D. Infinite, because the field is confined

Answer: A — Displacement current i_d = ε₀(dΦ_E/dt) depends on the rate of change of electric field; if E is constant, dE/dt = 0, so i_d = 0.

Q9. Derive the expression for displacement current and explain why Maxwell's modification to Ampere's law was necessary to ensure the consistency of electromagnetic theory. (This is a conceptual question requiring brief reasoning.)

  • A. Displacement current = ε₀(dΦ_E/dt); without it, different surfaces gave different B values for the same loop boundary ✓
  • B. Displacement current = μ₀(dΦ_E/dt); it was added to make Ampere's law dimensionally consistent
  • C. Displacement current = (dQ/dt); it replaces conduction current in capacitors
  • D. Displacement current = ε₀(d²Φ_E/dt²); it accounts for accelerating charges

Answer: A — When a capacitor charges, conduction current flows in the wire but no charge crosses between the plates; to maintain B consistency, the changing E-field must contribute as i_d = ε₀(dΦ_E/dt).

Q10. Two electromagnetic waves, one radio wave and one gamma ray, travel through vacuum. Which statement is correct?

  • A. The gamma ray travels faster than the radio wave
  • B. Both travel at the same speed c, but the gamma ray has much higher frequency and shorter wavelength ✓
  • C. The radio wave has higher energy because it has lower frequency
  • D. The gamma ray travels slower because it has higher frequency and loses energy faster

Answer: B — All EM waves travel at c in vacuum regardless of frequency; gamma rays have much shorter wavelengths and higher frequencies (higher energy) than radio waves.

Flashcards

What is displacement current, and why did Maxwell introduce it?

Displacement current = ε₀(dΦ_E/dt); Maxwell added it to Ampere's law because changing electric fields in a capacitor must produce magnetic fields for consistency.

State the equation for displacement current.

i_d = ε₀(dΦ_E/dt), where Φ_E is electric flux through a surface.

Why does Ampere's circuital law alone fail for a charging capacitor?

Different surfaces with the same boundary give different enclosed currents (wire surface vs. surface between plates), so magnetic field should be same but Ampere's law alone predicts zero between plates.

What is the relationship between changing E-field and B-field according to Maxwell?

A time-varying electric field generates a magnetic field, just as a current does; this is the essence of displacement current.

What was Hertz's 1885 contribution to electromagnetic waves?

Hertz experimentally demonstrated the existence of electromagnetic waves, proving Maxwell's prediction correct.

Name three types of electromagnetic waves and their wavelength ranges.

Radio waves (~10⁶ m), visible light (~500 nm), gamma rays (~10⁻¹² m); they span from longest to shortest wavelengths.

What is the speed of electromagnetic waves in vacuum?

c ≈ 3 × 10⁸ m/s, derived from Maxwell's equations using c = 1/√(ε₀μ₀).

How are E and B fields oriented in an electromagnetic wave?

E and B fields are perpendicular to each other and both perpendicular to the direction of wave propagation.

Why is the speed of light so important in Maxwell's theory?

The speed predicted by Maxwell's equations matched the measured speed of light, proving light is an electromagnetic wave and unifying optics with electromagnetism.

What is the key difference between conduction current and displacement current?

Conduction current flows due to movement of charges in conductors; displacement current arises from changing electric flux even without charge motion (e.g., in capacitor dielectric).

Important Board Questions

Define displacement current and state the equation that relates it to the rate of change of electric flux. [2 marks]

Displacement current = ε₀(dΦ_E/dt). Mention that it arises from changing E-field even without charge motion and produces magnetic field like conduction current.

A parallel plate capacitor with plate area 0.5 m² is connected to a time-varying voltage source. The electric field between the plates increases uniformly from 0 to 10⁴ V/m in 2 seconds. Calculate the displacement current. (ε₀ = 8.85 × 10⁻¹² F/m) [5 marks]

Find dE/dt = (10⁴ V/m)/(2 s) = 5 × 10³ V/(m·s). Then use i_d = ε₀A(dE/dt) = ε₀A × dE/dt. Show all steps: calculate rate of change, substitute into formula, compute numerical answer with units.

Explain why Maxwell's modification of Ampere's law (adding displacement current) was essential for the consistency of electromagnetic theory. Describe the problem that arose when applying the original Ampere's circuital law to a charging capacitor and how displacement current resolved it. [6 marks]

Original Ampere's law: ∮B·dl = μ₀i_conduction. Problem: a loop around the wire sees conduction current i, but a loop between the plates sees zero conduction current, yet B must be the same. Solution: displacement current i_d = ε₀(dΦ_E/dt) between plates makes both surfaces consistent. Explain this using Figures 8.1(a), (b), (c) from the text and state the modified law: ∮B·dl = μ₀(i_conduction + i_displacement).

Assertion-Reason Questions for Electromagnetic Waves

Q1. Assertion (A):

Displacement current is a real physical current that transports charge across the capacitor dielectric.

Reason (R):

Displacement current is defined as the rate of change of electric flux multiplied by ε₀, which represents the effect of changing electric fields, not actual charge transport.

Answer: D — A is false but R is true — The assertion is false. Displacement current is NOT real charge transport but rather a mathematical/physical construct representing how changing E fields produce magnetic fields. The reason correctly defines what displacement current actually is, but doesn't explain why the assertion is false.

Q2. Assertion (A):

Maxwell's modification to Ampere's law predicts that time-varying electric fields produce magnetic fields, analogous to how time-varying magnetic fields produce electric fields (Faraday's law).

Reason (R):

This symmetry between Faraday's law and the Ampere-Maxwell law is fundamental to understanding how electromagnetic waves are generated by coupled, changing E and B fields.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true, and the reason correctly explains WHY the assertion is important: it reveals the symmetry in Maxwell's equations that makes EM wave generation and propagation possible. This is the physical insight underlying the unified theory.

Q3. Assertion (A):

Between the plates of a charging capacitor, the magnetic field calculated using Ampere's original law (without displacement current) is zero.

Reason (R):

Because no conduction current flows through the region between capacitor plates, Ampere's law ∮B·dl = μ₀i gives B = 0 there.

Answer: C — A is true but R is false — The assertion is true—classical Ampere's law does give B = 0 between plates. However, the reason is incomplete/misleading. The actual reason is that the original law ONLY accounts for conduction current, but it's physically incorrect to conclude B = 0 because changing electric flux (displacement current) also produces B. The reason doesn't capture the misconception being resolved.

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Case-Based Questions (CBQ) for Electromagnetic Waves

Case Study:

An Indian telecommunications company is designing a new wireless charging system for smartphones using electromagnetic waves at 2.4 GHz frequency (same as WiFi). The system consists of a transmitter with a parallel plate capacitor that oscillates at this frequency, generating displacement currents that produce time-varying magnetic fields. Engineers need to ensure that the displacement current can efficiently couple energy to the receiving device. The capacitor has plate area A = 0.05 m² and during operation, the electric field between plates oscillates with amplitude E₀ = 10⁵ V/m and angular frequency ω = 2π × 2.4 × 10⁹ rad/s. The dielectric constant is κ = 1 (essentially air).

Q1. What is the maximum displacement current in this wireless charging capacitor? (ε₀ = 8.85 × 10⁻¹² F/m)

  • A. 67.3 mA
  • B. 6.73 A ✓
  • C. 0.673 A
  • D. 67.3 A

Answer: B

Q2. The oscillating electric field between the capacitor plates is E(t) = E₀ sin(ωt). The displacement current id will:

  • A. Be in phase with the electric field
  • B. Lead the electric field by 90° ✓
  • C. Lag the electric field by 90°
  • D. Be 180° out of phase with the electric field

Answer: B

Q3. Why is displacement current essential for wireless power transmission at 2.4 GHz frequency?

  • A. It produces time-varying magnetic fields that can radiate energy as EM waves ✓
  • B. It carries DC power across the capacitor plates
  • C. It reduces heat dissipation in the dielectric
  • D. It increases the capacitance of the device

Answer: A

Q4. If the engineers want to double the displacement current without changing the frequency, they could:

  • A. Increase the plate area A
  • B. Increase the frequency ω
  • C. Use a dielectric with higher κ
  • D. Both A and C are correct ✓

Answer: D

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Previous Year Questions (PYQ) for Electromagnetic Waves

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 55/1/1 · 1 mark

An electromagnetic wave passes from vacuum into a dielectric medium with relative electrical permittivity (3/2) and relative magnetic permeability (8/3). Then, its

  • wavelength is doubled and frequency remains unchanged.
  • wavelength is doubled and frequency is halved.
  • wavelength is halved and frequency remains unchanged.
  • wavelength and frequency both will remain unchanged.

CBSE 2026 · Set 55/1/1 · 1 mark

Electromagnetic waves used in a diagnostic tool in medicine have a wavelength range

  • 1 nm to 10⁻³ nm
  • 400 nm to 1 nm
  • 1 mm to 700 nm
  • 0.1 m to 1 mm

CBSE 2026 · Set 55/1/2 · 1 mark

Electromagnetic waves used for purification of water are

  • X-rays
  • Ultraviolet rays
  • Infrared rays
  • Ultrasonic rays

CBSE 2026 · Set 55/1/3 · 1 mark

Welders wear special glass goggles or face masks with glass windows to protect their eyes from

  • Infrared rays
  • Ultraviolet rays
  • X-rays
  • Microwaves

CBSE 2026 · Set 55/2/1 · 1 mark

An electromagnetic wave is propagating along x-axis. At any instant, the phase difference (in radian) between the electric field (E⃗) and the magnetic field (B⃗) associated with the wave is

  • zero
  • π/4
  • π/2
  • π

CBSE 2026 · Set 55/2/1 · 3 marks

Name the electromagnetic waves which are used for (i) detection of fractures in bones (ii) physiotherapy (iii) radar systems. Also write their wavelength range.

CBSE 2026 · Set 55/2/2 · 1 mark

The expression of magnetic fields associated with four electromagnetic waves are given below: I. B₁ = (4 × 10⁻⁶ T) sin [0.7 × 10³x + 1.4 × 10¹¹ t], II. B₂ = (2 × 10⁻⁷ T) sin [0.6 × 10³x + 1.5 × 10¹¹ t], III. B₃ = (3 × 10⁻⁵ T) sin [0.5 × 10³x + 1.5 × 10¹¹ t], IV. B₄ = (5 × 10⁻⁴ T) sin [0.2 × 10⁴x + 4.8 × 10¹¹ t]. Which wave is travelling in free space?

  • I
  • II
  • III
  • IV

CBSE 2026 · Set 55/2/2 · 3 marks

Name the electromagnetic waves which are used (i) as a diagnostic tool in medicine (ii) in remote switches for TV sets (iii) in water purifiers. Also write their wavelength range.

CBSE 2026 · Set 55/2/3 · 1 mark

The electric field and the magnetic field components of an electromagnetic wave travelling in vacuum are represented as E = m sin(αz − βt) and B = n sin(αz − βt) where m, n, α and β are constants characterising the wave. Then

  • m/n = c, β/α ≠ c
  • n/m = α/β = c
  • β/α = c, m/n ≠ c
  • m/n = β/α = c

CBSE 2026 · Set 55/2/3 · 3 marks

Name the electromagnetic waves which are (i) also known as 'heat waves', (ii) used in LASIK eye surgery, (iii) used to destroy cancer cells. Also write their wavelength range.

CBSE 2026 · Set 55(B)_Physics (Theory) · 3 marks

(a) A sinusoidal electromagnetic wave of frequency 40 MHz travels in free space in the x-direction. Calculate the wavelength and time period of the wave. (b) At some point and at some instant the electric field in the above electromagnetic wave has its maximum value of 750 N/C and is along the y-axis. Calculate the magnitude and direction of the magnetic field at this position and time.

CBSE 2026 · Set 55/3/1 · 1 mark

A plane electromagnetic wave travels through a medium and the magnetic field associated with it is given by B = 5 × 10⁻⁸ sin (3 × 10¹⁰ t − 150 x) T where x is in metres and t is in seconds. The velocity of the wave is :

  • 2·0 × 10⁸ ms⁻¹
  • 4·5 × 10⁷ ms⁻¹
  • 3·5 × 10⁷ ms⁻¹
  • 2·5 × 10⁸ ms⁻¹

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Class 12 Physics Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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Class 12 Physics Previous Year Questions, Chapter-wise

All 962 CBSE board questions from 2022–2026, sorted by chapter and question type. Free PDF download.

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Class 12 Physics Derivations

All 68 derivations, chapter-wise, step by step with diagrams and marked with the CBSE board years (2022–2026) each was asked in. Free PDF download.

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