Electromagnetic Waves Class 12 Notes

• Conduction current flows through a conductor because of moving charges; displacement current exists wherever the electric flux changes in time.
• Inside a charging capacitor the conduction current is zero but the displacement current keeps the magnetic field continuous.
• Maxwell's four equations describe how charges and fields produce the electric and magnetic fields (qualitative for boards; the displacement-current term is the exam focus).

• They are transverse: the electric field, magnetic field and direction of propagation are mutually perpendicular.
• The oscillating E and B fields are in phase and their amplitudes are related by
• $E_0=c{B}_0$
• .
• They travel at speed
• $c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}\approx 3\times 10^8\text{m/s}$
• in vacuum.
• They do not need a medium and carry energy as well as momentum.

• Intensity:
• $I=\frac{1}{2}c{\epsilon }_0{E}_0^2$
• , the average energy crossing unit area per second.
• Momentum carried:
• $p=\frac{U}{c}$
• , which produces radiation (light) pressure on a surface.
• Radiation pressure on a perfect absorber is
• $\frac{U}{c}/A$
• and twice that on a perfect reflector.

• In order of decreasing wavelength (increasing frequency and energy): radio, microwave, infrared, visible, ultraviolet, X-rays and gamma rays.
• Visible light spans about 400–700 nm, from violet to red.
• Radio waves: communication; microwaves: radar and oven; infrared: night vision and remote sensing.
• UV: sterilisation and vitamin-D synthesis; X-rays: medical imaging; gamma rays: nuclear processes and radiotherapy.

• AM/FM radio and television broadcast in the radio band; mobile phones use microwaves.
• Infrared lamps and remote controls; thermal imaging in night vision.
• X-rays for bone and dental imaging; gamma rays in cancer treatment and the sterilisation of food and medical equipment.
• Microwave ovens heat food by dielectric heating of water molecules (microwave frequency ≈ 2.45 GHz).

Example: A plane EM wave has an electric field amplitude $E_0=3\times 10^3\text{V/m}$. Find the amplitude of the magnetic field.

Solution: $B_0=\frac{E_0}{c}=\frac{3\times 10^3}{3\times 10^8}=1\times 10^{-5}\text{T}$. The fields are perpendicular to each other and to the direction of travel.

Exercises (NCERT Q8.1–Q8.10)

• 8.1) Figure 8.5 shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor is being charged by an external source. The charging current is constant and equal to 0.15 A. (a) Calculate the capacitance and the rate of change of potential difference between the plates. (b) Obtain the displacement current across the plates. (c) Is Kirchhoff's first rule (junction rule) valid at each plate of the capacitor? Explain.
• 8.2) A parallel plate capacitor made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with an (angular) frequency of 300 rad s⁻¹. (a) What is the rms value of the conduction current? (b) Is the conduction current equal to the displacement current? (c) Determine the amplitude of B at a point 3.0 cm from the axis between the plates.
• 8.3) What physical quantity is the same for X-rays of wavelength 10⁻¹⁰ m, red light of wavelength 6800 Å and radiowaves of wavelength 500 m?
• 8.4) A plane electromagnetic wave travels in vacuum along z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is 30 MHz, what is its wavelength?
• 8.5) A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding wavelength band?
• 8.6) A charged particle oscillates about its mean equilibrium position with a frequency of 10⁹ Hz. What is the frequency of the electromagnetic waves produced by the oscillator?
• 8.7) The amplitude of the magnetic field part of a harmonic electromagnetic wave in vacuum is B₀ = 510 nT. What is the amplitude of the electric field part of the wave?
• 8.8) Suppose that the electric field amplitude of an electromagnetic wave is E₀ = 120 N/C and that its frequency is ν = 50.0 MHz. (a) Determine B₀, ω, k, and λ. (b) Find expressions for E and B.
• 8.9) The terminology of different parts of the electromagnetic spectrum is given in the text. Use the formula E = hν (for energy of a quantum of radiation: photon) and obtain the photon energy in units of eV for different parts of the electromagnetic spectrum. In what way are the different scales of photon energies that you obtain related to the sources of electromagnetic radiation?
• 8.10) In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × 10¹⁰ Hz and amplitude 48 V m⁻¹. (a) What is the wavelength of the wave? (b) What is the amplitude of the oscillating magnetic field? (c) Show that the average energy density of the E field equals the average energy density of the B field. [c = 3 × 10⁸ m s⁻¹.]

Answers at a glance

1. 1
   Q8.1 → (a) C ≈ 80 pF; dV/dt ≈ 1.87 × 10¹⁰ V s⁻¹. (b) Displacement current = 0.15 A. (c) Valid only when the displacement current is included.
2. 2
   Q8.2 → (a) I_rms = 6.9 µA. (b) Yes, equal. (c) B_max ≈ 1.63 × 10⁻¹¹ T.
3. 3
   Q8.3 → Speed in vacuum, c = 3 × 10⁸ m s⁻¹, is the same for all three.
4. 4
   Q8.4 → E ⊥ B ⊥ ẑ (both transverse, mutually perpendicular, E × B along z); λ = 10 m.
5. 5
   Q8.5 → Wavelength band: 25 m to 40 m.
6. 6
   Q8.6 → Frequency of the EM waves = 10⁹ Hz.
7. 7
   Q8.7 → E₀ = 153 V m⁻¹ (≈ 1.53 × 10² V m⁻¹).
8. 8
   Q8.8 → (a) B₀ = 4.0 × 10⁻⁷ T, ω = 3.14 × 10⁸ rad s⁻¹, k = 1.05 rad m⁻¹, λ = 6.0 m. (b) E = 120 sin(1.05z – 3.14 × 10⁸t) x̂ V m⁻¹; B = 4.0 × 10⁻⁷ sin(1.05z – 3.14 × 10⁸t) ŷ T.
9. 9
   Q8.9 → Photon energies run from ≈ 10⁻⁹ eV (radio) to ≈ MeV (gamma); the energies are set by the quantum transitions of the sources — from accelerating charges up to nuclear re-arrangements.
10. 10
    Q8.10 → (a) λ = 1.5 × 10⁻² m. (b) B₀ = 1.6 × 10⁻⁷ T. (c) ū_E = ū_B = ¼ε₀E₀², because E₀ = cB₀ with c² = 1/(ε₀μ₀).

MCQ Questions

1 Mark Questions

1. 1
   Define displacement current and give its expression.
2. 2
   Who first predicted electromagnetic waves, and who experimentally produced them?
3. 3
   Write the relation between the amplitudes E₀ and B₀ of an electromagnetic wave.
4. 4
   Name one source of X-rays and one source of microwaves.

2 Mark Questions

1. 1
   For an electromagnetic wave, B₀ = 2 × 10⁻⁷ T. Find the corresponding amplitude of the electric field. (Ans: E₀ = cB₀ = 6 × 10⁻¹ V/m)
2. 2
   A radio station broadcasts at 150 kHz. Find the wavelength of the waves. (Ans: λ = c/f = 2000 m)
3. 3
   In an electromagnetic wave, E and B oscillate with frequencies — how are their phases related, and what is the ratio E₀/B₀? (Ans: in phase, ratio = c)

3 Mark Questions

1. 1
   Explain Maxwell's modification of Ampere's circuital law and justify the introduction of displacement current for a charging capacitor. (Ans: Id = ε₀ dφE/dt)
2. 2
   State four important properties of electromagnetic waves (transverse nature, speed c, E⊥B, carry energy/momentum).
3. 3
   Arrange the electromagnetic spectrum in order of increasing frequency and give one application of each band from radio to gamma rays.

5 Mark Questions

1. 1
   Show how Maxwell's correction of Ampere's circuital law led to the prediction of electromagnetic waves, and derive that their speed in vacuum is c = 1/√(μ₀ε₀). (Ans: combining Faraday's law with the displacement current gives the wave equation with speed wave = 1/√(μ₀ε₀))
2. 2
   Give the complete electromagnetic spectrum in order of increasing frequency, describing the source, method of production/detection and one application of each band, and explain why accelerated charged particles radiate electromagnetic waves.