Electromagnetic Waves

1. Introduction

Electromagnetic waves are oscillating electric and magnetic fields that propagate through space. Maxwell's theoretical prediction and Hertz's experimental confirmation revolutionized physics.

2. Maxwell's Equations

Maxwell unified electricity and magnetism through four equations:

1. Gauss's Law for Electricity: ∮ E · dA = Q/ε₀
2. Gauss's Law for Magnetism: ∮ B · dA = 0
3. Faraday's Law: ∮ E · dl = -dΦ_B/dt
4. Ampere-Maxwell Law: ∮ B · dl = μ₀(I + ε₀ dΦ_E/dt)

2.1 Displacement Current

The term I_d = ε₀ dΦ_E/dt is called the displacement current. It exists wherever the electric field changes with time, even in a vacuum.

'Ampere's law without the displacement current term was inconsistent with charging capacitors. Maxwell added this term to make it complete.'

3. Nature of EM Waves

Electromagnetic waves are transverse waves with E and B perpendicular to each other and to the direction of propagation.

c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s.

3.1 Energy Density

u_E = (1/2)ε₀E², u_B = (1/2)B²/μ₀. For EM waves, u_E = u_B.

3.2 Intensity

I = (1/2)ε₀cE₀² = cB₀²/2μ₀

3.3 Poynting Vector

S = (1/μ₀) (E × B) — gives the energy flux per unit area.

4. Electromagnetic Spectrum

5. Energy and Momentum of EM Waves

5.1 Energy Density

The total energy density of an electromagnetic wave is the sum of electric and magnetic contributions:

u_total = u_E + u_B = (1/2)ε₀E² + (1/2)B²/μ₀

For EM waves, the electric and magnetic contributions are equal: u_E = u_B = (1/2)u_total.

5.2 Intensity (Poynting Vector Magnitude)

I = u_total × c = ε₀E_rms² × c = (1/2)ε₀E₀²c

For sinusoidal waves: I = (1/2)cε₀E₀² = cB₀²/2μ₀

5.3 Radiation Pressure

When EM waves are absorbed by a surface: P = I/c. When EM waves are reflected: P = 2I/c (twice the momentum transfer).

This is the principle behind solar sails proposed for spacecraft propulsion.

5.4 Momentum of a Photon

Each photon carries momentum p = E/c = hf/c = h/λ.

'Even though photons have no mass, they carry momentum. This is purely a relativistic effect — the momentum of a massless particle is given by p = E/c.'

6. Electromagnetic Spectrum Details

1. Do not require a medium for propagation.
2. Travel at speed c = 3×10⁸ m/s in vacuum.
3. Exhibit reflection, refraction, interference, diffraction, and polarization.
4. Carry energy and momentum.

6. Worked Problems

Problem 1: The electric field of an EM wave is E = 100 sin(ωt - kx) V/m. Find the amplitude of the magnetic field. Solution: B₀ = E₀/c = 100/(3×10⁸) = 3.33×10^{-7} T.

Problem 2: An EM wave has frequency 10⁹ Hz. Find its wavelength in vacuum. Solution: λ = c/f = 3×10⁸/10⁹ = 0.3 m.

Problem 3: A capacitor with plates of area 0.1 m² is charged at dE/dt = 10¹² V/ms. Find displacement current. Solution: I_d = ε₀ dΦ_E/dt = ε₀ A dE/dt = 8.85×10^{-12} × 0.1 × 10¹² = 0.885 A.

7. Common Mistakes

'Students often think EM waves need a medium. They do not — they propagate through vacuum using self-sustaining oscillations of E and B fields.'

'Remember: higher frequency means shorter wavelength and higher energy per photon (E = hf).'

8. ISC Exam Focus

9. Self-Test Questions

1. What is displacement current? Show its necessity in Maxwell's equations.
2. State any four properties of electromagnetic waves.
3. The magnetic field of an EM wave is 2×10^{-8} T. Find the amplitude of the electric field.
4. Arrange the following in increasing order of wavelength: X-rays, infrared, microwaves, visible light, gamma rays.
5. Write the four Maxwell's equations in integral form and explain their physical significance.