Electromagnetic waves

Electromagnetic waves

1. From Maxwell to Waves (free space, $ρ=0,J=0$)

Maxwell's curl equations:

Taking $∇×(∇×E)$ with vector identity $∇×(∇×E)=∇(∇·E)-∇²E$ and $∇·E=0$ gives:

Same for $B$: $∇²B = μ0ε0 ∂²B/∂t²$ — wave equations with speed:

m/s

So light is EM wave, predicted before Hertz's experiment.

2. Plane Wave Solution

For wave propagating along $+z$ with $E$ along $x$:

with

Properties:

• Transverse: $E⊥B⊥k$ (propagation), $E×B$ along $k$ (right-hand).
• In phase: $E$ and $B$ peak together (free space, in phase).
• Ratio:

• Impedance of free space:

Ω.

3. Energy and Momentum (preview Poynting)

Energy stored $u =1/2(ε0E² + B²/μ0)$; for plane wave $ε0E² = B²/μ0$ ⇒

instantaneous. Time-averaged

.

4. Worked Checks

(a) Frequency f=1 GHz: λ=c/f=0.3 m, k=2π/0.3≈20.9 m⁻¹ ✓

(b) E0=10 V/m ⇒ B0=33 nT, H0=E0/Z0≈26.5 mA/m ✓

(c) Show E×B direction is propagation: \hat{x}×\hat{y}=\hat{z} ✓

5. Exam Points

• Derive wave equation from Maxwell and $c=1/√(μ0ε0)$ — 7 marks, very frequent.
• Write plane wave $E=E0 cos(kz-ωt)$, state $E⊥B⊥k$, $E0/B0=c$, $Z0=377 Ω$ — 5 marks, very frequent.
• Sketch $E,B$ vs $z$ at $t=0$ — 3 marks.
• Relation $k=2π/λ, ω=2πf$ — 3 marks.