Maxwell's Equations

Maxwell's Equations

1. The Four Equations (integral and differential forms — memorize both)

For fields $E, B, D=εE, H=B/μ$, free charge $ρ$, current $J$:

Name	Integral	Differential	Meaning
Gauss (electric)

|

→

| Electric flux = charge ($E$ originates on charges) |
| Gauss (magnetic) |

|

| No magnetic monopole ($B$ loops close) |
| Faraday |

|

| Changing $B$ induces $E$ (emf) |
| Ampère-Maxwell |

|

→

| Changing $E$ (displacement current $∂D/∂t$) plus current produces $B$ |

Vacuum: $D=ε0E, B=μ0H$. The displacement current $I_d = ε0 dΦ_E/dt$ is what Maxwell added to fix continuity.

2. Why Four Are Complete

They determine how $E,B$ are produced by charges/currents and how they inter-couple to propagate as waves. The two Gauss are constraints (initial conditions), the two curls are dynamics.

3. Wave Equation Consequence (next topic)

In free space $ρ=0,J=0$, taking curl of curl gives:

→ wave speed

m/s — Maxwell predicted EM waves, light is EM.

4. Worked Use

Need to find induced E due to changing B: e.g., B= B0 sin ωt \hat{z} inside circle radius R ⇒ ∮ E·dl = E·2πr = -π r² dB/dt = -π r² B0 ω cos ωt ⇒ E = -(B0 ω r/2) cos ωt \hat{φ} ✓

5. Exam Points

• State four Maxwell's equations in both forms — 7-10 marks, almost certain (most frequent EM question).
• Explain each physical meaning (one line) — 5 marks, very frequent.
• Show $c=1/√(μ0ε0)$ from them — 5 marks, very frequent (asked as derivation).
• Mention displacement current $∂D/∂t$ — 3-5 marks, very frequent.