Electromagnetic Waves

Lesson

Part 1 — a missing term in Ampere’s law, and how it predicts waves of electric and magnetic fields that travel at the speed of light.

In this lesson you will

Explain why Maxwell introduced the displacement current
Describe how changing electric and magnetic fields sustain each other as an electromagnetic wave
State the characteristics of EM waves: transverse nature, speed c = 1/√(μ₀ε₀), and E₀ = cB₀

Diagram

A puzzle in Ampere’s law

While a capacitor charges, current flows in the wires but not across the gap. Apply Ampere’s law to a loop round the wire:

A flat surface bounded by the loop is crossed by the wire: $\oint \vec B\cdot d\vec l = \mu_0 I$ .
A bag-shaped surface with the same boundary passes through the gap: no conduction current crosses it, so $\oint \vec B\cdot d\vec l = 0$ .
The same loop cannot have two different answers: something is missing.

Concept

Displacement current

Maxwell noticed that in the gap the electric field is changing. He added a displacement current $I_d = \varepsilon_0\dfrac{d\Phi_E}{dt}$ For the capacitor, $\Phi_E = EA = q/\varepsilon_0$ , so $I_d = dq/dt = I$ : the current is continuous through the gap.

Ampere–Maxwell law: $\oint \vec B\cdot d\vec l = \mu_0(I_c + I_d) = \mu_0 I_c + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$ .
A changing electric field produces a magnetic field, just as a changing magnetic field produces an electric field (Faraday).

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