Electromagnetic Waves

Lesson

Part 1 — a missing current in a capacitor gap leads to waves of electric and magnetic field that travel at the speed of light.

In this lesson you will

Explain why displacement current is needed for the continuity of current in a charging capacitor, and use i_d = ε₀ dΦE/dt
Sketch an electromagnetic wave showing E, B and the direction of travel
Use c = 1/√(μ₀ε₀), E₀ = cB₀ and c = fλ, and state the properties common to all EM waves

Concept

Ampère’s law meets a charging capacitor

Ampère’s circuital law, $\oint \vec B \cdot d\vec l = \mu_0 i$ , uses the current through any surface bounded by the loop. Take a loop round the lead of a capacitor that is being charged:

A flat disc spanning the loop is crossed by the wire: enclosed current $i$ .
A bag-shaped surface on the same loop, bulging through the gap, is crossed by no charge: enclosed current zero.
Same loop, two answers: the law as it stood is incomplete.

Diagram

Displacement current

Maxwell noticed that in the gap the electric flux is changing. He added a displacement current $i_d = \varepsilon_0\,\dfrac{d\Phi_E}{dt}$ For parallel plates, $\Phi_E = EA = \dfrac{q}{\varepsilon_0}$ , so $i_d = \dfrac{dq}{dt} = i$ : the current is continuous through the gap.

Ampère–Maxwell law: $\oint \vec B \cdot d\vec l = \mu_0(i + i_d)$ .
Basic idea only in ISC: use $i_d = \varepsilon_0\,d\Phi_E/dt$ as a formula.

Cookies