Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Ampère's law needs a second source term when currents are unsteady: the displacement current, the rate of change of the electric flux, which carries the current across a capacitor gap, produces a magnetic field like any current and competes with the conduction current in every medium.

In this section you will learn

what the displacement current $\partial D / \partial t$ is, and why Ampère's law is inconsistent without it as soon as a current is unsteady,
how the displacement current through a capacitor gap is the capacitor law $i = C d v / d t$ in disguise, and what magnetic field it produces in the gap, and
how to compare the conduction and the displacement current in a medium with the loss tangent $σ / (ω ε)$ and the crossover frequency, and when the displacement current may be dropped against the conduction current.

The second Maxwell equation, written with the new term on the right-hand side, reads

\nabla \times H = J + \frac{\partial D}{\partial t} . (5.2.1)

The two terms on the right stand side by side and have the same unit, $A/ m^{2}$ , so the equation says that $\partial D / \partial t$ is also a kind of current density: it produces a circulating magnetic field exactly as the conduction current density $J$ does. It is called the displacement current density, after the old name "displacement" for $D$ .

Definition · Displacement current

The displacement current density is $J_{d} = \partial D / \partial t [A/ m^{2}]$ , and the displacement current through a surface $S$ is

i_{d} = \int_{S} \frac{\partial D}{\partial t} \cdot d S = \frac{d Ψ}{d t} [A], (5.2.2)

the rate of change of the electric flux $Ψ = \int_{S} D \cdot d S$ through $S$ .

The everyday example of such a current is the current that "flows through" a capacitor in an alternating-current circuit (Figure 5.2.1). No charge ever crosses the gap between the plates, yet an ammeter anywhere in the loop reads the same current $i (t)$ , and the circuit behaves as if the current went straight through. What goes through the gap is the changing electric flux.

A capacitor in an alternating-current circuit. The conduction current i(t) stops at the plates, but the field between them changes in step with it, and the displacement current \int(\partial\vec D/\partial t)\cdot d\vec S through the gap equals i(t) at every instant (Section 5.2.2). — **Figure 5.2.1.** A capacitor in an alternating-current circuit. The conduction current $i (t)$ stops at the plates, but the field between them changes in step with it, and the displacement current $\int (\partial D / \partial t) \cdot d S$ through the gap equals $i (t)$ at every instant (Section 5.2.2).

5.2 The Displacement Current

Contents of Section 5.2

Home exercises · 1

Further exercises · 3