# Physics equations/24-Electromagnetic Waves

**Q:displacementCurrent**: The fourth problem is tedious:

## Problem[edit]

A circlular capactitor of radius 3.3 m has a gap of 12 mm, and a charge of 93 μC. The capacitor is discharged through a 9 kΩ resistor. What is what is the maximum magnetic field at the edge of the capacitor?

Though this is a tedious problem, it is significant because it can be solved with basic equations, and because it shows *why* Maxwell inserted the displacement term in the integral form of Maxwell's equations for magnetic field:

Define:

- to be the enclosed charge
- to be the enclosed charge.
- as the vacuum permittivity.
- as the magnetic permeability.

To under stand Maxwell's term, note that if the surface is taken *inside* the capacitor, it is enclosing electric field. But from Gauss's Law, we know that the electric field is directly proportional to charge. We need a term proportional to current, which is the time-derivative of charge. To recover Maxwell's term, we

## Deriving Maxwell's term[edit]

We start by assuming that the current could either be an actual current or a term corresponding to what the current would be if there were no capacitor. We denote this current with a tilde, and is called the displacement current.

(1)

The electric field, , is related to by Gauss' law

(2)

For a parallel plate capacitor:

(3)

Taking the time derivative,

(3)

Substituting into (1):

(4)

For our geometry, the electric field is constant over the surface of the capacitor. Hence we can convert to a surface integral:

(5)

Substituting yields our result:

(6)