A # displacement current

Displacement current is an apparent current postulated by James Clerk Maxwell to get rid of the unsavory discontinuity of the magnetic field around a capacitor (due to the halting of electrical current through the capacitor) in an oscillatory loop. Maxwell's confidence in the concept led him to his predicting electromagnetic waves and his positing that light was such a wave. The magnetic field surrounding the time varying electric field dD/dt was subsequently discovered. Modern understanding of the interrelationship of varying electrical and magnetic fields has eliminated the need for displacement current but this concept played a crucial historical role in the prediction and development of artificial production of electromagnetic radiation and all its applications.

## Theory

Consider a simple circuit containing a parallel-plate capacitor, a battery, and a switch. Immediately after the switch is closed a current flows through the conducting section of the circuit, resulting in an accumulation of positive charge on one plate of the capacitor and an equal accumulation of negative charge on the other plate. This is the usual conduction current with which we are familiar. Maxwell showed that it was necessary to assume that a current of the same value also flowed in the dielectric gap.

For a given charge density σ on the capacitor plates, the electric field in the dielectric gap is given by:

E = σ/ε

where ε is the permittivity of the dielectric.

From this

σ = εE = D

where D is the electric displacement.

During the charging of the capacitor, the charge density on the plates is changing at a rate /dt, and the value of the displacement in the dielectric gap changes accordingly:

/dt = dD/dt      (1)

or

/dt = ε0 dE/dt + dP/dt      (2)

The right-hand terms in equations (1) and (2) are descriptive of the dielectric gap and have (because the left-hand term has) the dimensions of current per unit cross-sectional area. (N.b. the electric current through any cross-sectional area is the net charge that is transferred through it per second.) Let us examine, in equation (2), the two components of the current:

dP/dt is the rate of change of polarization. This is associated with the actual motion of charge in the dielectric, corresponding as it does to the rotation of permanent dipoles, or the formation of induced ones. In short, it arises from a "displacement" of charge in the dielectric and possess a current character that is easily recognized.

ε0 dE/dt represents the current that is associated with a change in the electric field strength, even when there is a vacuum between the capacitor plates. It does not correspond to a movement of charge and is therefore difficult to interpret physically. However, regardless of the difficulty of interpretation, something occurring in the dielectric space is equivalent to a current of this magnitude. At least, there are magnetic effects associated with ε0 dE/dt that are identical with those which would arise from the passage of actual current of this amount.

The sum of these two current components, namely dD/dt (equation (1)), was called by Maxwell the displacement current, the name obviously stemming from the physical effect associated with the dP/dt component. It was in this context that D, the electric displacement, was introduced. The total current in the gap, A dD/dt (where A is the area of the capacitor plates), is thus equal to A dσ/dt, or dq/dt, the current flowing elsewhere in the circuit.

This recognition that the electric current is constant in all parts of a circuit, even in the space between the plates of a capacitor, was one of Maxwell's major contributions to physics. Without it, he could not have developed his electromagnetic theory of light.