# 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 *dσ*/*dt*, and the value of the displacement
in the dielectric gap changes accordingly:

*dσ*/*dt* = *dD*/*dt*
(1)

or

*dσ*/*dt* = *ε*_{0} *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.

*ε*_{0} *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 *ε*_{0} *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.