Displacement Current - Definition, Formula, FAQs

What is Displacement Current?

A displacement current is a current that occurs when an electric field changes in time, for example in a capacitor that is charging or discharging. It doesn’t actually involve any flow of charges at all, but it causes an electromagnetic field to change smoothly, like a current. This concept has an explanation for how electric and magnetic fields interact when there are no moving charges.

Formula of Displacement Current

The displacement current was introduced by James Clerk Maxwell to explain the continuity of current in a circuit, especially between the plates of a capacitor.

$
I_d=\varepsilon_0 \frac{d \Phi_E}{d t}
$

where

• $I_d=$ displacement current
• $\varepsilon_0=$ permittivity of free space
• $\Phi_E=$ electric flux
• $\frac{d \Phi_E}{d t}=$ rate of change of electric flux

For a Capacitor

$
I_d=\varepsilon_0 A \frac{d E}{d t}
$

Also read -

• NCERT Notes For All Subjects
• NCERT Solutions for All Subjects
• NCERT Exemplar Solutions for All Subjects

The Maxwell Ampere Law

The Maxwell-Ampere Law is an extension of Ampere's circuital law given by James Clerk Maxwell. It explains how a magnetic field is produced not only by an electric current but also by a changing electric field.

Statement
The magnetic field around a closed loop is produced by conduction current as well as displacement current passing through the surface bounded by the loop.

The equation of Ampere Maxwell is:

In the Ampère-Maxwell law, the magnetic field $\vec{B}$ around a closed loop is related to the current passing through that loop. The law is expressed as:

$
\oint \vec{B} \cdot d \vec{s}=\mu_0 I
$

where:

• $\oint \vec{B} \cdot d \vec{s}$ represents the line integral of the magnetic field $\vec{B}$ around a closed path,
• $\mu_0$ is the permeability of free space (a constant that measures how much the magnetic field can penetrate free space),
• $I$ is the electric current enclosed by the loop.

The Problem with Ampère’s Law in Certain Situations

Ampère’s Law works well for steady (unchanging) currents, but it has a limitation when applied to circuits with changing electric fields, such as those with a condenser (capacitor).

Example with a Capacitor (Condenser)
Consider a capacitor connected to a voltage source, with:

• One plate is connected to a positive charge $Q^{+}$,
• The other plate is connected to a negative charge $Q^{-}$

As the capacitor gets charged, an electric current will flow in the circuit, but no current actually flows between the plates of the capacitor because they are separated by an insulating gap. If we try to apply Ampère’s Law in this case to a loop between the capacitor plates, we run into a problem because we do not have any current there.