Question

What is meant by 'displacement current' and write modified equation of Ampere's law.

Hint:

Remember the symmetry introduced by Maxwell: Faraday's law of induction states that a changing magnetic field produces an electric field. The concept of displacement current completes this symmetry by stating that a changing electric field produces a magnetic field. These two ideas are the foundation of electromagnetic waves.

Answer 1

Step 1: Understanding the Concept:
James Clerk Maxwell discovered an inconsistency in Ampere's Circuital Law, particularly when applied to situations with time-varying electric fields, such as the charging or discharging of a capacitor. To resolve this, he introduced the concept of displacement current.

Step 2: Explanation of Displacement Current:
Displacement current (\(I_D\)) is not a current in the traditional sense of flowing charges (which is conduction current, \(I_C\)). Instead, it is a theoretical current that is associated with a changing electric field.
Consider the gap between the plates of a capacitor being charged. While charges flow in the connecting wires (\(I_C\)), no charge actually crosses the gap. However, as charge accumulates on the plates, the electric field (\(E\)) between the plates changes with time. This changing electric field produces a magnetic field in the gap, just as a real current would.
Maxwell defined this displacement current as being proportional to the rate of change of electric flux (\(\Phi_E\)) through a surface.
The displacement current is given by the formula: \[ I_D = \epsilon_0 \frac{d\Phi_E}{dt} \] where \(\Phi_E = \int \vec{E} \cdot d\vec{A}\) is the electric flux and \(\epsilon_0\) is the permittivity of free space. The existence of displacement current implies that a changing electric field is a source of a magnetic field.

Step 3: Modified Ampere's Law (Ampere-Maxwell Law):
The original Ampere's law was \(\oint \vec{B} \cdot d\vec{l} = \mu_0 I_C\). Maxwell modified this by adding the displacement current term to the total current.
The modified law, known as the Ampere-Maxwell Law, states that the line integral of the magnetic field around a closed loop is proportional to the sum of the conduction current (\(I_C\)) and the displacement current (\(I_D\)) passing through the surface enclosed by the loop.
The equation is: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 (I_C + I_D) \] Substituting the expression for \(I_D\), we get: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_C + \epsilon_0 \frac{d\Phi_E}{dt} \right) \]