Question

What is meant by displacement current? A capacitor is being charged by a battery. Show that Ampere-Maxwell law justifies continuity and constancy of the current flowing in the circuit.

Answer 1

Displacement current:
The displacement current was introduced by James Clerk Maxwell to explain the continuity of current in circuits involving capacitors. It is given by the rate of change of electric flux through a given surface. This term is necessary to maintain the symmetry of Ampere's law for circuits with capacitors, where no physical current flows through the capacitor plates but the current still appears in the circuit. Mathematically, the displacement current \( I_D \) is given by: \[ I_D = \epsilon_0 \frac{d\Phi_E}{dt} \] where \( \epsilon_0 \) is the permittivity of free space, and \( \Phi_E \) is the electric flux, which is the product of the electric field \( E \) and the area \( A \) through which it passes. For a capacitor being charged by a battery, the current flowing into the capacitor is the same as the displacement current between the plates of the capacitor. This current does not flow through the dielectric but is a result of the changing electric field. Ampere-Maxwell law:
The Ampere-Maxwell law states: \[ \oint_{\partial S} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{free}} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} \] where \( I_{\text{free}} \) is the conduction current, and the second term represents the displacement current \( I_D \). This law ensures the continuity of current because the displacement current term accounts for the changing electric field in the capacitor, which gives rise to a "current" flowing through the dielectric, ensuring that the current remains continuous. Since the total current \( I \) remains constant in the circuit, we have: \[ I = I_{\text{battery}} = I_{\text{capacitor}} = I_{\text{displacement}} \] Thus, the continuity and constancy of the current are justified by the Ampere-Maxwell law, even in the presence of a capacitor.