Question

A parallel plate capacitor made of circular plates is being charged such that the surface charge density on its plates is increasing at a constant rate with time. The magnetic field arising due to displacement current is:

• A. constant between the plates and zero outside the plates
• B. non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates
• C. zero between the plates and non-zero outside
• D. zero at all places

Hint:

Displacement current acts as a source of magnetic field just like conduction current. Apply Ampère-Maxwell's law considering the displacement current between the capacitor plates. The magnetic field depends on the distance from the axis and is continuous at the edges of the plates.

Answer 1

The Correct Option is B

A changing electric field within a parallel plate capacitor generates a magnetic field. A constant rate of increase in surface charge density produces a displacement current, \( I_d \), across the capacitor gap. Maxwell's equations state that a time-varying electric field induces a magnetic field.

Applying the Ampère-Maxwell law: \[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( I + I_d \right)\] In the region between the plates, real current \(I = 0\), leaving only the displacement current, \( I_d \), defined as: \[ I_d = \epsilon_0 \frac{d\Phi_E}{dt} \] where \(\Phi_E\) is the electric flux. Given that the surface charge density (\(\sigma\)) is increasing: \[ I_d = \epsilon_0 \frac{d(\sigma A)}{dt} = \epsilon_0 A \frac{d\sigma}{dt} \] Due to the problem's symmetry, a circular path along the imaginary cylindrical surface connecting the plate peripheries, where the field is maximal, is chosen. Consequently, the magnetic field strength diminishes radially from this surface within the plates and remains non-zero outside them due to the persistent changing electric field.

Therefore, the magnetic field is non-zero everywhere, reaching its maximum at the imaginary cylindrical surface that links the peripheries of the plates. The correct choice is: non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates.