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 parallel plate capacitor can generate a magnetic field due to a changing electric field. When the surface charge density increases at a constant rate, a displacement current, \( I_d \), is produced across the gap between the plates. According to Maxwell's equations, a changing electric field induces a magnetic field.

We'll apply the Ampère-Maxwell law: \[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( I + I_d \right)\] For the region between the plates, there is no real current (\(I = 0\)), but there is a displacement current, \( I_d \) given by: \[ I_d = \epsilon_0 \frac{d\Phi_E}{dt} \] where \(\Phi_E\) is the electric flux. Since the surface charge density (\(\sigma\)) is increasing, we have: \[ I_d = \epsilon_0 \frac{d(\sigma A)}{dt} = \epsilon_0 A \frac{d\sigma}{dt} \] The symmetry of the problem suggests using a circular path along the imaginary cylindrical surface joining the peripheries of the plates, where the field is maximal. Thus, the magnetic field strength decreases radially from this surface inside the plates and remains non-zero outside the plates as the changing electric field persists.

Thus, the magnetic field is non-zero everywhere, specifically maximal at the imaginary cylindrical surface connecting the peripheries of the plates. Therefore, the correct option is: non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates.