A parallel plate capacitor made of circular plates is being charged such that the surface charge density on its plates is increas…

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:

1. A. Zero at all places
2. B. Constant between the plates and zero outside the plates
3. C. Non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates (Correct answer)
4. D. Zero between the plates and non-zero outside

Correct Answer

Option C — Non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates

Detailed Solution & Explanation

The correct answer is Option C.

Key Points

• Surface charge density $\sigma=\dfrac{q}{A}$ with $\dfrac{d\sigma}{dt}=$ constant, so the displacement current $i_d=A\dfrac{d\sigma}{dt}$ is constant.
• The system behaves like a uniform cylindrical current distribution of radius $R$ (the plate radius).
• Inside ($r<R$): $B\propto r$ (increases with $r$); outside ($r>R$): $B\propto\dfrac{1}{r}$ (decreases with $r$).
• Hence $B$ is non-zero everywhere and is maximum at $r=R$, the cylindrical surface joining the plate peripheries.