The magnetic field inside a circular parallel plate capacitor of...

According to the Ampere-Maxwell law, the magnetic field $B$ inside the capacitor at a radial distance $r$ is given by $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{d,\text{enclosed}}$.

Since the enclosed displacement current $I_{d,\text{enclosed}} = J_d (\pi r^2)$, we have $B(2\pi r) = \mu_0 J_d \pi r^2 \Rightarrow B = \dfrac{\mu_0 J_d r}{2}$.

We are given that $B$ is constant in time at any fixed point $r$. This implies that the displacement current density $J_d$ must be constant in time.

The displacement current density is related to the electric field by $J_d = \epsilon_0 \dfrac{dE}{dt}$.

For $J_d$ to be a non-zero constant, the rate of change of the electric field $\dfrac{dE}{dt}$ must be constant. Therefore, the electric field $E$ must vary linearly with time.

Answer: $E$ varies linearly with time