Question #43087

This problem tests the rigorous application of the Ampere-Maxwell law alongside the local continuity equation for a non-uniform system.

A common lethal trap is misapplying Gauss's law by assuming $E = Q_{enc}/(\epsilon_0 \pi r^2)$, which incorrectly treats the electric field as uniform over the disk, or miscalculating the divergence of the surface current by omitting the $1/r$ cylindrical geometric factor (assuming $\nabla \cdot \vec{K} = \partial K_r/\partial r$).

Both errors lead to an incorrect displacement current density of $2\beta r t$ instead of the true $3\beta r t$, resulting in the trap option $\frac{2}{3} \mu_0 \beta t r^2$.

Students must recognize that for a parallel-plate capacitor ($d \ll R$), the electric field is determined strictly by the local surface charge density ($E = \sigma/\epsilon_0$), forcing them to correctly derive $\sigma(r,t)$ via the full 2D divergence operator before closing the Amperean loop.