# GRE GR 1777, Problem 005 Solution

Physics GRE GR 1777

5. Electrodynamics (Maxwell’s Equations)

Solution) In Maxwell’s equations, Ampère’s law (with Maxwell’s correction) is

$\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$.

We can write this equation as

$\nabla \times \vec{B} - \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} = \mu_0 \vec{J}$,

where $\vec{J}$ is the current (the electric displacement current). And $\frac{\partial \vec{E}}{\partial t}$ is change of electric fields with respect to the time (rate of change of the electric flux).

If we assume that $\nabla \times \vec{B} = 0$, and the current flows through a surface S, then we can clearly see that the electric displacement current through S is proportional to the rate of change of the electric flux through S.

$\vec{J}_{D} = \epsilon_0 \frac{\partial \vec{E}}{\partial t} + \frac{\partial \vec{P}}{\partial t}$.