GRE GR 1777, Problem 005 Solution

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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.

Therefore, the answer is (E).

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Another (and better) solution)

The definition of electric displacement current is

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

If you know this definition, the problem would be solved more easily.

Reference: https://en.wikipedia.org/wiki/Displacement_current

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