10.1 Electrostatics with Conductors

As learned in the first unit, charge redistributes on the surface of a conductor. Because of Gauss’s law, constructing a gaussian surface inside the conductor leads to the conclusion that the electric field inside the conductor is 0

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As shown above, any gaussian surface we construct in the interior of the conductor has no electric flux since ${q_{enc} = 0}$. Since the voltage ${\Delta V = \int \vec{E} \cdot d\vec{r}}$, the fact that the electric field is 0 means that the conductor is held at a constant electric potential throughout the conductor. Any excess charge on the conductor must reside on its surface; if excess charge were in the interior, it would create an internal field and the charges would move until they reached the surface and the field inside dropped to zero again.

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In the figure above we have a conductor in electrostatic equilibrium in a uniform electric field ${\vec{E}_0}$. This leads to polarization within the conductor. The free electrons will move left, since electrons move in the direction opposing the electric field. This creates a ``cavity'' of no electrons on the right slab which can be modeled as positive charge. The charges create another electric field ${\vec{E}_1}$ pointing from the positive charges on the bottom slab to the negative charge. The net electric field inside the conductor is ${\vec{E}_{net} = \vec{E}_0 + \vec{E}_1}$. ${\vec{E}_0}$ and ${\vec{E}_1}$ point in different directions, and we can find the magnitude of ${\vec{E}_{net}}$ with the following reasoning.

Since the conductor is stationary instead of accelerating, it is in electrostatic equilibrium (the electrons in the conductor aren't accelerating). If the electrons aren't accelerating, that means the electric field on the inside of the conductor is 0 and hence ${|\vec{E}_0| = |\vec{E}_1|}$. From the same reasoning as before, the conductor is at a constant electric potential. This idea of 0 electric field inside the conductor is known as electrostatic shielding.