# UY1: Gauss’s Law For Conductors

Claim: When excess charge is placed on a solid conductor and is at rest (equilibrium), it resides entirely on the surface, not in the interior of the material.

Reason: The electric field within the conductor must be zero. If there is an electric field, the charges will move. As the electric field within the conductor is 0, by Gauss’s law, there must be no charges enclosed within the Gaussian surface.

Claim: If there is a cavity inside the conductor and there is no charge within the cavity, the net charge on the surface of the cavity must be zero, because $\vec{E} = 0$ everywhere on the Gaussian surface. In fact, there can’t be any charge anywhere on the cavity surface.

Reason: By Gauss’s Law, no net electric flux = no charge enclosed.

Claim: The direction of the $\vec{E}$ field at a point just outside any conductor is always perpendicular to the surface.

Proof:

Consider a Gaussian surface in the form of a small cylinder – one end with area A lies within the conductor and the other just outside. The charge enclosed is related to the surface charge density:

\begin{aligned} \frac{Q_{encl}}{\epsilon_{0}} &= \frac{\sigma A}{\epsilon_{0}} \\ \oint \vec{E} . d \vec{A} &= \frac{\sigma A}{\epsilon_{0}} \\ E_{\perp} \times A &= \frac{\sigma A}{\epsilon_{0}} \\ E_{\perp} &= \frac{\sigma}{\epsilon_{0}} \end{aligned}

### Charge in cavity in conductor

Suppose we place a small body with a charge q inside a cavity within a conductor. The conductor is uncharged and is insulated from the charge q.

Since E inside a conductor = 0, $\oint E.dA = 0$. There is no electric flux through a Gaussian surface which is enclosing the cavity. Since the cavity holds a positive charge, this means that the surface of the cavity holds a negative charge which exactly cancels the positive charge so as to give a zero E inside the conductor.

However, if you take a Gaussian surface enclosing the whole conductor, you must find electric flux through the surface befitting of a positive charge enclosed. Hence, positive charges must be distributed on the surface of the conductor.

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