Conductors and Shielding

A conductor is simply a material, like any metal, where some charges are free to wander around instead of being stuck to their atoms. The name says what it's good at: it conducts, or carries, charge from place to place. That freedom is also what makes metal boxes work as shields, keeping electric fields away from delicate electronics. The two key facts, $\mathbf{E}_{\text{inside}} = \mathbf{0}$ and $E_{\text{surface}} = \sigma/\varepsilon_0$, look formal, but they just say: there's no field inside the metal (the free charges have already canceled it), and the field just outside is set by $\sigma$ (how much charge sits on each square meter of surface). Everything else in this lesson is Gauss's law applied to those two facts; bear with it.

What a metal does to a field. Put a block of copper in an electric field. Its free electrons feel a force and drift against the field, piling up on one face and leaving the positive ions uncovered on the other. That separated charge makes its own field, pointing the other way, and the drift continues until the two fields cancel exactly. In a metal this is over in far less than a nanosecond. Once the charges stop moving (electrostatic equilibrium), the field inside the metal is zero, because if it weren't, the free charges would still be moving.

Three consequences, each from Gauss's law.

1. Excess charge lives on the surface. Draw a Gaussian surface just inside the metal. $\mathbf{E} = \mathbf{0}$ at every point on it, so the flux is zero and it encloses no net charge. That's true of every such surface, so the interior is neutral and any extra charge sits on the outer skin.
2. The field just outside is perpendicular, with strength $\sigma/\varepsilon_0$. A component along the surface would push the surface charges along it, so at equilibrium there isn't one. For the strength, take a small pillbox with one face just outside and one just inside. Flux leaves only through the outer face: $EA = \sigma A/\varepsilon_0$, so $E = \sigma/\varepsilon_0$. That's twice the $\sigma/2\varepsilon_0$ of a lone sheet of charge, because inside the metal the field is zero, so all of the flux has to leave through one side.
3. An empty cavity has no field. Wrap a Gaussian surface inside the metal around a hollow with no charge in it: zero flux, so the cavity's wall carries no net charge. Could it carry $+$ in one spot and $-$ in another? Then a field line would run across the cavity from $+$ to $-$, and a loop following that line and returning through the metal would have $\oint\mathbf{E}\cdot d\mathbf{l} > 0$. That's impossible for an electrostatic field, which is conservative. So the cavity is field-free no matter what happens outside: a Faraday cage.

A charge inside a hollow conductor. Put $+q$ in the cavity. A Gaussian surface in the metal must enclose zero net charge, so $-q$ gathers on the cavity wall, and since the metal started neutral, $+q$ appears on the outside. On a spherical conductor that outer charge spreads evenly, so the field outside is exactly that of a point charge at the center, wherever $q$ sits inside. Shielding works one way: the metal hides where the charge is, but not how much. Ground the conductor and the outer $+q$ drains away; then nothing outside can tell the charge is there at all.

Sharp points. Charge on a conductor crowds toward sharp edges and points, where $\sigma$, and with it the surface field $\sigma/\varepsilon_0$, is largest. That's how lightning rods and spark plugs work, and why high-voltage hardware is smooth and round. The next unit, Electric Potential, shows why: a whole conductor sits at one potential, and a small curved piece reaches it with a little charge on a tiny area.

A common mistake. It's tempting to think the field inside metal is always zero, as if metal simply blocks fields. It isn't: $\mathbf{E} = \mathbf{0}$ holds only at equilibrium. A wire carrying a current has a field inside it, and that field is what keeps the current flowing, as the DC circuits unit will show. Two other slips: surface charge on a conductor is usually not uniform, and the field at a conductor's surface is $\sigma/\varepsilon_0$, not the lone sheet's $\sigma/2\varepsilon_0$.

Shielding is everywhere in electronics: the braided outer conductor of a coaxial cable, the metal cans soldered over the radio chips in a phone, the copper-lined rooms where engineers test devices for interference, and the screened rooms around MRI scanners. Cars and airliners struck by lightning protect the people inside the same way: the charge races over the metal skin while the field inside stays small. Sharp-point physics runs lightning rods, the corona wires in laser printers and air purifiers, and the rule that high-voltage hardware must be smooth and rounded.