Conductors & Earthing

Inside a conductor the field is zero. Feed that one fact into Gauss’s law and out come all the rules about where charge sits on a conductor, and why earthing keeps you safe.

Charge lives on the surface#

Once the free electrons in a conductor have settled, E = 0 at every point inside the metal. Draw a closed surface lying entirely inside the metal: the flux through it is zero, so by Gauss’s law it encloses no net charge.

• Every volume inside a conductor is electrically neutral. Any charge you inject moves to the surface.
• If the conductor is hollow with nothing in the cavity, the charge goes to the outer surface, and the field in the cavity is zero (this needs a little more mathematics to prove).

A charge inside a cavity

Now put +q in the cavity. A Gaussian surface through the metal around the cavity still has E = 0 on it, so it must enclose zero charge: −q appears on the inner wall. If the conductor was neutral, the balancing +q appears on the outer surface.

The field just outside a conductor#

A conductor’s surface is an equipotential, so the field just outside is perpendicular to it. Take a pillbox with one face (area ΔS) just outside, the other inside the metal. Only the outside face has flux, EΔS, and the charge enclosed is σΔS:

The field jumps from σ/ε₀ to zero across a skin only 4–5 atomic layers thick; that skin is what we mean by “the surface” of a conductor.

Earthing a conductor#

The earth is a huge conductor. Adding or removing a little charge barely changes its potential, so we take it as zero, the same as a point at infinity (its real surface charge, about −1 nC/m², can be ignored for most purposes). Connect a conductor to the earth and charge flows until the conductor is at zero potential too: it is earthed (grounded).

• A charged metal sphere at Q/4πε₀R, earthed: all its charge flows away.
• A shell with +Q at its centre holds −Q inside and +Q outside, and sits at Q/4πε₀R. Earthed, the outer +Q drains away; the inner −Q stays, held by the central charge.