Conductors in electrostatic equilibrium

When the charges in a metal stop moving, Gauss's law pins down everything: zero field inside, all the charge on the surface, and the shielding that makes a Faraday cage work.

Source: Serway & Jewett, Physics for Scientists and Engineers, 7th ed., §24.4.

Before you start

What you need first

Topic 1 — a conductor has charges (electrons) free to move.
Topic 6–7 — Gauss's law, and that a surface with $E=0$ on it encloses no net charge.
Topic 3 — a charge in a field feels a force $F=qE$.

What you'll be able to do

Explain why $E = 0$ inside a conductor in equilibrium.
Say where the charge sits, and find the field just outside, $E = \sigma/\varepsilon_0$.
Explain why charge crowds at sharp points.
Explain electrostatic shielding (the Faraday cage).

The starting point

The field inside is zero

A conductor is full of free charges. If there were any field inside, those charges would feel a force $F=qE$ and keep moving. Equilibrium means they have stopped — so the field inside must be exactly zero.

$$ E_{\text{inside}} = 0 $$

The charges arrange themselves precisely so their own field cancels any outside field, all through the body of the metal.

Zero field inside; the excess charge sits on the outer surface.

All the extra charge sits on the surface

Draw a Gaussian surface just inside the metal. Since $E=0$ everywhere on it, the flux is zero — so by Gauss's law it encloses no net charge. That holds for any surface inside the metal, so none of the excess charge can be in the body:

$$ \text{excess charge} \;\longrightarrow\; \text{entirely on the outer surface} $$

This is why, from Topic 7, a charged metal ball acts like a point charge outside and has $E=0$ inside — all its charge lives on the surface.

The field just outside the surface

Right at the surface the field must point straight out (any sideways part would push the surface charges along until it vanished). A small pillbox — one face just outside, one just inside (where $E=0$) — gives, with local surface density $\sigma$:

$$ E = \frac{\sigma}{\varepsilon_0} $$

where:
Symbol	Meaning	SI unit
E	field just outside the surface (perpendicular to it)	N/C
σ	local surface charge density at that spot	C/m²
ε₀	permittivity of free space $=8.85\times10^{-12}$	C²/(N·m²)

Watch the factor. A conductor surface gives $E=\sigma/\varepsilon_0$ — twice the isolated sheet's $\sigma/2\varepsilon_0$ (Topic 7). The pillbox here has only one face outside the metal (the inside face sits where $E=0$), so there is no factor of $\tfrac12$.

📐 Worked example 1

A charged metal ball

A metal ball of radius $R = 0.10$ m carries $Q = 6.0\ \mu\text{C}$. (a) Find the field just outside its surface. (b) Find the field at its centre.

1Just outside, the ball looks like a point charge at $r = R$: use $E = kQ/R^2$:

$$ E = \frac{(8.99\times10^{9})(6.0\times10^{-6})}{(0.10)^2} $$

2Work it out:

$$ E \approx 5.4\times10^{6}\ \text{N/C} $$

(b) At the centre — anywhere inside the metal — the field is exactly $\mathbf{0}$.

Charge crowds at sharp points

The surface charge is not spread evenly unless the conductor is a sphere. It piles up where the surface curves sharply — at points and edges. More $\sigma$ there means a stronger field $E=\sigma/\varepsilon_0$ just outside.

This is why lightning rods are pointed: the field at the tip gets strong enough to ionise the air and quietly bleed charge away.

Sparse charge on the blunt end; crowded charge and a strong field at the point.

Shielding: the Faraday cage

Hollow out the conductor. With no charge in the cavity, a Gaussian surface in the metal around it still has $E=0$, so the cavity walls hold no net charge and the field inside the cavity is zero — even when there is a strong field outside.

The metal soaks up the outside field on its surface and keeps the inside calm. That is why a car or a metal mesh shields what's inside from outside fields — a Faraday cage.

Outside field stops at the metal; the cavity inside stays field-free.

✏️ Try it yourself

A solid metal ball of radius $R = 0.20$ m carries $Q = -8.0\ \mu\text{C}$.

(a) Find the field just outside its surface, and its direction.
(b) Find the field at a point $0.10$ m from the centre (inside the metal).

(a) Step 1. Just outside, use $E = k|Q|/R^2 = \dfrac{(8.99\times10^{9})(8.0\times10^{-6})}{(0.20)^2}$. Step 2. $E \approx 1.8\times10^{6}\ \text{N/C}$, pointing inward (toward the ball, because the charge is negative). (b) The point is inside the metal, so $E = 0$.

Common mistakes

Mistake	Fix
Thinking charge spreads through the body of the metal.	In equilibrium all excess charge is on the outer surface; the inside is neutral with $E=0$.
Using $\sigma/2\varepsilon_0$ at a conductor surface.	A conductor surface gives $E = \sigma/\varepsilon_0$ — no factor of $\tfrac12$ (only one pillbox face is outside).
Expecting the field inside the metal to depend on the outside field.	It is always $0$ in equilibrium — the surface charge rearranges to guarantee it.
Thinking a Faraday cage needs to be solid.	Even a mesh shields well; the charges still rearrange to keep the cavity field near zero.

Recap — the whole topic on one screen

Idea	What you own now
Inside	$E = 0$ everywhere inside a conductor in equilibrium.
Charge	All excess charge sits on the outer surface.
Just outside	$E = \dfrac{\sigma}{\varepsilon_0}$, perpendicular to the surface.
Sharp points	Charge crowds there → strongest field (lightning rods).
Cavity	Field-free inside → electrostatic shielding (Faraday cage).

Next topic · new theme

Theme 2 — Electric potential & capacitance

That finishes the electric field. Next we ask how much energy it takes to move a charge through a field — the idea of electric potential (voltage) — which leads to capacitors, the components that store charge and energy.

→ Topic 9 · Electric potential & potential difference

Mistake	Fix
Thinking charge spreads through the body of the metal.	In equilibrium all excess charge is on the outer surface; the inside is neutral with \(E=0\).
Using \(\sigma/2\varepsilon_0\) at a conductor surface.	A conductor surface gives \(E = \sigma/\varepsilon_0\) — no factor of \(\tfrac12\) (only one pillbox face is outside).
Expecting the field inside the metal to depend on the outside field.	It is always \(0\) in equilibrium — the surface charge rearranges to guarantee it.
Thinking a Faraday cage needs to be solid.	Even a mesh shields well; the charges still rearrange to keep the cavity field near zero.

Idea	What you own now
Inside	\(E = 0\) everywhere inside a conductor in equilibrium.
Charge	All excess charge sits on the outer surface.
Just outside	\(E = \dfrac{\sigma}{\varepsilon_0}\), perpendicular to the surface.
Sharp points	Charge crowds there → strongest field (lightning rods).
Cavity	Field-free inside → electrostatic shielding (Faraday cage).