Electrostatics of conductors

Inside a conductor

Narrated video. Transcript:

A metal is full of free electrons that can drift through it. In electrostatics, every charge has come to rest. That one fact decides how conductors behave.

Put a neutral metal block in an external field. The free electrons drift against the field and pile up on one face, leaving positive charge on the opposite face.

These induced charges make a field of their own, opposite to the outside field. The electrons keep drifting until the two fields cancel exactly inside the metal.

So, first result: inside a conductor, the electrostatic field is zero. If it were not, the free electrons would still be moving.

Second: just outside a charged conductor, the field is normal to the surface. If it had a part along the surface, charges would slide along it.

Third: there is no excess charge inside a conductor. Draw any small closed surface inside the metal. E is zero all over it, so by Gauss’s law the charge enclosed is zero.

So any extra charge on a conductor sits on its surface. Watch the field lines of the outside field: they end on the induced charges, and meet the surface at right angles.

Fourth: the whole conductor is at one potential. E is zero inside and has no part along the surface, so no work is needed to move a charge anywhere in it or on it.

Surface charge and shielding

Narrated video. Transcript:

Fifth result: the field just outside a charged conductor is E = σ/ε₀, along the outward normal, where σ is the surface charge density at that spot.

To see why, draw a tiny pillbox that pokes through the surface, with flat ends of area δS. Only the outer end has any flux, E δS: the field is zero inside, and along the curved side it has no normal part.

The charge inside the box is σ δS. Gauss’s law gives E δS = σ δS/ε₀, so E = σ/ε₀. More charge per unit area means a stronger field there.

Example: a metal sphere of radius 10 cm carries 2 nC. Its surface charge density is 2 × 10⁻⁹ ÷ (4π × 0.01), about 1.6 × 10⁻⁸ C/m². So E just outside is σ/ε₀, about 1800 V/m.

Check with Coulomb’s law: kQ/R² = 9 × 10⁹ × 2 × 10⁻⁹ ÷ 0.01 = 1800 V/m. The sphere’s potential is kQ/R = 180 V, and it is 180 V everywhere inside too.

Sixth: electrostatic shielding. Take a conductor with an empty cavity inside. Whatever the charges and fields outside, the field inside the cavity is zero.

The charges on the outer surface arrange themselves to cancel the outside field everywhere within. So the cavity is shielded from external fields.

That is why sensitive instruments are kept in metal boxes, and why you are safe inside a car in a thunderstorm: its metal body is a shield. Stay inside, and do not touch the metal.

Try it: a conductor in a field

Interactive simulation. Things to try:

• The grey ball is a metal sphere in a uniform field E₀. The field lines bend to meet its surface at right angles.
• Drag q₀ inside the ball: |E| reads zero and V is the same everywhere inside.
• The + and − marks show the induced charge: negative where field lines arrive, positive where they leave. In total it is zero.
• Give the sphere a charge Q, or turn E₀ down to zero to see an isolated charged sphere.

At the centre of a charged shell

A hollow metal sphere of radius 10 cm carries a charge of 2 nC. What are the field and the potential at its centre?

1. E = 0, V = 0
2. E = 0, V = 180 V
3. E = 1800 V/m, V = 180 V
4. E = 1800 V/m, V = 0

From field to surface charge

The field just outside a charged conductor is 9 × 10⁴ N/C. What is the surface charge density there? (ε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻²)

1. 8.0 × 10⁻⁷ C/m²
2. 1.6 × 10⁻⁶ C/m²
3. 4.0 × 10⁻⁷ C/m²
4. 1.0 × 10¹⁶ C/m²

Summary

• Inside a conductor: E = 0, V = constant
• Just outside the surface: E = (σ/ε₀) n̂
• Charged sphere (outside / inside): V = kQ/r / V = kQ/R

• In electrostatics, the field inside a conductor is zero and the field at its surface is normal to it.
• Excess charge resides only on the surface; the interior carries no net charge.
• The whole conductor, inside and surface, is at one potential.
• The field inside a cavity in a conductor is zero: electrostatic shielding.