9 min readConcept sheet2 or 3-mark questions on the electrostatic properties of a conductor, electrostatic shielding and the field at its surface

Conductors in electrostatics: free charges and bound charges

A conductor's free electrons move until the field inside is zero. So E = 0 inside, extra charge sits on the surface, and the whole conductor has one potential.

Free and bound chargesConductors have free charges (electrons in metals, ions in electrolytes); insulators have bound charges held inside their atoms

Inside a conductorE = 0, no net charge, and the same potential everywhere

Just outside a charged conductorE = (σ/ε₀) n̂, normal to the surface (σ = local surface charge density)

Electrostatic shieldingThe field inside an empty cavity of a conductor is zero, whatever the field outside

Free charges vs bound charges

Free charges (conductors)Charges that can move through the whole material. In metals, the outer electrons leave their atoms and wander freely. In electrolytes (salt water, acids), positive and negative ions move. Even a tiny field makes them drift.

Bound charges (insulators)Electrons stay held to their own atoms or molecules. A field can only shift them slightly inside each molecule. They cannot travel through the material, so there is no conduction.

Six results for a conductor in electrostatics

E = 0 inside. If there were a field, the free electrons would keep moving. They stop only when the field inside is zero.
E is normal to the surface. A part along the surface would move charges along it.
No net charge inside. By Gauss's theorem, a closed surface inside, where E = 0, encloses zero charge. Any extra charge sits on the outer surface.
The potential is constant throughout the conductor and equal to its surface value, because E = 0 inside and E has no part along the surface.
E = σ/ε₀ just outside, where σ is the local surface charge density.
The field inside an empty cavity is zero, even when the conductor sits in a strong outside field: this is electrostatic shielding.

Derivation: field just outside a charged conductor

1
Draw a small cylinder (pillbox) of end area δS, half inside and half outside the conductor's surface.
2
Inside the conductor E = 0, so the inner end has no flux.
3
Just outside, E is normal to the surface, so the curved side has no flux.
4
Flux through the outer end = E δS. Charge enclosed = σ δS.
5
Gauss's theorem: E δS = σ δS/ε₀.
6★
So, E = σ/ε₀, normal to the surface. This is twice the field of a free sheet, because all the field lines leave on one side.

★ key step

Electrostatic shielding

Take a conductor with an empty cavity inside it. Whatever the charges or fields outside, the field inside the cavity is zero. The charges on the conductor rearrange themselves to cancel any field in the cavity. This is electrostatic shielding. Delicate instruments are kept in metal boxes for this reason. Cables carrying weak signals have a metal braid round the inner wire. During lightning, the inside of a car, bus or train coach is shielded by its metal body.

Worked example: field at the surface of a conductor

Problem

A charged metal sphere of radius 10 cm has a uniform surface charge density of 8.85 × 10⁻⁷ C m⁻². Find (a) the field just outside it, (b) the field at its centre and (c) its total charge. (ε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻²)

Working

(a) E = σ/ε₀ = 8.85 × 10⁻⁷ ÷ 8.85 × 10⁻¹² = 1 × 10⁵ N C⁻¹, pointing outward.
(b) The centre is inside the conductor, so E = 0.
(c) q = σ × 4πR² = 8.85 × 10⁻⁷ × 4 × 3.14 × (0.10)².
q = 8.85 × 10⁻⁷ × 0.1256 ≈ 1.11 × 10⁻⁷ C.

Answer

(a) 1 × 10⁵ N C⁻¹, outward; (b) zero; (c) about 1.1 × 10⁻⁷ C

In real life

During a monsoon thunderstorm, sitting inside a car, bus or train coach is far safer than standing under a tree: the metal body shields the inside. Cable TV and internet coaxial cables use a metal braid to shield the signal from outside electrical noise.

Exam angle

3 marks: “State any four electrostatic properties of a conductor, with a reason for each.” E = 0 inside (free charges would otherwise move); E is normal at the surface (no part along it); net charge only on the surface (Gauss's theorem with E = 0 inside); potential constant throughout (E = 0 inside).

2 marks: “Show that the electric field just outside a charged conductor is σ/ε₀.” Pillbox across the surface: no flux through the inner end (E = 0) or the side (E is normal). E δS = σ δS/ε₀, so E = σ/ε₀.

2 marks: “What is electrostatic shielding? Why is it safer to sit inside a car during a lightning storm?” The field inside an empty cavity of a conductor is zero, whatever the outside field. The car's metal body acts as such a conductor and shields the people inside.

Remember it

Conductor rules: zero inside, charge outside, field normal at the surface, one potential throughout.