Short on time? This page is the recap. For the reasoning, Figure 1 and the worked examples, use the full study guide. This topic is part of the calculus-based course.
Recap
- An ideal conductor has electrons that move freely. Positive net charge means missing electrons, modelled as positive carriers on the surface.
- Free charges move until the field inside cancels. This electrostatic equilibrium is reached effectively at once.
- E = 0 inside the metal → by Gauss’s law, no net charge inside → all excess charge is on the surface.
- E = 0 inside → ΔV = −∫E·dl = 0 → the whole conductor, surface included, is one equipotential.
- The field just outside is perpendicular to the surface, with E = σ/ε₀.
- σ is greatest at points and edges, so the field is strongest there.
- In an external field, a conductor polarizes: induced − and + charges appear on opposite sides, the net charge stays the same, and the inside stays field-free.
- A closed conducting shell shields its cavity from outside fields.
Key relationships
| Situation | Result |
|---|---|
| Inside the metal of any conductor in equilibrium | E = 0; V = constant |
| Just outside a conducting surface | E = σ/ε₀, perpendicular to the surface |
| Thin isolated sheet (for comparison) | E = σ/(2ε₀) |
| Isolated conducting sphere, charge Q | σ = Q/(4πR²); E = Q/(4πε₀r²) for r ≥ R; E = 0 inside |
| Empty cavity in a conductor | no charge on the wall; E = 0 in the cavity |
| Charge q in a cavity of a conductor with net charge Q | wall: −q; outer surface: Q + q |
Assumptions
- Electrostatic equilibrium: no charge is moving.
- “Ideal” conductor: charges move freely; insulators behave differently.
- Spherical results need spherical symmetry and no nearby charges.
Mistakes to avoid
- V = 0 inside because E = 0. V is constant, not zero.
- σ/(2ε₀) next to a conductor. It is σ/ε₀.
- Charge −q on the outer surface. The induced −q is on the cavity wall.
- Uniform σ on every shape. Only an isolated sphere has uniform σ.
- “Shielding works both ways.” A charge inside still produces a field outside unless the shell is grounded.
Quick self-check
- Just outside a conductor, σ = 4.0 × 10⁻⁸ C/m². What is E? (4.5 × 10³ N/C, perpendicular to the surface)
- +2.0 nC is placed in the cavity of a neutral metal shell. What are the charges on the inner and outer surfaces? (−2.0 nC and +2.0 nC)
- A metal sphere is at 500 V. What is the potential at its centre? (500 V)
- A closed, neutral metal box stands in a uniform external field of 5.0 × 10³ N/C. What is E in the empty space inside? (Zero: charges induced on the outer surface cancel the outside field everywhere within it)
Next: practice questions.