Conductors in Equilibrium
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
In a conductor, charges move freely until they stop pushing each other around. In that equilibrium: E = 0 inside the metal, any excess charge sits on the surface, the whole conductor is at one potential, and the field just outside is perpendicular to the surface.
1. The four rules
2. Consequences
Cavities: a charge q in a cavity induces −q on the cavity wall. The outer surface gets whatever is needed to make the total right.
Shielding: the field outside depends only on the outer surface charge. Moving the charge around inside the cavity doesn’t change anything outside.
Sharp points: charge crowds onto sharply curved parts of the surface, so σ and the field just outside are largest there. That’s why lightning rods are pointed.
Inside a hollow charged conductor, E = 0 and V equals the surface potential (not zero).
3. Example, then try it
A hollow metal sphere of radius 0.10 m carries 5.0 nC. Find E and V inside it, and E just outside.
Inside: E = 0, and V equals the surface value: kQ/R = (9.0 × 10⁹)(5.0 × 10⁻⁹) ÷ 0.10 = 450 V.
Just outside: E = kQ/R² = 4500 N/C, the same as σ/ε0.
A +2.0 nC point charge sits in a cavity inside a neutral thick metal shell. Find the charge on the cavity wall and on the outer surface. The shell is then given an extra −5.0 nC. Find the new surface charges and the field 0.50 m from the center.
Neutral shell: cavity wall −2.0 nC (so E = 0 in the metal); outer surface +2.0 nC.
Add −5.0 nC: the cavity wall must still be −2.0 nC. The shell’s total is now −5.0 nC, so the outer surface has −5.0 − (−2.0) = −3.0 nC.
Field outside: total enclosed charge = +2.0 − 5.0 = −3.0 nC. E = (9.0 × 10⁹)(3.0 × 10⁻⁹) ÷ 0.25 = 108 N/C, pointing inward.
Answer: −2.0 nC and +2.0 nC; then −2.0 nC and −3.0 nC; 108 N/C inward.
4. Common mistakes
A charged metal object has one sharply pointed end and one rounded end. Where is the electric field just outside its surface strongest?
Charge concentrates where the surface curves most sharply, so σ, and E = σ/ε0, is largest at the point.
Ready to practice?
5 questions on Conductors in Equilibrium, each with a hint and an explanation for every wrong answer.