Conductors in Electrostatic Equilibrium

When a conductor reaches electrostatic equilibrium (no net motion of charges), several important properties emerge. These properties follow from Gauss's Law and the definition of a conductor.

Definition

A conductor is in electrostatic equilibrium when:

  • There is no net motion of charge within the conductor
  • All charges are at rest
  • The system has reached a steady state

Key Properties

1. E = 0 Inside the Conductor

Reasoning:

  • If $E \neq 0$ inside, free electrons would experience force $\vec{F} = q\vec{E}$
  • Electrons would accelerate and move
  • But we're in equilibrium — no motion
  • Therefore, $E = 0$ everywhere inside

Mathematical proof using Gauss's Law:

  • Consider Gaussian surface just inside conductor surface
  • Since $E = 0$ everywhere on this surface, flux = 0
  • Therefore, enclosed charge = 0
  • All excess charge resides on the outer surface

2. All Excess Charge on Surface

Any net charge placed on a conductor distributes itself on the outer surface.

For hollow conductors:

  • Charge resides on outer surface only
  • Inner surface has no net charge (unless there's charge inside the cavity)

3. E is Perpendicular to Surface

At the surface of a conductor, $\vec{E}$ must be perpendicular to the surface.

Reasoning:

  • If $\vec{E}$ had a parallel component, charges would move along the surface
  • But we're in equilibrium — no motion
  • Therefore, $\vec{E}$ has only perpendicular component

4. Field Just Outside the Conductor

$$E_{surface} = \frac{\sigma}{\varepsilon_0}$$

Where $\sigma$ is the local surface charge density.

Derivation: Use Gaussian "pillbox" with one face just outside, one just inside:

  • Inside face: $E = 0$, so no flux
  • Curved side: $\vec{E} \perp$ surface, so no flux
  • Outside face: $\Phi = EA$
  • Enclosed charge: $Q = \sigma A$

From Gauss's Law: $$EA = \frac{\sigma A}{\varepsilon_0} \Rightarrow E = \frac{\sigma}{\varepsilon_0}$$

Summary Table

Location Electric Field Charge
Inside conductor $E = 0$ None (excess)
Just outside surface $E = \frac{\sigma}{\varepsilon_0}$ $\sigma$ on surface
Far from conductor Like point charge $Q$ total

Induced Charges

Charge Inside a Cavity

If charge $+q$ is placed inside a cavity within a conductor:

  • $-q$ is induced on the inner surface
  • $+q$ is induced on the outer surface
  • $E = 0$ in the conducting material

Example: Spherical Shell

Sphere (radius $a$, charge $+q$) inside hollow sphere (inner radius $b$, outer radius $c$, initially neutral):

Region Field Explanation
$r < a$ $E = 0$ Inside inner conductor
$a < r < b$ $E = \frac{kq}{r^2}$ Outside point charge
$b < r < c$ $E = 0$ Inside outer conductor
$r > c$ $E = \frac{kq}{r^2}$ Net charge = $+q$

Induced charges:

  • Inner surface of hollow sphere: $-q$
  • Outer surface of hollow sphere: $+q$

Example Calculation

Problem: A conducting sphere of radius $R = 0.50$ m carries charge $Q = 2.0 \times 10^{-6}$ C. Find: (a) Field inside the conductor (b) Field just outside the surface

Solution: (a) $E_{inside} = 0$ (property of conductors)

(b) Surface charge density: $$\sigma = \frac{Q}{4\pi R^2} = \frac{2.0 \times 10^{-6}}{4\pi (0.50)^2} = 6.37 \times 10^{-7} \text{ C/m}^2$$

$$E_{surface} = \frac{\sigma}{\varepsilon_0} = \frac{6.37 \times 10^{-7}}{8.85 \times 10^{-12}} = 7.2 \times 10^{4} \text{ N/C}$$

Applications

  1. Electrostatic shielding (Faraday cage) — $E = 0$ inside protects contents
  2. Lightning rods — Charge concentrates at sharp points (high $\sigma$ → high $E$ → corona discharge)
  3. Van de Graaff generator — Charge transported to outer surface

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