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UY1: Gauss's Law For Conductors

Use Gauss's law to explain why electrostatic charge resides on conductor surfaces and why fields are normal to the surface.

  • University Physics Year 1
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Learning objectives

  • Construct electric-field and potential models for discrete and continuous charge distributions.
Why this matters + quick links

This page gives the UY1 working model/result for Gauss’s Law For Conductors. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

In electrostatic equilibrium, Gauss’s law gives strong results for conductors without detailed integration.

1) At a glance

  • Prerequisites: electrostatic equilibrium, flux integral, symmetry
  • Outcomes: justify E = 0 inside conductors and derive E_⊥ = σ/ε₀ at the surface
  • Key results: vector E = 0 in conductor material (electrostatics) and E_⊥ = σ/ε₀ just outside the surface
  • Key facts: free charge moves until internal electric field vanishes
  • Common trap: applying these results to non-equilibrium situations with current flow

Motivation / intuition

Conductors have mobile charges. If an internal electric field existed at rest, charges would keep moving, so the only stable electrostatic state is one where the field inside the conducting material is zero and charge lives on surfaces.

2) Setup

Assume a conductor in electrostatic equilibrium (charges at rest).

Use two Gaussian surfaces:

  • one fully inside conductor material
  • one tiny cylindrical pillbox crossing the conductor surface

3) Core derivation/explanation

A) Field inside conductor material

Inside the conductor, if vector E ≠ 0, free charges would keep moving. At equilibrium: vector E = 0 (inside conductor material) Then for any Gaussian surface fully inside, ∮ vector E · d vector A = 0 ⇒ Q_encl = 0 So excess charge cannot remain in the bulk; it resides on surfaces.

B) Field just outside conductor

Use a thin pillbox with one face inside (where E = 0) and one just outside. Side flux is negligible for vanishing height.

Gauss’s law: E_⊥ A = (σ A)/ε₀ ⇒ E_⊥ = σ/ε₀ Hence field just outside the surface is perpendicular to the conductor.

C) Cavity statements

  • If a cavity has no internal charge, net charge on cavity wall is zero.
  • If a cavity contains charge q, induced charge on inner cavity wall is -q so that field in conductor material stays zero.
  • For an initially neutral isolated conductor, + q appears on outer surface.
Quick checks (units + limits/sign)
  • Units: [σ/ε₀] = N/C, consistent with an electric field.
  • Limits/signs: if σ > 0, the field just outside points outward normal; if σ = 0, then E_⊥ = 0 (no normal field jump).

4) Worked example(s)

A neutral hollow conductor contains a point charge + 5 nC in its cavity (not touching walls).

By the cavity rule:

  • induced charge on inner wall: -5 nC
  • charge on outer surface: + 5 nC
  • net conductor charge stays zero

This is required by both Gauss’s law and charge conservation.

5) Practice set (with hints + answers)

  1. In electrostatic equilibrium, what is vector E inside conductor material?
  2. Surface charge density is σ = 2.0 × 10⁻⁶ C/m². Find E_⊥ just outside.
  3. A neutral conductor cavity contains charge -q. What charge appears on inner cavity wall?

Hints

  1. Think about motion of free charges.
  2. Use E_⊥ = σ/ε₀.
  3. Inner wall charge must cancel enclosed cavity charge for zero field in metal.

Answers

  1. vector E = 0.
  2. E_⊥ ≈ 2.26 × 10⁵ N/C.
  3. + q.

6) Summary + next steps

  • Excess charge on a conductor at equilibrium lies on surfaces.
  • Internal conductor field is zero; surface field is normal with magnitude σ/ε₀.
  • Cavity charges induce balancing inner-wall charge and corresponding outer-surface charge.

Next: Electric Field And Potential Of Charged Conducting Sphere Previous: Usage Of Gauss’s Law Back To Electromagnetism (UY1)