Skip to main content
United StatesPhysics C: Electricity and MagnetismSyllabus dot point

What are the properties of a conductor in electrostatic equilibrium, and why do they hold?

Topic 10.1 Electrostatics with Conductors: describe the field, charge and potential of a conductor in electrostatic equilibrium using Gauss's law.

A calculus-based answer to AP Physics C E&M Topic 10.1, covering the zero interior field, surface charge, equipotential conductors, the field just outside a conductor, and shielding, all justified by Gauss's law.

Generated by Claude Opus 4.810 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section
  1. What this topic is asking
  2. The zero interior field
  3. Charge on the surface
  4. Equipotential and the external field
  5. Try this

What this topic is asking

The College Board (Topic 10.1) wants you to describe a conductor in electrostatic equilibrium: the field inside it is zero, any net charge sits on the surface, the conductor is an equipotential, and the field just outside is perpendicular to the surface. Every one of these follows from Gauss's law and the definition of equilibrium.

The zero interior field

If any field existed inside, it would exert a force on the free electrons, which would move, contradicting equilibrium. They rearrange precisely until the interior field is everywhere zero. This single fact drives the rest.

Charge on the surface

Draw a Gaussian surface anywhere inside the conductor's material. Because E⃗=0\vec{E} = 0 there, the flux is zero, so qenc=0q_{enc} = 0: no net charge sits in the interior. Any excess charge must therefore reside on the surface. For a solid conductor that means the outer surface; for one with a cavity, charge can also appear on the cavity wall (see shielding).

Equipotential and the external field

With zero field inside, the line integral ΔV=−∫E⃗⋅dr⃗\Delta V = -\int\vec{E}\cdot d\vec{r} between any two interior points is zero, so the whole conductor sits at one potential: it is an equipotential, surface included. Just outside the surface, the field must be perpendicular (any tangential component would drive surface charges along the surface, violating equilibrium), and a pillbox Gaussian surface gives

Eoutside=σε0E_{outside} = \frac{\sigma}{\varepsilon_0}

with σ\sigma the local surface charge density, which is larger where the surface curves more sharply.

Try this

Q1. State the electric field deep inside a charged solid conductor in equilibrium. [1 point]

  • Cue. Zero; the free charges arrange so the interior field cancels.

Q2. A conductor's surface has local charge density σ=2.0×10−6\sigma = 2.0\times10^{-6} C per m squared. Find the field just outside (ε0=8.85×10−12\varepsilon_0 = 8.85\times10^{-12}). [2 points]

  • Cue. E=σε0=2.0×10−68.85×10−12=2.3×105E = \dfrac{\sigma}{\varepsilon_0} = \dfrac{2.0\times10^{-6}}{8.85\times10^{-12}} = 2.3\times10^5 N/C, perpendicular to the surface.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2022 (style)1 marksSection I (multiple choice). A solid conducting sphere carries net charge QQ. In electrostatic equilibrium, where does this charge reside? (A) uniformly through the volume (B) entirely on the outer surface (C) at the center (D) it depends on QQ. Justify your reasoning.
Show worked answer →

A 1-point MCQ on charge location in a conductor. The answer is (B).

A Gaussian surface drawn just inside the conductor encloses zero net charge, because the interior field is zero (∮E⃗⋅dA⃗=0=qenc/ε0\oint\vec{E}\cdot d\vec{A} = 0 = q_{enc}/\varepsilon_0). So no net charge can be in the interior; it all sits on the outer surface. The trap is (A), which would be true for an insulator, not a conductor.

AP 2024 (style)5 marksSection II (FRQ, conceptual and quantitative). A neutral hollow conducting sphere has a point charge +q+q placed at the center of its cavity. (a) Use Gauss's law to find the charge induced on the inner cavity wall. (b) Find the charge on the outer surface. (c) Describe the field in the conductor's material and just outside the outer surface.
Show worked answer →

A 5-point FRQ on induced charges and shielding.

(a) Inner wall (2 points): a Gaussian surface inside the conductor's material has zero field, so it encloses zero net charge. The +q+q at the center must be cancelled by −q-q induced on the inner wall.
(b) Outer surface (1 point): the conductor is neutral overall, so +q+q appears on the outer surface (conservation of charge: −q-q inner ++q+ +q outer =0= 0 net).
(c) Fields (2 points): inside the conductor's material the field is zero; just outside, the field is radial with E=kqr2E = \dfrac{kq}{r^2}, as if the charge sat at the center.

Markers reward the −q-q inner wall from Gauss, the +q+q outer from conservation, and the field description.

Related dot points

Sources & how we know this