AP® Physics C: Electricity and Magnetism · Unit 10 · Topic 10.1
The Field at a Conductor's Surface
Just outside a conductor in equilibrium, the electric field is always perpendicular to the surface and has magnitude sigma/epsilon0, where sigma is the local surface charge density.
A conductor in electrostatic equilibrium has no electric field inside it, and free charge can slide around on it without resistance. If the field just outside had any component running along the surface, that component would shove the mobile surface charges sideways. They'd move until that sideways push vanished. So equilibrium forces the external field to point straight out (or in), perpendicular to the surface, with nothing left over tangentially.
To get the size of that field, build a small pillbox Gaussian surface: a flat can straddling the conductor's surface, one face just inside the metal, one face just outside, with a tiny area A. The inside face contributes zero flux, since E is zero inside a conductor. The sides contribute nothing either, because the external field is perpendicular to the surface, so it's parallel to the sides and never pierces them. Only the outer face contributes flux, E times A. The enclosed charge is sigma times A, so Gauss's law gives E A = sigma A / epsilon0, and the area cancels: E = sigma/epsilon0.
This result is what makes sharp points on conductors dangerous or useful. Charge crowds onto regions of high curvature, so sigma spikes at a sharp tip, and E = sigma/epsilon0 spikes with it. That's why lightning rods are built pointed: the field near the tip gets large enough to ionize the surrounding air and bleed charge off gently, before a full bolt can build. This surface-field result also sets up capacitors, where two charged conductor surfaces facing each other each obey E = sigma/epsilon0, and the uniform field between them is what stores the energy in the gap.
Key terms
- surface charge density (sigma)
- Charge per unit area sitting on a conductor's surface, measured in coulombs per square meter.
- electrostatic equilibrium
- A state where free charges in a conductor have stopped moving, so the interior field is zero.
- pillbox Gaussian surface
- A thin, flat closed surface straddling a charged surface, used to relate E to sigma via Gauss's law.
- tangential field component
- The part of an electric field that runs parallel to a surface rather than into or out of it.
Watch it explained
- How it worksWalk me through the pillbox argument giving E equal to sigma over epsilon zero at a surface.
- Common mistakeWhy doesn't a field component along the surface survive on a conductor in equilibrium?
- Worked exampleFind the field just outside a conductor whose local density is 8 nC per square meter.
- Ties togetherHow does the sharp point result explain why lightning rods are built with a point?
Check yourself
End of topic.