AP® Physics C: Electricity and Magnetism · Unit 10 · Topic 10.1
Conductors as Equipotentials
Inside a conductor in equilibrium the electric field is zero, so moving a charge anywhere inside it takes no work. That makes every point of the conductor, interior and surface, sit at the same potential.
Free charges in a conductor move until they cancel the field inside completely. If any field remained, the free electrons would feel a force and keep sliding until it vanished. Potential difference between two points is defined by how much work it takes to move a charge between them through the field, so with zero field inside, that work is zero no matter the path. Every interior point and the whole surface end up at one shared value of V.
This does not mean charge spreads evenly. Charge piles up more densely where the surface curves sharply, so a lumpy or pointed conductor has uneven surface charge density even though V is constant everywhere on it. For a sphere the symmetry makes life easy: all the charge sits on the outer surface, and outside the sphere it behaves exactly like a point charge, so V = kQ/R at the surface and everywhere inside, since there is no further potential drop to cross.
Connect two spheres with a thin wire and charge flows until both reach the same potential, redistributing itself according to each sphere's radius. This is the bridge into capacitors: a capacitor is just two conductors, each an equipotential surface on its own, held at a potential difference by equal and opposite charge. Understanding why a single conductor flattens to one potential is what makes the voltage between two separated conductors meaningful and calculable.
Key terms
- Equipotential
- A surface or region where every point has the same electric potential.
- Electrostatic equilibrium
- The state of a conductor once free charges have stopped moving and the interior field is zero.
- Surface charge density
- Charge per unit area on a conductor's surface, which can vary even though potential does not.
- Potential (V)
- Electric potential energy per unit charge at a point, measured in volts.
Watch it explained
- How it worksHow does zero field inside a conductor force the potential to stay flat throughout it?
- Worked exampleFind the potential at the center and at the surface of a 10 cm sphere carrying 25 nC.
- Going furtherWhat happens to the potential of two metal spheres joined by a long thin wire?
- Common mistakeWhy doesn't a big metal sphere hold the same charge as a small one at the same voltage?
Check yourself
End of topic.