AP® Physics C: Electricity and Magnetism · Unit 10 · Topic 10.3
Spherical and Cylindrical Capacitors
Capacitance for spheres and cylinders comes from using Gauss's law to find E between the conductors, then integrating to get V, then dividing Q by V. An isolated sphere counts infinity as its "other plate."
Capacitance measures how much charge a conductor arrangement stores per volt of potential difference. For spherical and cylindrical capacitors, the recipe is always the same three steps. First, put charge Q on the inner conductor and use Gauss's law with a spherical or cylindrical surface to find the electric field E in the gap. Second, integrate E along a radial path from inner to outer conductor to get the potential difference V. Third, compute C = Q/V, and the Q cancels out because E was proportional to Q in the first place. What's left is a formula built only from geometry and epsilon0.
A lone metal sphere still has a capacitance because it can still hold charge at a nonzero potential, and nothing in the definition C = Q/V requires a second plate nearby. The sphere's field lines simply terminate at infinity instead of on an inner or outer partner, so infinity plays the role of the other conductor, sitting at V = 0. That's why the isolated sphere formula C = 4 π epsilon0 R looks like the spherical capacitor formula with the outer radius sent to infinity.
This topic sits right after parallel plate capacitors and extends the same Gauss's-law method to curved geometry, then feeds into combinations of capacitors and energy storage, where U = Q2/(2C) uses whichever capacitance formula fits the shape. Notice also that when the gap between concentric cylinders or spheres becomes very thin compared to the radius, both formulas collapse toward the parallel plate result C = epsilon0 A / d, confirming the three formulas are really one idea viewed at different scales.
Key terms
- Capacitance
- The ratio C = Q/V, measuring stored charge per unit potential difference between two conductors.
- Gauss's law
- A law relating the electric flux through a closed surface to the enclosed charge, used here to find E in the gap.
- Isolated sphere
- A single charged conductor with no nearby partner, where infinity acts as the implicit second plate at zero potential.
- Coaxial cylinders
- Two cylindrical conductors sharing a central axis, with charge on the inner one and the outer one grounded or oppositely charged.
Watch it explained
- How it worksWalk me through getting the capacitance of two concentric spheres out of Gauss's law.
- Worked exampleFind the capacitance of a 2 m coaxial cable with inner radius 1 mm and outer radius 4 mm.
- Going furtherWhat does the spherical capacitor formula turn into when the gap becomes very thin?
- Worked exampleFind the capacitance of an isolated 10 cm metal sphere sitting alone in empty space.
Check yourself
End of topic.