AP® Physics C: Electricity and Magnetism · Unit 10 · Topic 10.3
Capacitance and the Parallel-Plate Capacitor
A capacitor stores separated charge on two nearby conductors, and its capacitance C = Q/V measures how much charge it holds per volt. For parallel plates, geometry alone fixes C = epsilon0 A / d.
Two metal plates never touch, but connect them to a battery and charge piles up anyway: positive on one plate, negative on the other, pulled there by the battery and held there by mutual attraction across the gap. No charge crosses the gap. It just separates and waits, and the energy needed to pull it apart gets stored in the electric field between the plates. That field is what gives the arrangement its name: a capacitor, something with a capacity to hold charge at a given voltage.
Make the plates large and close together and something nice happens: the field between them becomes uniform, pointing straight from positive plate to negative plate, with field lines only fringing near the edges. Treat each plate as an infinite sheet of charge with surface charge density sigma = Q/A. Each sheet produces a field of magnitude sigma/(2 epsilon0) pointing away from it; between the plates these two fields add, outside they cancel. That gives E = sigma/epsilon0 = Q/(epsilon0 A). Integrate E over the separation d to get the voltage: V = Ed = Qd/(epsilon0 A). Since capacitance is defined as C = Q/V, the charge Q cancels and C = epsilon0 A / d falls out, depending only on plate area and separation.
This sets up everything else in the unit: once you can find C from geometry, you can predict Q = CV for any applied voltage, and you can see how inserting a dielectric or changing plate spacing will change C before you ever touch the charge or voltage. Capacitance is the geometric constant that connects whatever charge and voltage a capacitor happens to have.
Key terms
- Capacitance
- The ratio C = Q/V, measuring how much charge a conductor pair stores per volt of potential difference.
- Parallel-plate capacitor
- Two flat conducting plates separated by a small gap, used to store charge and energy in a uniform electric field.
- Surface charge density (sigma)
- Charge per unit area on a plate, sigma = Q/A, which sets the strength of the field it produces.
- Uniform electric field
- A field with the same magnitude and direction everywhere, as found between closely spaced oppositely charged plates away from the edges.
Watch it explained
- How it worksHow does the field between two oppositely charged plates come out perfectly uniform?
- How it worksWalk me through deriving C equal to epsilon zero A over d for a parallel-plate capacitor.
- Worked exampleFind the capacitance of two 20 cm square plates separated by 1 mm of air.
- Common mistakeWhy doesn't a capacitor holding more charge on it have a larger capacitance?
Check yourself
End of topic.