AP Physics C E&M- 10.3 Capacitors - Study Notes- New Syllabus

Parallel-Plate Capacitor

A parallel-plate capacitor consists of two conducting plates of area \(A\), separated by a distance \(d\), with a dielectric material (or vacuum) between them. It is used to store electric charge and energy.

Capacitance:

Capacitance relates the magnitude of charge stored on each plate to the potential difference across the plates:

\( \mathrm{C = \dfrac{Q}{V}} \)

For a parallel-plate capacitor with a dielectric:

\( \mathrm{C = \kappa \varepsilon_0 \dfrac{A}{d}} \)

• \(\mathrm{Q}\) = charge on one plate
• \(\mathrm{V}\) = potential difference between plates
• \(\mathrm{A}\) = plate area
• \(\mathrm{d}\) = plate separation
• \(\mathrm{\varepsilon_0 = 8.85 \times 10^{-12} \, C^2/(N \cdot m^2)}\)
• \(\mathrm{\kappa}\) = dielectric constant of material between plates

Electric Field Between Plates:

The electric field between the plates (ignoring edge effects) is uniform:

\( \mathrm{E = \dfrac{V}{d} = \dfrac{\sigma}{\varepsilon}} \)

• \(\mathrm{\sigma = Q/A}\) = surface charge density
• \(\mathrm{\varepsilon = \kappa \varepsilon_0}\) = permittivity of dielectric

Electric Potential Energy:

A charged capacitor stores energy equal to the work required to separate charges onto the plates:

\( \mathrm{U = \tfrac{1}{2} C V^2 = \tfrac{1}{2} Q V = \dfrac{Q^2}{2C}} \)

Summary of Properties:

• Capacitance: proportional to plate area, inversely proportional to separation.
• Dielectric increases capacitance by factor \(\kappa\).
• Capacitance defined by \( \mathrm{C = Q/V} \).
• Electric field between plates is uniform (except at edges).
• Capacitor stores electrical potential energy in the separated charges.