Capacitance

Source lecture(s): SC134 Lec6

Intuition

A capacitor stores charge by separating opposite charges onto two isolated conductors.

Formal Definition

\[C = \frac{Q}{\Delta V}\]

Mathematical Formulation

Parallel plate: \(C = \varepsilon_0 A/d\) Series: \(1/C_{\text{eq}} = \sum 1/C_i\) Parallel: \(C_{\text{eq}} = \sum C_i\) Energy: \(U = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\)

Derivation

Integrate \(dW = V\,dq\) to assemble charge from \(0\) to \(Q\).

Worked Example

\(10\,\mu\text{F}\) capacitor at 12 V stores \(U = 0.5\times10\mu\text{F}\times144 = 720\,\mu\text{J}\).

Common Mistakes

Assuming \(C\) depends on \(Q\) or \(V\) (it is geometry/material only).
Forgetting factor \(1/2\) in energy.

Quiz

Q1. Does capacitance depend on charge?

Answer

No—depends only on geometry and dielectric.