Exam Notes 4.1

Capacitance

A capacitor stores charge Q on two plates when a potential difference V is applied. Charge and potential difference are proportional. The capacitance C is the "charge stored per unit p.d." measured in Coulombs per Volt (Farads).

Q = VC

Capacitor Discharge

Capacitor discharge is exponential for voltage, charge, and current.

The time constant \tau = RC is the time it takes for V, Q, and I to decrease to 37% of their initial values.

V = V_0 e^{-\frac{t}{RC}}

Taking the natural log of both sides allows rearrangement to solve for t, R, or C. The gradient of a \ln(V) against t graph is \pm \frac{1}{RC}.

Capacitor Structure

A capacitor consists of two plates separated by a dielectric (insulating layer).

C = \frac{A \epsilon0 \epsilonr}{d}

Where:

• A is the area of the plates.

• d is the separation of the plates.

• \epsilon_0 is the permittivity of free space.

• \epsilon_r is the relative permittivity (dielectric constant).

A dielectric increases capacitance because its polar molecules align with the electric field, creating an opposing field, reducing potential difference V, allowing the same charge to be stored at a lower V. Removing a dielectric requires work, which increases the stored energy (if disconnected).

• If connected to a power supply, V is constant, and Q changes.

• If disconnected, Q is constant, and V changes.