Combining capacitors and stored energy

Capacitors can be joined in two ways. The rules are the opposite of those for resistors.

• Series (one after another): every capacitor carries the same charge Q; the voltages add. 1/C = 1/C₁ + 1/C₂ + ... The total is smaller than the smallest one.
• Parallel (side by side): every capacitor has the same voltage V; the charges add. C = C₁ + C₂ + ...

Example: 2 µF and 3 µF. In series: 1/C = 1/2 + 1/3 = 5/6, so C = 1.2 µF. In parallel: C = 5 µF.

For a mixed network, simplify step by step from the far end, as with resistors, redrawing the circuit after each step.

Energy stored: charging a capacitor means pushing charge against the growing voltage. The stored energy is:

U = ½CV² = ½QV = Q²/(2C)

Choose the form that uses what stays fixed. Example: 4 µF at 10 V stores ½ × 4 × 10⁻⁶ × 100 = 2 × 10⁻⁴ J.

The energy lives in the electric field between the plates. Per unit volume, the energy density is u = ½ε₀E².

Sharing charge: when a charged capacitor is connected to an uncharged one, charge flows until both have the same voltage. The total charge stays the same, but some energy is lost, as heat and radiation in the connecting wires. Find the common voltage from V = (total Q) / (total C).