Energy stored in a capacitor
A charged capacitor stores U = ½CV² = Q²/2C = ½QV. A 900 pF capacitor charged to 100 V stores ½ × 900 × 10⁻¹² × 100² = 4.5 × 10⁻⁶ J.
Capacitors
In series the charge is the same on each and 1/C = 1/C₁ + 1/C₂ + 1/C₃. In parallel the voltage is the same and C = C₁ + C₂ + C₃. Energy U = ½CV². Leave a box empty to use fewer capacitors.
In parallel9 × 10⁻⁴ μF
Charge0.09 μC
Energy4.5 μJ
- One capacitor: C = 9 × 10⁻⁴ μF
- Charge from the cell: Q = CV = 9 × 10⁻⁴ × 100 = 0.09 μC
- Energy stored: U = ½CV² = ½ × 9 × 10⁻⁴ × 100² = 4.5 μJ
Why it works
- While charging, at some moment the capacitor holds a charge q and the voltage across it is q/C.
- Moving a further small charge dq onto it takes work dW = (q/C) dq.
- Adding this up from 0 to the final charge Q gives W = Q²/2C.
- The work is stored as energy in the electric field between the plates: U = Q²/2C.
- With Q = CV this is also U = ½CV² = ½QV.
Energy stored in a capacitor: quick answers
Where is the energy of a capacitor stored?
In the electric field between its plates. Per unit volume the energy density is ½ε₀E².
Why is it ½QV and not QV?
The voltage rises from 0 to V while charging, so on average each bit of charge is moved through V/2.
What if the charged 900 pF capacitor is joined to an uncharged 900 pF one?
The charge shares equally and the voltage halves to 50 V. The energy falls to 2.25 × 10⁻⁶ J; the other half is lost as heat and radiation in the wires.