Energy Stored in Capacitors: Formula, Derivation and Uses

A charged capacitor holds energy in the electric field between its plates. Charging means pushing charges together against their repulsion; the work done is stored and returned on discharge through a bulb, motor or flash tube. That is why a small battery, charging slowly, can still power a brilliant flash: the capacitor saves energy and releases it all at once.

Stored energy grows with the square of voltage, so voltage is the powerful lever, limited only by dielectric breakdown. Defibrillators, flashes, backup supplies and pulsed lasers are all designed around this storage formula.

Key Concepts

• Where energy lives: In the field in the dielectric, not on the plates.
• Why one-half appears: Voltage grows from 0 to V during charging, so the average voltage is V divided by 2; work equals average voltage times total charge.
• Energy density u: Energy per unit volume of field, set by field strength and medium.
• Dielectric insertion: At fixed voltage, energy rises with K. At fixed charge, energy falls. The battery or your hand supplies or absorbs the difference.

Formulas

• Stored energy: U equals one half C V squared equals Q squared divided by (2C) equals one half Q V. C in farads, V in volts, Q in coulombs, U in joules. All three forms are identical.
• Energy density: u equals one half epsilon0 E squared (times K with a dielectric). E in V/m, u in joules per cubic metre. Assumes a uniform field.

Worked Example

A 10 microfarad capacitor is charged to 400 V. Find U and Q, then U at 800 V.

1. C equals 1.0 x 10^-5 F. V squared equals 160,000.
2. U equals 0.5 x 1.0 x 10^-5 x 160,000 equals 0.80 J.
3. Q equals C V equals 1.0 x 10^-5 x 400 equals 4.0 x 10^-3 C. Check: one half Q V equals 0.5 x 4.0 x 10^-3 x 400 equals 0.80 J. Matches.
4. Doubling V quadruples V squared, so U at 800 V equals 4 x 0.80 equals 3.2 J.

Common Misconceptions

• Misconception: U equals Q V. Correction: It is one half Q V, because voltage builds up during charging.
• Misconception: Doubling charge doubles energy. Correction: U equals Q squared divided by 2C, so doubling Q quadruples U.
• Misconception: A dielectric always increases stored energy. Correction: Only at fixed voltage. Isolated (fixed Q), energy falls by factor K and the field pulls the slab in.

Real-World Applications

• Defibrillators: Roughly 360 J released through the chest in milliseconds to restore heart rhythm.
• Camera flash: Slow battery charging, microsecond bright discharge.
• Supercapacitors: Bridging power cuts in servers, buses and regenerative braking.
• Pulsed power: Capacitor banks drive lasers and particle accelerators with huge brief pulses.

U equals one half C V squared: three equivalent forms, one message. Raise voltage and energy climbs steeply; that stored field energy is what flashes, defibrillators and pulse machines spend in an instant.