Energy Stored in a Capacitor

When you charge a capacitor, you are moving charge from one plate to the other against an electric field that gets stronger as more charge piles up. The first bit of charge moves almost for free, since the plates start neutral and there's barely any voltage to push against. But every later bit of charge has to fight a bigger voltage, because V = q/C grows as q grows. So the work needed per unit charge keeps climbing the whole time you're charging it.

To get total work, you add up V dq over the whole charging process, from q = 0 up to the final charge Q. Since V = q/C, that integral is the integral of (q/C) dq from 0 to Q, which works out to Q²/(2C). Using Q = CV, this can also be written as (1/2)CV² or (1/2)QV. All three forms say the same thing: the energy stored is half of what you'd get by naively multiplying the final charge by the final voltage, because the voltage wasn't at its final value the whole time.

This result connects directly to the field picture of capacitors from earlier in the unit: that same energy can be found instead from the electric field filling the gap, which is where the idea of energy density in a field comes from. It also sets up what happens when you physically change a capacitor's geometry, since pulling plates apart or sliding in a dielectric changes C, and tracking whether Q or V stays fixed tells you whether the stored energy rises or falls.