Energy stored by a capacitor

The energy (measured in Joules) stored in a capacitor is equal to the work done to charge it. Consider a capacitance C, holding a charge +q on one plate and -q on the other. Moving a small element of charge  d q {\displaystyle \mathrm {d} q} from one plate to the other against the potential difference V = q/C requires the work  d W {\displaystyle \mathrm {d} W} :

d W = q C d q {\displaystyle \mathrm {d} W={\frac {q}{C}}\,\mathrm {d} q}

where

W is the work measured in joules

q is the charge measured in coulombs

C is the capacitance, measured in farads

We can find the energy stored in a capacitance by integrating this equation. Starting with an uncharged capacitance (q=0) and moving charge from one plate to the other until the plates have charge +Q and -Q requires the work W:

W c h a r g i n g = ∫ 0 Q q C d q = 1 2 Q 2 C = 1 2 C V 2 = W s t o r e d {\displaystyle W_{charging}=\int _{0}^{Q}{\frac {q}{C}}\,\mathrm {d} q={\frac {1}{2}}{\frac {Q^{2}}{C}}={\frac {1}{2}}CV^{2}=W_{stored}}

Combining this with the above equation for the capacitance of a flat-plate capacitor, we get:

W s t o r e d = 1 2 C V 2 = 1 2 ϵ A d V 2 {\displaystyle W_{stored}={\frac {1}{2}}CV^{2}={\frac {1}{2}}\epsilon {\frac {A}{d}}V^{2}} .

where

W is the energy measured in joules

C is the capacitance, measured in farads

V is the voltage measured in volts