Answer to Question #202054 in Electric Circuits for Kai

Question

Let the energy stored in the capacitor be U. Show that U is given by the expression: U=½Q²/C

[Hint: The power P in the capacitor is given by P=dV/dt =IV where I = dQ/dt. This is a simple exercise on integration.]

Answer 1

Explanations & Calculations

$\qquad\qquad \begin{aligned} \small P &=\small\frac{dU}{dt}=Vi\\ \small \frac{dU}{dt}&=\small V \frac{dQ}{dt}\\ \small dU&=\small V dQ\\ &=\small \frac{Q}{C}\cdot dQ\\ \small \int_0^U dU&=\small \frac{1}{C}\cdot\int_0^Q Q dQ\\ U&=\small \frac{Q^2}{2C} \end{aligned}$ for a typical capacitor.