能量存储在电容器中

A capacitor has the ability to store charge, so it should be no surprise that a capacitor also has the ability to store energy. But how much energy can a capacitor store? To determine this, we can see how much energy it would take to charge a capacitor.

Suppose we have a capacitor of capacitance $C,$ and we wish to charge it up to a charge of $Q.$ We can break up the total charge $Q$ into small charge elements $dq,$ and then transfer the charge from one plate to the other one element at a time. From this, we can then see how much energy is needed to add each charge element.

Suppose that the capacitor already has a charge $q$ on it. By the definition, the potential difference across the capacitor is $\frac{q}{C}.$ Now, we move charge $dq$ from one plate to the other. Since the energy is simply the potential difference times the charge, we get: $$dE = \frac{q}{C}dq.$$ Now, all we need to do is integrate from $0$ to $Q$: $$E = \int_{0}^{Q}\frac{q}{C}dq$$ $$E = \frac{Q^2}{2C}$$ This is the energy stored in a capacitor! We usually express this in a number of different forms for our convenience. Let $V$ be the potential difference across the fully charged capacitor. Using the fact that $C = \frac{Q}{V}$ by the definition of capacitance, we get: $$E = \frac{Q^2}{2C} = \frac{1}{2}QV = \frac{1}{2}CV^2.$$