# Energy Stored in Capacitor

Let a parallel plate capacitor of capacitance $C$ has a charge $Q$ ($+Q$ on one plate and $-Q$ on another). The potential difference between its plate is given by
\begin{align}
V=\frac{Q}{C}.
\end{align}

The electrostatic energy stored in the capacitor is given by
\begin{align}
U &=\frac{1}{2}CV^2 \\
&=\frac{Q^2}{2C} \\
&=\frac{1}{2}QV.
\end{align}

The energy $U$ of an electrostatic field $E$ per unit volume $V$ is given by
\begin{align}
\frac{U}{V}=\frac{1}{2}\epsilon_0 E^2.
\end{align}

## Problems from IIT JEE

**Problem (IIT JEE 2002): **
Two identical capacitors have the same capacitance $C$. One of them is charged to potential $V_1$ and the other to $V_2$. The negative ends of the capacitors are connected together. When the positive ends are also connected, the decrease in energy of the combined system is,

- $\frac{1}{4}C\left(V_{1}^{2}-V_{2}^{2}\right)$
- $\frac{1}{4}C\left(V_{1}^{2}+V_{2}^{2}\right)$
- $\frac{1}{4}C\left(V_{1}-V_{2}\right)^2$
- $\frac{1}{4}C\left(V_{1}+V_{2}\right)^2$

**Solution:**
Initially, total electrostatic energy stored in two capacitors is $U_i=\frac{1}{2}CV_1^2+\frac{1}{2}CV_2^2$.

Initial charges on two capacitors are $q_1=CV_1$ and $q_2=CV_2$. Let $q_1^\prime$ and $q_2^\prime$ be charges on capacitors when same terminals of two capacitors are connected to each other (see figure). By charge conservation,
\begin{align}
q_1^\prime+q_2^\prime=q_1+q_2.
\end{align}
The potential $V^\prime$ across two capacitors is equal, i.e.,
\begin{align}
&q_1^\prime /C=q_2^\prime /C.
\end{align}
Solve above equations to get,
\begin{align}
q_1^\prime & =q_2^\prime \\
&=(q_1+q_2)/2, \text{and} \\
V^\prime &=(V_1+V_2)/2. \nonumber
\end{align}
Thus, final electrostatic energy stored in the two capacitors is,
\begin{align}
U_f &=\tfrac{1}{2}CV^2+\tfrac{1}{2}CV^2 \\
&=\tfrac{1}{4}C(V_1^2+V_2^2+2V_1V_2), \nonumber
\end{align}
and loss in energy is
\begin{align}
U_i-U_f=\frac{1}{4}C(V_1-V_2)^2.
\end{align}

## Related

- Capacitance
- Parallel plate capacitor
- Capacitors in series and parallel
- Charging and discharging of a capacitor