Question

Two spherical conductors each of capacity $C$ are charged to potential $V$, and these are then connected by means of fine wire. What is the loss of energy?

Answer 1

Loss of energy:

1. Consider two capacitors having the capacitance of $C_1$ and$C_2$, and potential of $V_1$ and $V_2$ respectively.
2. It is found that the initial energy before combining the capacitors is greater than the final energy after combining the capacitors.
3. Hence, there is some energy loss because of combining the capacitors.
4. The energy loss can be expressed as follows:

$∆U=\frac{C_1{C}_2{\left(V_1+V_2\right)}^2}{2\left(C_1+C_2\right)}$

Here, $∆U$ is the energy loss due to the combining of capacitors.

5. Since the capacitance of both the capacitors is $C$, and potential is $V$, the energy loss can be calculated as follows:

$$\begin{gathered} ∆U=\frac{C\times C\times {\left(V+V\right)}^2}{2\left(C+C\right)} \\ ∆U=\frac{C^2\times 4V^2}{2C} \\ ∆U=2C{V}^2 \end{gathered}$$

Hence, the loss in energy is $2C{V}^2$.