Question

A certain capacitor is charged to a potential difference$\mathit{V}$. If you wish to increase its stored energy by 10%, by what percentage should you increase$\mathit{V}$?

Short answer

Percentage increase in potential difference V is 4.9%.

Step-by-step solution

The given data

Increased stored energy,

$$\begin{gathered} ∆U/U=10\% \\ =0.1 \end{gathered}$$

Understanding the concept of the stored energy

In the capacitor, electrostatic potential energy is stored. Therefore, it has to do with the charge and voltage between the capacitor's plates. Here, we need to use the formula of energy stored in the capacitor to find the percentage of the required increase in potential difference.

Formula:

The energy stored between the capacitors,$U=\frac{1}{2}C{V}^2$ …(i)

Step 3: Calculation of the increased percentage of potential difference

If the increase in the stored energy is by small amount $∆U$, then we must have to increase the potential difference by small amount$∆V$. Thus, the increased equation is given using equation (i) as follows:

$$\begin{gathered} \left(U+∆U\right)=\frac{1}{2}C{\left(V+∆V\right)}^2 \\ U\left(1+\frac{∆U}{U}\right)=\frac{1}{2}C{V}^2{\left(1+\frac{∆V}{V}\right)}^2 \\ U\left(1+\frac{∆U}{U}\right)=U{\left(1+\frac{∆V}{V}\right)}^2\left(\mathrm{from}\mathrm{equation}\left(\mathrm{i}\right)\right) \\ \sqrt{1+\frac{∆U}{U}}=1+\frac{∆V}{V} \\ \frac{∆V}{V}=\sqrt{1+\frac{∆U}{U}}-1 \\ =\sqrt{1+0.1}-1 \\ =\sqrt{1.1}-1 \\ =0.049 \\ =4.9\% \end{gathered}$$

Hence, the value of the increased percentage in the potential difference is 4.9%.