Question

Keeping the voltage of the charging source constant, what would be the percentage change in the energy stored in a parallel plate capacitor if the separation between its plates were to be decreased by 10%?

Answer 1

The energy which is stored in a capacitor may be defined as the function of the voltage across the capacitor and the value of capacitance. And the capacitance C is inversely proportional to the distance between the parallel plates.
Also given that the separation distance between two plates is decreased by 10%. Hence calculate the capacitance in the two cases. That is, in the initial case and when the distance between the plates is decreased by 10%. By taking their ratio we will get the new capacitance. Then by substituting the value of initial and new capacitance we will get the change stored in a parallel plate capacitor.

Complete step-by-step solution: -
Energy can be calculated using the equation,
$E=\dfrac{1}{2}C{{V}^{2}}$……………..(1)
Then the capacitance C is given by,
$C=\dfrac{{{\varepsilon }_{0}}A}{d}$………………….(2)
From this equation it is clear that capacitance C is inversely proportional to the distance between the parallel plates.
Since the separation distance between two plates is decreased by 10%. Then,
${{C}_{1}}=\dfrac{{{\varepsilon }_{_{0}}}A}{{{d}_{1}}}$………………….(3)
$\begin{aligned} & {{d}_{1}}=d-0.1d \\\ & {{d}_{1}}=0.9d \\\ \end{aligned}$
Dividing equation (3) by equation (2) we get,
$\dfrac{{{C}_{1}}}{C}=\dfrac{d}{{{d}_{1}}}$
$\dfrac{{{C}_{1}}}{C}=\dfrac{d}{0.9d}$
$\therefore {{C}_{1}}=1.11C$
Then the change in energy is,
${{E}_{1}}-E=\dfrac{1}{2}\times C{{V}^{2}}\left( 1.11-1 \right)$
${{E}_{1}}-E=\dfrac{1}{2}\times C{{V}^{2}}\left( 0.11 \right)$
${{E}_{1}}-E=\left( 0.11 \right)\times E$
Thus the percentage change in energy is 11%.

Note: The energy which is stored in a capacitor may be defined as the function of the voltage across the capacitor. The voltage will be higher when it is in a parallel connection so we can say that in parallel connection it can store more energy than in a series connection.