An ideal parallel-plate capacitor having circular plates of diameter D that are a distance d apart stores energy U when it is connected across a fixed potential difference.If you want to triple the amount of energy stored in this capacitor by changing only the size of its plates,the diameter should be changed to

Question

Quizplus · Accepted Answer

The energy stored in a parallel-plate capacitor is given by:
$$ U = \frac{1}{2}\epsilon_0 \frac{\pi}{4} D^2 \frac{V^2}{d} $$
where $\epsilon_0$ is the permittivity of free space, $D$ is the diameter of the plates, $d$ is the distance between the plates, and $V$ is the potential difference between the plates.

To triple the energy stored, we need to increase $U$ by a factor of 3. Therefore, we can write:
$$ \frac{1}{2}\epsilon_0 \frac{\pi}{4} D^2 \frac{(3V)^2}{d} = 3U $$
Simplifying and solving for $D$, we get:
$$ D = \sqrt{3} \sqrt[4]{\frac{24U d}{\pi \epsilon_0 V^2}} $$
We want to change only the diameter of the plates, so $d$ and $V$ must be constant. Therefore, we can see that the only option that satisfies this condition is choice C), where $D = D\sqrt{3}$.

An Ideal Parallel-Plate Capacitor Having Circular Plates of Diameter D 3\sqrt { 3 }3​

An Ideal Parallel-Plate Capacitor Having Circular Plates of Diameter D $\sqrt { 3 }$