Question

A parallel plate capacitor has a uniform electric field ‘ E → ’ in the space between the plates. If the distance between the plates is ‘d’ and the area of each plate is ‘A’, the energy stored in the capacitor is : (ε₀=permittivity of free space)

• A. \(\frac{E^2Ad}{\epsilon_0}\)
• B. \(\frac{1}{2}\epsilon_0E^2\)
• C. \(\epsilon_0EAd\)
• D. \(\frac{1}{2}\epsilon_0E^2Ad\)

Answer 1

The Correct Option is D

To find the energy stored in a parallel plate capacitor with a uniform electric field \(\textbf{E}\) between the plates, we need to understand the relationship between the electric field, the charge, and the potential difference.

The relevant parameters given are: 

• The electric field \(\textbf{E}\).
• The distance between the plates: \(d\).
• The area of each plate: \(A\).
• Permittivity of free space: \(\epsilon_0\).

Step 1: Understand the potential difference.

The potential difference \(V\) between the plates is related to the electric field \(E\) and the distance \(d\) by:

\(V = E \cdot d\)

Step 2: Calculate the charge on the capacitor.

The capacitance \(C\) of a parallel plate capacitor is given by the formula:

\(C = \frac{\epsilon_0 \cdot A}{d}\)

Therefore, the charge \(Q\) on the capacitor can be related to the capacitance and potential difference by:

\(Q = C \cdot V = \frac{\epsilon_0 \cdot A}{d} \cdot E \cdot d = \epsilon_0 \cdot A \cdot E\)

Step 3: Calculate the energy stored in the capacitor.

The energy \(U\) stored in a capacitor is given by:

\(U = \frac{1}{2} \cdot C \cdot V^2\)

Substitute the expressions for \(C\) and \(V\):

\(U = \frac{1}{2} \cdot \frac{\epsilon_0 \cdot A}{d} \cdot (E \cdot d)^2\)

\(= \frac{1}{2} \cdot \epsilon_0 \cdot A \cdot E^2 \cdot d\)

Thus, the energy stored in the capacitor is:

\(U = \frac{1}{2} \epsilon_0 E^2 A d\)

Conclusion:

The correct option for the energy stored in the capacitor is:

\(\frac{1}{2}\epsilon_0E^2Ad\)

Hence, the correct answer is the fourth option: \(\frac{1}{2}\epsilon_0E^2Ad\).