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 solve this question, we need to determine the energy stored in a parallel plate capacitor. Let's go through the steps and reasoning:

1. Understanding the problem: We have a parallel plate capacitor with an electric field \( E \) between the plates. The distance between the plates is \( d \), and the area of each plate is \( A \). We need to find the energy stored in this capacitor.
2. Relevant Formula: The energy \( U \) stored in a capacitor is given by the formula: \(U = \frac{1}{2}CV^2\) where \( V \) is the potential difference across the capacitor. For parallel plate capacitors: \(C = \frac{\epsilon_0 A}{d}\) and \(V = Ed\) (since the electric field \( E = \frac{V}{d} \)).
3. Substituting the values: Now, using the expression for \( V \), we have: \(V = Ed\) Therefore: \(CV^2 = \left(\frac{\epsilon_0 A}{d} \right)(Ed)^2 = \epsilon_0 A E^2 d\)
4. Calculating the energy: Substituting the value of \( CV^2 \) in the energy formula: \(U = \frac{1}{2}\epsilon_0 A E^2 d\)
5. Conclusion: Thus, the energy stored in the capacitor is: \(\frac{1}{2}\epsilon_0E^2Ad\)
6. Incorrect Options:
   • \(\frac{E^2Ad}{\epsilon_0}\) suggests incorrect dependency on permittivity.
   • \(\frac{1}{2}\epsilon_0E^2\) lacks dependency on area \( A \) and distance \( d \).
   • \(\epsilon_0EAd\) lacks the factor \( \frac{1}{2} \) which comes from the energy formula.

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