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 area of each plate is 'A', the energy stored in the capacitor is

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

Hint:

Remember that \( Ad \) represents the volume of the space between the capacitor plates. Energy stored per unit volume (energy density) is a fundamental property of the electric field itself.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the expression of the total electrostatic energy stored in a parallel plate capacitor in terms of the internal electric field and physical dimensions.
Step 2: Key Formula or Approach:
1. Energy stored in a capacitor: \( U = \frac{1}{2} CV^2 \)
2. Capacitance of parallel plates: \( C = \frac{\epsilon_0 A}{d} \)
3. Potential difference: \( V = E \times d \)
Step 3: Detailed Explanation:
Substitute the expressions for \( C \) and \( V \) into the energy formula:
\[ U = \frac{1}{2} \left( \frac{\epsilon_0 A}{d} \right) (Ed)^2 \] \[ U = \frac{1}{2} \left( \frac{\epsilon_0 A}{d} \right) (E^2 d^2) \] \[ U = \frac{1}{2} \epsilon_0 E^2 Ad \] Alternatively, using the concept of energy density (\( u \)):
Energy density \( u = \frac{1}{2} \epsilon_0 E^2 \)
Total energy \( U = u \times \text{Volume} = \left( \frac{1}{2} \epsilon_0 E^2 \right) \times (Ad) \)
\[ U = \frac{1}{2} \epsilon_0 E^2 Ad \] Step 4: Final Answer:
The energy stored in the capacitor is \(\frac{1}{2} \epsilon_0 E^2 Ad\).