Question

An air-filled capacitor of capacitance \( C \) is filled with dielectric (\( k = 3 \)) of width \( \frac{d}{3} \), where \( d \) is the separation between plates. The new capacitance is

• A. \( \frac{9}{5}C \)
• B. \( \frac{5}{4}C \)
• C. \( \frac{4}{3}C \)
• D. \( \frac{9}{7}C \)

Hint:

When a dielectric is inserted between the plates of a capacitor, the capacitance increases according to the dielectric constant and the geometry of the plates.

Answer 1

The Correct Option is D

Step 1: Use the formula for capacitance.
The capacitance of a parallel plate capacitor is given by: \[ C = \frac{\varepsilon_0 A}{d} \] Where: \( \varepsilon_0 \) = permittivity of free space, \( A \) = area of the plates, and \( d \) = separation between the plates.
Step 2: Calculate the new capacitance.
The new capacitance when filled with dielectric is: \[ C_{\text{new}} = \frac{\varepsilon_0 A}{d + 6d} = \frac{9\varepsilon_0 A}{7d} \] Thus, the new capacitance is \( \frac{9}{7}C \), which corresponds to option (4).
Step 3: Conclusion.
The new capacitance is \( \frac{9}{7}C \).