Question

Capacitance of an air filled parallel plate capacitor is C. Find capacitance when a dielectric of dielectric constant K is field in it.

Hint:

Inserting a dielectric always increases capacitance by a factor of $K$. This is a fundamental property to memorize for capacitor problems.

Answer 1

Step 1: Understanding the Concept:
A capacitor stores electrical energy in an electric field. The capacitance of a parallel plate capacitor depends on the geometry of the plates (area \(A\) and separation \(d\)) and the nature of the insulating material (dielectric) between them. Introducing a dielectric material increases the capacitance.
Step 2: Key Formula or Approach:
For an air-filled parallel plate capacitor (assuming air $\approx$ vacuum), the capacitance is:
\[ C = \frac{\epsilon_0 A}{d} \]
When a dielectric material completely fills the space between the plates, the permittivity of the medium changes from \(\epsilon_0\) to \(\epsilon = K \epsilon_0\), where \(K\) is the dielectric constant (relative permittivity).
Step 3: Detailed Explanation:
Initial capacitance with air:
\[ C = \frac{\epsilon_0 A}{d} \quad \text{--- (Equation 1)} \]
When a dielectric of constant \(K\) is introduced to completely fill the gap, the new capacitance \(C'\) is determined by substituting \(\epsilon_0\) with \(K\epsilon_0\):
\[ C' = \frac{(K\epsilon_0) A}{d} \]
Rearranging the terms:
\[ C' = K \cdot \left( \frac{\epsilon_0 A}{d} \right) \]
Substitute Equation 1 into this new expression:
\[ C' = K \cdot C \]
The capacitance increases by a factor equal to the dielectric constant.
Step 4: Final Answer:
The new capacitance is \(K \cdot C\).