Question

Two parallel plate air capacitors of same capacity ' C ' are connected in parallel to a battery of e.m.f. ' $E$ '. Then one of the capacitors is completely filled with dielectric material of constant ' K '. The change in the effective capacity of the parallel combination is

• A. $\frac{\text{C}}{(\text{K}-1)}$
• B. $\frac{\text{KC}}{\text{K}-1}$
• C. $\text{KC} + 1$
• D. $\text{C}(\text{K} - 1)$

Hint:

Effective capacity in parallel is simply the sum of individual capacities.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When capacitors are connected in parallel, their equivalent capacity is the simple sum of their individual capacities.
Introducing a dielectric material into a capacitor increases its capacitance by a factor of the dielectric constant $K$.
We need to calculate the initial equivalent capacity and the final equivalent capacity to find the change.
Step 2: Key Formula or Approach:
Equivalent capacity in parallel: $C_{\text{eq}} = C_1 + C_2$.
Capacity with dielectric: $C' = KC$.
Change in capacity: $\Delta C_{\text{eq}} = C_{\text{eq, final}} - C_{\text{eq, initial}}$.
Step 3: Detailed Explanation:
Initial State:
Two capacitors, each of capacity $C$, are in parallel.
Initial effective capacity is: \[ C_{\text{initial}} = C + C = 2C \] Final State:
One of the capacitors is completely filled with a dielectric of constant $K$.
Its new capacity becomes $C' = KC$.
The other capacitor remains an air capacitor with capacity $C$.
These two are still in parallel. The new effective capacity is: \[ C_{\text{final}} = C + KC \] Change in Capacity:
The change in effective capacity is the difference between the final and initial states: \[ \Delta C = C_{\text{final}} - C_{\text{initial}} \] \[ \Delta C = (C + KC) - 2C \] Combine the terms with $C$: \[ \Delta C = KC - C \] Factor out the common term $C$: \[ \Delta C = C(K - 1) \] Step 4: Final Answer:
The change in effective capacity is $C(K - 1)$.