Question

A parallel plate air capacitor consists of two metal plates of area \(A\) and separation \(d\). A slab of thickness \(t\) and dielectric constant \(2k\) is inserted between the plates with its faces parallel to the plates. The effective capacitance of the system is (\(\epsilon _{0}=\) permittivity of free space)

• A. \(\frac{\epsilon _{0}A}{(d-t)+\frac{t}{2k}}\)
• B. \(\frac{\epsilon _{0}A}{(d-t)-\frac{t}{k}}\)
• C. \(\frac{\epsilon _{0}A}{(2d-t)-\frac{t}{2k}}\)
• D. \(\frac{\epsilon _{0}A}{(d-2t)-\frac{t}{k}}\)

Hint:

A slab of thickness t and dielectric constant K replaces a gap t by an effective gap t/K.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
With a dielectric slab of thickness \(t\) and constant \(K\), the effective separation becomes \(d-t+\dfrac{t}{K}\).

Step 2: Substitute K
Here \(K=2k\), so the effective separation is \((d-t)+\dfrac{t}{2k}\).

Step 3: Capacitance
\(C=\dfrac{\epsilon_0A}{(d-t)+\dfrac{t}{2k}}\)

Final Answer:
The capacitance is epsilon naught A over d minus t plus t over 2k. \[ \boxed{\text{(A) }\dfrac{\epsilon_{0}A}{(d-t)+\frac{t}{2k}}} \]