Question

In a parallel plate air capacitor of plate separation '\(d\)', a dielectric slab of thickness '\(t\)' is introduced between the plates. The capacitance becomes one-third of the original value. The dielectric constant of the slab will be

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

Hint:

With a slab, C = e0 A / (d - t + t/K).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
With a dielectric slab of thickness \(t\) and constant \(K\) between plates a distance \(d\) apart, the capacitance is
\[ C' = \frac{\varepsilon_0A}{d - t + \frac tK} \]
Without the slab, \(C = \frac{\varepsilon_0A}{d}\).

Step 2: Use the given ratio:
\(C' = \frac{C}{3}\) means \(d - t + \frac tK = 3d\).

Step 3: Solve for K:
\[ \frac tK = 3d - d + t = 2d + t \Rightarrow K = \frac{t}{2d + t} \]

Step 4: Why the other options are wrong.
\(\frac{t}{d+t}\) would correspond to a capacitance ratio of \(\frac12\). The other two forms do not satisfy \(d - t + \frac tK = 3d\).

Final Answer:
The dielectric constant is \(\frac{t}{2d + t}\), option (B). \[ \boxed{\frac{t}{2d+t}} \]