MHT CET · Physics · Capacitance

In a parallel plate air capacitor of plate separation \(d\), a dielectric slab of thickness \(t\) is introduced between the plates \((t < d)\). The capacitance becomes one-third of the original value. The dielectric constant of the slab will be

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

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{t}{2 d+t}\)

Step-by-step Solution

Detailed explanation

Equivalent Capacitance of the dielectric-air system is given by
\(\frac{1}{C^{\prime}}=\left(\frac{d-t}{\varepsilon_0 A}\right)+\left(\frac{t}{K \varepsilon_0 A}\right)=\frac{d-t\left(1-\frac{1}{K}\right)}{\varepsilon_0 A}\)
If new capacitance is given by one third of the old value:
\(\begin{aligned}
& C=\frac{\varepsilon_0 A}{d-t\left(1-\frac{1}{K}\right)}=\frac{\varepsilon_0 A}{3 d} \\
& \Rightarrow d-t+\frac{t}{K}=3 d \\
& \Rightarrow K=\frac{t}{2 d+t}
\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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