MHT CET · Physics · Capacitance

Air capacitor has capacitance of \(1 \mu \mathrm{~F}\). Now the space between two plates of capacitor is filled with two dielectrics as shown in figure. The capacitance of the capacitor is [ \(\mathrm{d}=\) distance between two plates of capacitor, \(\mathrm{K}_1\) and \(\cdot \mathrm{K}_2\) are dielectric constants of first dielectric and second dielectric respectively]

A \(3 \mu \mathrm{~F}\)
B \(6 \mu \mathrm{~F}\)
C \(8 \mu \mathrm{~F}\)
D \(12 \mu \mathrm{~F}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(3 \mu \mathrm{~F}\)

Step-by-step Solution

Detailed explanation

Capacitance of a parallel plate capacitor,
\(\mathrm{C}=\frac{\varepsilon_0 \mathrm{~A}}{\mathrm{~d}}=1 \mu \mathrm{~F}\) ... (given)
After inserting the dielectrics,
\(\mathrm{C}_1=\mathrm{K}_1 \cdot \frac{\varepsilon_0 \mathrm{~A}}{2 \mathrm{~d}}=4 \times \frac{1}{2}=2 \mu \mathrm{~F}\)
... (given, \(\mathrm{K}_1=4\) )
\(\begin{array}{ll}
& \mathrm{C}_2=\mathrm{K}_2 \cdot \frac{\varepsilon_0 \mathrm{~A}}{2 \mathrm{~d}}=2 \times \frac{1}{2}=1 \mu \mathrm{~F} \\
\therefore \quad & \mathrm{C}_{\text {eff }}=\mathrm{C}_1+\mathrm{C}_2=3 \mu \mathrm{~F}
\end{array}\)
...(given, \(\mathrm{K}_2=4\) )

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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