MHT CET · Physics · Capacitance

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

A $\frac{C}{2} [\frac{K - 1}{K + 1}]$
B $\frac{C}{4} [\frac{K - 1}{K + 1}]$
C $\frac{C}{2} [\frac{K + 1}{K - 1}]$
D $\frac{C}{2} {[\frac{K - 1}{K + 1}]}^{2}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{C}{2} [\frac{K - 1}{K + 1}]$

Step-by-step Solution

Detailed explanation

\frac{1}{C_{1}} = \frac{1}{C} + \frac{1}{C} = \frac{2}{C} \Rightarrow C_{1} = \frac{C}{2}

\frac{1}{C_{2}} = \frac{1}{C} + \frac{1}{K C}

\frac{1}{C_{2}} = \frac{1}{C} [1 + \frac{1}{K}]

∴ C_{2} = \frac{C}{(1 + \frac{1}{K})} = \frac{C K}{(K + 1)}

∆ C = \frac{C K}{(K + 1)} - \frac{C}{2} = C [\frac{K}{K + 1} - \frac{1}{2}]

= C [\frac{2 K - K - 1}{2 (K + 1)}] = \frac{C}{2} [\frac{K - 1}{K + 1}]

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.

Same subject

Explore more questions on app

From MHT CET

Explore more questions on app

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

Step-by-step Solution

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