MHT CET · Physics · Capacitance

Two identical capacitors have the same capacitance ' \(C\) '. One of them is charged to potential ' \(V_1\) ' and the other to \(V_2\). The negative ends of the capacitors are connected together. When positive ends are also connected, the decrease in energy of the combined system is

A \(\frac{1}{4} \mathrm{C}\left(\mathrm{V}_1-\mathrm{V}_2\right)^2\)
B \(\frac{1}{2} \mathrm{C}\left(\mathrm{V}_1^2+\mathrm{V}_2^2\right)\)
C \(\frac{1}{2} \mathrm{C}\left(\mathrm{V}_1^2-\mathrm{V}_2^2\right)\)
D \(\frac{1}{2} \mathrm{C}\left(\mathrm{V}_1+\mathrm{V}_2\right)^2\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{1}{4} \mathrm{C}\left(\mathrm{V}_1-\mathrm{V}_2\right)^2\)

Step-by-step Solution

Detailed explanation

\(
\mathrm{Q}_1=\mathrm{CV}_1, \mathrm{Q}_2=\mathrm{CV}_2, \mathrm{Q}=\mathrm{Q}_1+\mathrm{Q}_2=2 \mathrm{CV}
\)
After connection,
\(
\begin{aligned}
& \mathrm{CV}_1+\mathrm{CV}_2=2 \mathrm{CV} \\
& \therefore \mathrm{V}=\frac{\mathrm{V}_1+\mathrm{V}_2}{2}
\end{aligned}
\)
Decrease in energy \(=\frac{1}{2} \mathrm{CV}_1^2+\frac{1}{2} \mathrm{CV}_2^2-\frac{1}{2}(2 \mathrm{C})\left(\frac{\mathrm{V}_1+\mathrm{V}_2}{2}\right)^2\)
\(
=\frac{1}{4} \mathrm{C}\left(\mathrm{V}_1-\mathrm{V}_2\right)^2
\)

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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