KCET · Physics · Capacitance

A capacitor of capacitance \(5 \mu \mathrm{~F}\) is charged by a battery of emf 10 V . At an instant of time, the potential difference across the capacitors is 4 V and the time rate of change of potential difference across the capacitor is \(0.6 \mathrm{Vs}^{-1}\). Then, the time rate at which energy is stored in the capacitor at any instant is

A \(12 \mu \mathrm{~W}\)
B \(3 \mu \mathrm{~W}\)
C Zero
D \(30 \mu \mathrm{~W}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(12 \mu \mathrm{~W}\)

Step-by-step Solution

Detailed explanation

Given, battery emf, \(e=10 \mathrm{~V}\)
\(C=5 \mu \mathrm{~F}=5 \times 10^{-6} \mathrm{~F}\)
\(V=4 \mathrm{~V} \Rightarrow d V / d t=0.6 \mathrm{Vs}^{-1}\)
We know that stored energy,
\(U=\frac{1}{2} C V^2\)
\(\therefore \quad \frac{d U}{d t}=\frac{1}{2} C \cdot 2 V \frac{d V}{d t}=C V \frac{d V}{d t}\)
\(=5 \times 10^{-6} \times 4 \times 0.6\)
\(=12 \times 10^{-6} \mathrm{~W}=12 \mu \mathrm{~W}\)

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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