MHT CET · Physics · Alternating Current

An inductor of inductance \(2 \mu \mathrm{H}\) is connected in series with a resistance, a variable capacitor and an a.c. source of frequency 5 kHz . The value of capacitance for which maximum current is drawn into the circuit is \(\frac{1}{\mathrm{x}} \mathrm{F}\), where the value of ' \(x\) ' is (Take \(\pi^2=10\) )

A 500
B 1000
C 2000
D 4000

Verified Solution

Answer & Solution

Correct Answer

(C) 2000

Step-by-step Solution

Detailed explanation

For maximum current to be drawn the circuit should be in resonance,
\(\therefore \mathrm{X}_{\mathrm{L}}=\mathrm{X}_{\mathrm{C}} \quad \Rightarrow 2 \pi \mathrm{fL}=\frac{1}{2 \pi \mathrm{fC}}\)
\(\therefore \mathrm{C}=\frac{1}{4 \pi^2 \mathrm{f}^2 \mathrm{~L}}=\frac{1}{4 \times 10 \times 25 \times 10^6 \times 2 \times 10^{-6}}\)
\(\therefore \mathrm{C}=\frac{1}{2000} \mathrm{~F} \quad \Rightarrow \mathrm{x}=2000\)

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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