MHT CET · Physics · Alternating Current

A series LCR circuit with resistance \(R=500 \mathrm{ohm}\) is connected to an a.c. source of \(250 \mathrm{~V}\). When only the capacitance is removed, the current lags behind the voltage by \(60^{\circ}\). When only the inductance is removed, the current leads the voltage by \(60^{\circ}\). The impedance of the circuit is

A \(\frac{500}{\sqrt{3}} \Omega\)
B \(500 \sqrt{3} \Omega\)
C \(250 \Omega\)
D \(500 \Omega\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(500 \Omega\)

Step-by-step Solution

Detailed explanation

When capacitance is removed
\(
\begin{aligned}
& \tan \phi=\tan 60^{\circ}=\frac{X_L}{R} \\
& \therefore \frac{X_L}{R}=\sqrt{3} \text { or } X_L=R \sqrt{3}
\end{aligned}
\)
When inductance is removed,
\(
\begin{aligned}
& \frac{X_C}{R}=\tan 60^{\circ}=\sqrt{3} \\
& \therefore X_C=R \sqrt{3} \\
& \therefore X_C=X_L
\end{aligned}
\)
Impedance, \(\mathrm{Z}=\sqrt{\mathrm{R}^2+\left(\mathrm{X}_{\mathrm{L}}=\mathrm{X}_{\mathrm{C}}\right)^2}=\mathrm{R}=500 \Omega\)

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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