MHT CET · Physics · Alternating Current

An inductance coil has a resistance of \(100 \Omega\). When are a.c. signal of frequency \(100 \mathrm{Hzis}\) applied to the coil, the voltage leads the current by \(45^{\circ}\). The inductance of the coil in henry is \(\left[\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}\right]\)

A \(\frac{1}{\pi}\)
B \(\frac{5}{2 \pi}\)
C \(\frac{2}{\pi}\)
D \(\frac{1}{2 \pi}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{1}{2 \pi}\)

Step-by-step Solution

Detailed explanation

The phase angle in an LR circuit is given by, since \(\phi=45^{\circ}\),
\(\tan \phi=\frac{X_L}{X_R}=1\)
Resistance of inductive coil \(X_L=X_R=100 \Omega\)
Thus, inductance of coil \(L=\frac{X_L}{2 \pi f}\)
\(\therefore L=\frac{100}{2 \pi \times 100 \mathrm{~s}^{-1}}=\frac{1}{2 \pi} \mathrm{H}\)

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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