A series LCR circuit is subjected to an ac signal of $$200$$ V, $$50$$ Hz. If the voltage across the inductor $$(L = 10 \text{ mH})$$ is $$31.4$$ V, then the current in this circuit is :

We are given a series LCR circuit with an AC signal of $$200$$ V and $$50$$ Hz. The inductor has inductance $$L = 10$$ mH, and the voltage across the inductor is $$V_L = 31.4$$ V. We need to find the current in the circuit.

Recall that the voltage across an inductor in an AC circuit is related to the current by $$V_L = I \times X_L$$, where $$X_L$$ is the inductive reactance.

The inductive reactance is given by the formula $$X_L = \omega L = 2\pi f L$$. Substituting the given values ($$f = 50$$ Hz, $$L = 10 \text{ mH} = 10 \times 10^{-3}$$ H) yields $$X_L = 2\pi \times 50 \times 10 \times 10^{-3}$$, so $$X_L = 2\pi \times 0.5 = \pi \approx 3.14 \, \Omega$$.

Using the relationship $$V_L = I \times X_L$$ and solving for $$I$$ gives $$I = \frac{V_L}{X_L} = \frac{31.4}{3.14} = 10 \text{ A}$$.

The correct answer is Option (3): 10 A.