Question

An alternating voltage is applied to a series LCR circuit. If the current leads the voltage by $45^\circ$, then $(\tan 45^\circ = 1)$

• A. $X_{L}=X_{C}-R$
• B. $X_{L}=X_{C}+R$
• C. $X_{C}=\sqrt{X_{L}^{2}+R^{2}}$
• D. $X_{L}=\sqrt{X_{C}^{2}+R^{2}}$

Hint:

Physics Tip: When the current leads the voltage, the circuit is capacitive in nature ($X_C>X_L$). If the voltage leads the current, it is inductive ($X_L>X_C$).

Answer 1

The Correct Option is B

Concept: Physics (Alternating Current) - Phase Relationship in LCR Circuits.

Step 1: Recall the phase angle formula. In a series LCR circuit, the phase angle $\phi$ between the voltage and the current is given by the relation: $$ \tan \phi = \frac{X_{L} - X_{C}}{R} \text{} $$

Step 2: Substitute the given phase angle. The problem states that the current leads the voltage by $45^{\circ}$. Using the given value $\tan 45^{\circ} = 1$: $$ \tan 45^{\circ} = \frac{X_{L} - X_{C}}{R} \text{} $$ $$ 1 = \frac{X_{L} - X_{C}}{R} \text{} $$

Step 3: Rearrange to find the relation. Cross-multiplying gives: $$ R = X_{L} - X_{C} \text{} $$ $$ X_{L} = X_{C} + R \text{} $$ $$ \therefore \text{The correct relation is } X_{L} = X_{C} + R. \text{} $$