Question

In an LCR series a.c. circuit, the voltage across each of the components L, C and R is 60 V. The voltage across the LC combination is

• A. 120 V
• B. 60 V
• C. zero V
• D. $60\sqrt{3}$ V

Hint:

Whenever the voltage across the inductor equals the voltage across the capacitor ($V_L = V_C$), the circuit is operating at electrical resonance. At resonance, the reactive components cancel each other out completely, making the net voltage across the LC combination exactly zero.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the resultant voltage across the combination of the inductor ($L$) and the capacitor ($C$) in a series LCR circuit, where the individual voltage across each component is given as 60 V.

Step 2: Key Formula or Approach:
In a series LCR circuit, the alternating current $I$ is the same through all components. The phase relationships of the voltages with respect to the current are:
1. The voltage across the inductor ($V_L$) leads the current by $\frac{\pi}{2}$ ($90^\circ$).
2. The voltage across the capacitor ($V_C$) lags the current by $\frac{\pi}{2}$ ($90^\circ$).
Because $V_L$ and $V_C$ are in opposition ($180^\circ$ out of phase), the net voltage across the LC combination is given by the magnitude of their difference:
$$V_{LC} = |V_L - V_C|$$

Step 3: Detailed Explanation:
From the problem description, we are given that the root-mean-square voltages are:
$$V_L = 60\ \text{V}$$ $$V_C = 60\ \text{V}$$ Substituting these values into the phasor difference equation for the LC combination:
$$V_{LC} = |60\ \text{V} - 60\ \text{V}| = 0\ \text{V}$$ Because the two voltages are equal in magnitude and exactly opposite in phase, they completely cancel each other out.

Step 4: Final Answer:
The voltage across the LC combination is zero V, which corresponds to option (C).