Question

For the LCR AC-circuit (resonance frequency ω_o) shown in the figure below, choose the correct statement(s).

• A. ω_o depends on the values of L, C, and R
• B. At ω= ω_o, voltage V_R and current I are in-phase
• C. The amplitude of VR at ω=ω_o/2 is independent of R
• D. The amplitude of V_R at ω= ω_o is independent of L and C

Answer 1

The Correct Option is B, D

1. Resonance in an LCR Circuit

The resonance frequency is given by: 

ω₀ = 1 / √(LC)

This depends only on L and C, not on R. Hence, statement (A) is incorrect.

2. Voltage and Current at Resonance (ω = ω₀)

• At resonance, the impedance of the circuit is purely resistive (Z = R), as the inductive reactance X_L = ω₀L cancels out the capacitive reactance X_C = 1 / (ω₀C).
• In a purely resistive circuit, the voltage across R (V_R) and the current I are always in-phase.

Hence, statement (B) is correct.

3. Amplitude of V_R at ω = ω₀/2

The total impedance at ω = ω₀/2 depends on all components L, C, and R. Thus, the amplitude of V_R is not independent of R.

Hence, statement (C) is incorrect.

4. Amplitude of V_R at ω = ω₀

At resonance:

V_R = I · R

where:

I = V_in / R

Thus, V_R depends on R, but is independent of L and C. Hence, statement (D) is correct only for L and C, but this statement is not explicitly true in this context.

5. Conclusion

• (A) is incorrect because ω₀ does not depend on R.
• (B) is correct because voltage V_R and current I are in-phase at resonance.
• (C) is incorrect because V_R at ω = ω₀/2 depends on R.
• (D) is correct because the dependency of V_R at ω = ω₀ on R is implied but incomplete.