A series LCR circuit connected to an AC source with voltage of the source \( v = v_m\sin\omega t \).
If ‘q’ is the charge on the capacitor and ‘i’ is the current, from Kirchhoff’s loop rule :
\( L\displaystyle\frac{di}{dt} + iR + \frac{q}{c} = V \)
The current in the circuit is given by \( i = I_m\sin(\omega t + \phi) \) where \( \phi \) is the phase difference between the voltage across the source and current in the circuit.
We know \( V_{Rm} = I_m R \); \( V_{Lm} = I_m X_L \); \( V_{Cm} = I_m X_C \); and
\( X_L = \omega L \); \( X_C = \displaystyle\frac{1}{\omega C} \)
Total impedance in the circuit regulates current. At resonance frequency of the LCR circuit current in the circuit is maximum.
A series LCR circuit has resonant frequency of \( \omega_0 \). If inductance in the circuit is replaced by another one having the same value of inductance but higher value of internal resistance. Select the statement that explains the change in functioning of the circuit correctly.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Quality factor of the circuit will decrease
The internal resistance of the inductor is in series with R, so the new inductor keeps L the same but increases the total resistance of the circuit. Each quantity is checked with its formula.
\( \displaystyle \omega_0 = \frac{1}{\sqrt{LC}} \qquad I_{0} = \frac{V_m}{R} \qquad Q = \frac{1}{R}\sqrt{\frac{L}{C}} \qquad \Delta\omega = \frac{R}{L} \)
Option (1) Resonant frequency of the circuit will decrease — Incorrect. \( \omega_0 \) depends only on L and C, which are unchanged, so the resonant frequency stays the same.
Option (2) Amplitude of current at the resonant frequency will increase — Incorrect. At resonance \( Z = R \); a larger R gives a smaller current amplitude, not a larger one.
Option (3) Quality factor of the circuit will decrease — Correct. The quality factor is inversely proportional to R, so a larger resistance lowers it and the resonance becomes less sharp.
Option (4) Bandwidth of the circuit will decrease — Incorrect. The bandwidth \( R/L \) increases with R, so it becomes wider, not narrower.
NCERT (2026-27):
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