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.
In a series LCR circuit if frequency of the ac source is increased from resonance value \( f_0 \) to \( 2f_0 \). Choose the statement depicting correct situation in the circuit.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Impedance of the circuit would increase
At the resonant frequency \( f_0 \), \( X_L = X_C \) and the impedance has its minimum value \( Z = R \). Away from resonance the reactances no longer cancel.
At \( 2f_0 \) the inductive reactance doubles and the capacitive reactance halves:
\( X_L' = 2X_L \)
\( X_C' = X_C/2 = X_L/2 \)
\( \displaystyle X_L' - X_C' = 2X_L - \frac{X_L}{2} = \frac{3}{2}X_L \ne 0 \)
\( \displaystyle Z' = \sqrt{R^{2} + \left(\frac{3}{2}X_L\right)^{2}} \gt R \)
So the impedance increases and the current falls.
Option (2) Potential difference across the resistance would increase — Incorrect. The current falls, so the potential difference \( I R \) across the resistance decreases.
Option (3) rms value of the current in the circuit will increase — Incorrect. \( I_{\mathrm{rms}} = V_{\mathrm{rms}}/Z \), and \( Z \) has increased, so the rms current decreases.
Option (4) Potential difference across the series combination of L & C will become zero — Incorrect. The voltages across L and C cancel only at resonance; at \( 2f_0 \) the combination has a net voltage \( I (X_L' - X_C') \ne 0 \).
NCERT (2026-27):
Resonance – “… So if ω is varied, then at a particular frequency ω0, Xc = XL, and the impedance is minimum
\( \displaystyle \left(Z = \sqrt{R^2 + 0^2} = R\right) \)
This frequency is called the resonant frequency: … At resonant frequency, the current amplitude is maximum; im = vm/R.”