Read the following passage and answer the questions.
A student studies a series LCR circuit like the tuning circuit of a radio. She connects a 50 Ω resistor, a 20 mH inductor and a 50 μF capacitor in series with an ac source. The rms voltage of the source stays at 10 V. Its angular frequency \(\omega\) can be changed over a wide range. An ac ammeter in the circuit reads the rms current. An ac voltmeter can be joined across each element in turn. She starts with \(\omega\) at a low value and raises it slowly. At each step she notes the ammeter reading and the three voltmeter readings. She then repeats the whole run with a 100 Ω resistor in place of the 50 Ω one. The inductor, the capacitor and the source stay the same.
Consider the following statements about the circuit:
(A) With the 50 Ω resistor, the rms current at resonance is 0.2 A.
(B) At resonance, the power factor of the circuit is zero.
(C) At angular frequencies below the resonant value, the circuit is predominantly inductive.
(D) The circuit would show no resonance if the capacitor were taken out of it.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (D) only
When \( X_L=X_C \), the impedance of the circuit equals \( R \). The current is then in phase with the source voltage.
(A) With the 50 Ω resistor, the rms current at resonance is 0.2 A. — True. At resonance the current is fixed by the resistance alone:
\( \begin{aligned} I &= \frac{V}{R} \\ &= \frac{10}{50} \\ &= 0.2\ \mathrm{A} \end{aligned} \)
(B) At resonance, the power factor of the circuit is zero. — False. At resonance \( \phi=0 \), so the power factor \( \cos\phi=1 \) and the power dissipated in the resistor is the largest.
(C) At angular frequencies below the resonant value, the circuit is predominantly inductive. — False. Below \( \omega_0 \), \( X_C=1/\omega C \) is larger than \( X_L=\omega L \), so the circuit is predominantly capacitive and the current leads the voltage.
(D) The circuit would show no resonance if the capacitor were taken out of it. — True. Resonance needs both an inductor and a capacitor, so that their voltages, opposite in phase, can cancel. An RL circuit has no frequency at which this happens.
NCERT (2026-27):
AC Voltage Applied to a Series LCR Circuit – “At resonant frequency, the current amplitude is maximum; im = vm/R.”
Power in AC Circuit: the Power Factor – “Case (iv) … At resonance Xc – XL = 0, and φ = 0. Therefore, cosφ = 1 and P = I2Z = I2 R. …”
Phasor-diagram solution – “If XC > XL, φ is positive and the circuit is predominantly capacitive. Consequently, the current in the circuit leads the source voltage. …”
AC Voltage Applied to a Series LCR Circuit – “It is important to note that resonance phenomenon is exhibited by a circuit only if both L and C are present in the circuit. … This means that we cannot have resonance in a RL or RC circuit.”