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.
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): When the 50 Ω resistor is replaced by the 100 Ω resistor, the resonant angular frequency of the circuit does not change.
Reason (R): At resonance, the inductive reactance of the circuit is equal to its capacitive reactance.
In the light of the above statements, choose the most appropriate answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Both (A) and (R) are correct and (R) is the correct explanation of (A).
Assertion (A) When the 50 Ω resistor is replaced by the 100 Ω resistor, the resonant angular frequency of the circuit does not change. — True. The inductor and the capacitor are the same in both runs, and the resonant angular frequency depends only on them.
Reason (R) At resonance, the inductive reactance of the circuit is equal to its capacitive reactance. — True. Resonance is the condition \( X_L=X_C \).
Why Reason (R) explains Assertion (A)?
The condition \( X_L=X_C \) reads \( \omega_0L=1/\omega_0C \), which gives \( \omega_0=1/\sqrt{LC} \).
The resistance does not appear in this condition, so changing \( R \) from 50 Ω to 100 Ω leaves \( \omega_0 \) unchanged; only the height of the current peak changes.
NCERT (2026-27):
AC Voltage Applied to a Series LCR Circuit – “So if ω is varied, then at a particular frequency ω0, Xc = XL, and the impedance is minimum … This frequency is called the resonant frequency:”
\( \displaystyle X_c=X_L \quad \text{or} \quad \frac{1}{\omega_0C}=\omega_0L \quad \text{or} \quad \)\( \displaystyle \omega_0=\frac{1}{\sqrt{LC}} \)\( \qquad (7.28) \)