An LCR series circuit containing resistance \( R_1 \), inductance \( L_1 \) and capacitance \( C_1 \) gives resonance at the same frequency \( f \) as a second similar combination \( R_2 \), \( L_2 \) and \( C_2 \). If the two circuits are connected in series, then the frequency of the combined circuit is.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( f \)
The resonant frequency of a series LCR circuit is \( \omega = 1/\sqrt{LC} \), independent of \( R \). Both circuits have the same \( \omega \), so
\( \displaystyle L_1 = \frac{1}{\omega^2 C_1}, \qquad L_2 = \frac{1}{\omega^2 C_2} \)
In series the inductances add and the reciprocals of the capacitances add:
\( \displaystyle L = L_1 + L_2 = \frac{1}{\omega^2}\left(\frac{1}{C_1} + \frac{1}{C_2}\right), \qquad \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} \)
\( \displaystyle L = \frac{1}{\omega^2 C} \;\Rightarrow\; \frac{1}{\sqrt{LC}} = \omega \)
So the combined circuit resonates at the same frequency \( f \).