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) → $f$
Equivalent capacitance in series = $\frac{C_1 C_2}{C_1+C_2}$
Equivalent inductance in series = $L_1+L_2$
Frequency of combined circuit $\omega=\sqrt{\frac{1}{L C}}$
$=\sqrt{\frac{C_1+C_2}{C_1 C_2\left(L_1+L_2\right)}}$
$=\sqrt{\frac{C_1+C_2}{L_1 C_1 C_2+L_2 C_1 C_2}}$
As frequency of individual circuit is same $L_1 C_1=L_2 C_2$.
$\omega=\sqrt{\frac{C_1+C_2}{L_1 C_1 C_2+L_1 C_1 C_1}}$
$=\sqrt{\frac{C_1+C_2}{L_1 C_1\left(C_1+C_2\right)}}$
$=\sqrt{\frac{1}{L_1 C_1}}$
$\omega=\omega$ of individual circuit.
So, frequency is also same
(3) is correct ans.