A series LCR circuit is given below:
The value of \( C_o \), for circuit to be in resonance is: (Given \( \pi^{2} = 10 \))
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1.5 μF
A series LCR circuit is in resonance when the inductive and capacitive reactances are equal, \( \omega^{2}LC_{\mathrm{eq}} = 1 \). From the figure: \( L = 5\,\mathrm{H} \), \( C = 0.5\,\mu\mathrm{F} \) with \( C_o \) connected in parallel with it, and the source has \( \omega = 100\pi\,\mathrm{rad\,s^{-1}} \).
\( \displaystyle C_{\mathrm{eq}} = \frac{1}{\omega^{2}L} = \frac{1}{(100\pi)^{2}\times5} = \frac{1}{10^{4}\times10\times5} = 2\times10^{-6}\,\mathrm{F} = 2\,\mu\mathrm{F} \)
Capacitors in parallel add:
\( C_{\mathrm{eq}} = C + C_o \Rightarrow C_o = 2\,\mu\mathrm{F} - 0.5\,\mu\mathrm{F} = 1.5\,\mu\mathrm{F} \)