In an LCR series circuit, the resonance frequency is $f_0$. If the capacitor is replaced by two capacitors of the same capacitance $C$ connected in parallel, the new resonance frequency will be:

Solution

1. The initial resonance frequency of the LCR circuit is given by $f_0=\frac{1}{2\pi \sqrt{LC}}$.
2. When two capacitors of capacitance $C$ are connected in parallel, the equivalent capacitance becomes $$\begin{array}{rl}{C}_{eq}& =C+C\\ & =2C\end{array}$$.
3. The new resonance frequency $f^{\prime }$ is calculated as $$\begin{array}{rl}{f}^{\prime }& =\frac{1}{2\pi \sqrt{L(2C)}}\\ & =\frac{1}{\sqrt{2}}\cdot \frac{1}{2\pi \sqrt{LC}}\end{array}$$.
4. Substituting the initial frequency, we get $f^{\prime }=\frac{f_0}{\sqrt{2}}$.
5. Hence the answer is (B).