Question

LCR circuit, the resonance frequency of circuit increases two times of the initial circuit by changing $C$ and $C^{\prime}$ and $R$ from $100\, \Omega$ to $400\, \Omega$, while the inductance was kept the same. The ratio $C / C^{\prime}$, is

• A. 2
• B. 8
• C. 16
• D. 4

Answer 1

Answer: (D)

Given,
in series $L C R$ circuit,
initial capacitance, $C_{1}=C$
final value of capacitance, $C_{2}=C^{\prime}$
inductance, $L_{1}=L_{2}=L$
and resistance, $R_{1}=100 \,\Omega$
$R_{2}=400 \,\Omega$
$\therefore $ Resonance frequency,
$f_{1}=\frac{1}{2 \pi \sqrt{L_{1} C_{1}}} \Rightarrow f_{1}=\frac{1}{2 \pi \sqrt{L C}}$
and $f_{2}=\frac{1}{2 \pi \sqrt{L_{2} C_{2}}} \Rightarrow f_{2}=\frac{1}{2 \pi \sqrt{L \cdot C^{\prime}}}$
Given,$f_{2}=2 f_{1} \Rightarrow \frac{1}{2 \pi \sqrt{L C^{\prime}}}=\frac{2}{2 \pi \sqrt{L C}} $
$\frac{1}{\sqrt{C^{\prime}}}=\frac{2}{\sqrt{C}} \Rightarrow \frac{1}{C^{\prime}}=\frac{4}{C} \Rightarrow \frac{C}{C^{\prime}}=4$