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 $LCR$ 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,
$f_1=\frac{1}{2\pi \sqrt{L_1{C}_1}}\Rightarrow f_1=\frac{1}{2\pi \sqrt{LC}}$
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,$f_2=2f_1\Rightarrow \frac{1}{2\pi \sqrt{L{C}^{\prime }}}=\frac{2}{2\pi \sqrt{LC}}$
$\frac{1}{\sqrt{C^{\prime }}}=\frac{2}{\sqrt{C}}\Rightarrow \frac{1}{C^{\prime }}=\frac{4}{C}\Rightarrow \frac{C}{C^{\prime }}=4$