Question

The capacitor of an oscillatory circuit ina container. when the container is evacuated, the resonance frequency of the circuit is $10kHz$. when the container is filled with a gas, the resonance frequency changes by $50Hz$. The dielectric constant of the gas is :

• A. $1.01$
• B. $2.001$
• C. $3.01$
• D. $1.002$

Answer 1

The correct option is D $1.002$

$f=\frac{1}{\sqrt{wc}}$

$f^1=\frac{1}{\sqrt{wkc}}$

$\frac{1}{\sqrt{wc}}-\frac{1}{\sqrt{wkc}}=50HZ.$

$\frac{1}{\sqrt{wc}}\left(1-\frac{1}{\sqrt{k}}\right)=50HZ$

$=10\times {10}^3\left(1-\frac{1}{\sqrt{k}}\right)=50$

$=1-\frac{1}{\sqrt{k}}=\frac{1}{1000}$

$=\frac{1}{\sqrt{k}}=1-\frac{1}{1000}=\frac{999}{1000}$

$\sqrt{k}=\frac{999}{1000}$

$k=\frac{1000}{999}\times \frac{1000}{999}$

$\approx 1.002$

Hence$,$ option $\left(D\right)$ is correct$.$