# A capacitor of an oscillating circuit is enclosed in a container. When the container is...

## Question:

A capacitor of an oscillating circuit is enclosed in a container. When the container is evacuated, the frequency is 10 kHz. When the container is filled with a gas, the frequency changes by 50 Hz. The dielectric constant of gas is _____.

## Dielectric material:

Dielectric is a material used for the electrical obstruction. The use of dielectric is to increase the charges on the surface of capacitor's plate when placed in an electric field. Dielectric is also used to separate the positive charges and negative charges.

Given data

• Frequency of the circuit is {eq}f = 10\;{\rm{kHz}}.{/eq}
• Change in frequency is {eq}\Delta f = 50\;{\rm{Hz}}.{/eq}
• Dielectric constant of the gas is {eq}K.{/eq}

The expression for the frequency is,

{eq}f = \dfrac{1}{{2\pi \sqrt {LC} }} {/eq}

Substituting the given values,

{eq}{10^4} = \dfrac{1}{{2\pi \sqrt {LC} }}......\left( 1 \right) {/eq}

When the container is filled with gas, the expression for the frequency becomes,

{eq}f + \Delta f = \dfrac{1}{{2\pi \sqrt {\dfrac{{LC}}{K}} }} {/eq}

Substituting the given values,

{eq}{10^4} + 50 = \dfrac{1}{{2\pi \sqrt {\dfrac{{LC}}{K}} }}......\left( 2 \right){/eq}

On dividing expression 2 by expression 1,

{eq}\begin{align*} \sqrt K &= \dfrac{{10000 + 50}}{{10000}}\\ K &= 1.01 \end{align*} {/eq}

Therefore the dielectric constant of gas is {eq}1.01.{/eq}