Question

two identical charged spheres are suspended in air by strings of equal lengths and make an angle of 30* with each other .when suapended in a liquid of density 0.8 g/cc., the angle remains the same . What is the dielectric constant of liquid ?Take density of the material of the sphere = 1.6 g/cc

Answer 1

Is the dielectric constant 2?

Answer 2

The dielectric constant of liquid is K = 2.

Two  identical charged spheres suspended in air by strings of equal lengths makes angle 30 degree, which is same when they are suspended in liquid with density 0.8 g/cc.  

There will be two equation formed accordingly for the two medium. In air, $\bold{T \cos \theta=m g \text { and } T \sin \theta=F}$. Thereby, from both we get $\bold{\tan \theta=\frac{F}{m g}}$. Which is the equation for air.

In liquid, $\mathrm{T}^{\prime} \cos \theta+\mathrm{F}^{\prime}=\mathrm{mg}$

$\Rightarrow \mathrm{T}^{\prime} \cos \theta=\mathrm{mg}-\mathrm{F}^{\prime} \text { and } \mathrm{T}^{\prime} \sin \theta=\mathrm{F}^{\prime}$

$\Rightarrow \text { Thereby }, \tan \theta=\frac{\mathrm{F}^{n}}{\mathrm{mg}-\mathrm{F}^{\prime}}$

$\Rightarrow \text { Comparing both through tan } \theta \text { value, we get } \frac{-\mathrm{F}}{\mathrm{mg}}=\frac{\mathrm{F}^{*}}{\mathrm{mg}-\mathrm{F}^{\prime}}$

$\begin{array}{l}{\Rightarrow \mathrm{F}^{\prime \prime}=\frac{\mathrm{F}}{\mathrm{k}} \text { as already known }} \\ {\Rightarrow \mathrm{F} \frac{\mathrm{F}}{\mathrm{mg}}=\frac{F}{K}\left(\frac{\mathrm{mg}-\mathrm{mg} \rho}{\mathrm{d}}\right)} \\ {\Rightarrow \mathrm{K} \mathrm{mg}\left(\frac{1-\rho}{\mathrm{d}}\right)=\mathrm{mg}}\end{array}$

$\begin{array}{l}{\Rightarrow \mathrm{K}\left(\frac{1-\rho}{\mathrm{d}}\right)=1} \\ {\Rightarrow \mathrm{K}\left(\frac{1-0.8}{1.6}\right)=1} \\ {\Rightarrow \mathrm{k} \times \frac{1}{2}=1} \\ {\Rightarrow \mathrm\bold{{K}=2}}\end{array}$