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Question

Consider two hollow concentric spheres, $$S_{1}$$ and $$S_{2}$$, enclosing charges $$2Q$$ and $$4Q$$ respectively as shown in the figure. (i) Find out the ratio of the electric flux through them. (ii) How will the electric flux through the sphere $$S_{1}$$ change if a medium of dielectric constant $$'\epsilon r'$$ is introduced in the space inside $$S_{1}$$ in place of air? Deduce the necessary expression.
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Solution

Given two hollow concentric spheres $$S_{1}$$ and $$S_{2}$$ enclosing charges $$Q$$ and $$2Q$$.
i) We have to find the ratio of electric flux through them.

So, from Gauss law, flux through a surface is equal to \dfrac{Total\;charge\;enclosed}{\varepsilon _{0}} 

So, electric flux passing through sphere $$S_{1}$$
$$\phi_{1}=\dfrac{2Q}{\varepsilon _{0}}$$

And, electric flux passing through sphere $$S_{2}$$ 
$$\phi_{2}=\dfrac{2Q+4Q}{\varepsilon _{0}}=\dfrac{6Q}{\varepsilon _{0}}$$

$$\therefore$$ Ratio of flux through $$S_{1}$$ and $$S_{2}$$ is
$$\dfrac{\phi_{1}}{\phi_{2}}=\dfrac{2Q\times \varepsilon _{0}}{\varepsilon _{0}\times 6Q}$$
$$\dfrac{\phi_{1}}{\phi_{2}}=\dfrac{1}{3}$$

ii) Now, we have to find the electric flux through sphere $$S_{1}$$ if a medium of dielectric constant '$$\varepsilon _{r}$$' is introduced in space inside $$S_{1}$$ in place of air.
Using Gauss theorem, 
$$\phi_{1}=\oint \overrightarrow{E}.\overrightarrow{dS}=\dfrac{2Q}{\varepsilon _{0}}$$

Now when a material with dielectric constant $$\varepsilon _{r}$$ is introduced then, $$\varepsilon _{r}=\dfrac{\varepsilon }{\varepsilon _{0}}$$
Now, flux through $$S_{1}$$ is $$\phi_{1}'=\dfrac{2Q}{\varepsilon }$$
But $$\varepsilon =\varepsilon _{r}\varepsilon _{0}$$
Where $$\varepsilon =$$ permittivity of medium 
$$\varepsilon _{0}=$$ permittivity of air

So, $$\phi_{1}'=\dfrac{2Q}{\varepsilon _{r}\varepsilon _{0}}$$
But $$\dfrac{2Q}{\varepsilon _{0}}=\phi_{1}$$
So, $$\phi_{1}'=\dfrac{\phi_{1}}{\varepsilon _{r}}$$
So, now flux is reduced $$\varepsilon _{r}$$ times when placed in a dielectric medium of dielectric constant $$\varepsilon _{r}$$

Physics

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