S₁ and S₂ are two hollow concentric spheres with charge q and 2q. Space between S₁ and S₂ is filled with a dielectric of dielectric constants.

The ratio of flux through S₂ and flux through S₁ is K. Then, the value of $\frac{9K}{11}$ is ________.

With air in the space, between S₁ and S₂.

${\varphi }_{S_1}=\frac{q}{{\epsilon }_0}\text{ and }{\varphi }_{S_2}=\frac{2q}{{\epsilon }_0}$

When a medium of K= 5 is filled in space

${\varphi }_{S_1}^{\mathrm{\prime }}=\frac{Q}{K{\epsilon }_0}=\frac{{\varphi }_{S_1}}{5}=\frac{q}{5{\epsilon }_0}$

And in this case, flux though S₂

$\begin{array}{l}{\varphi }_{S_2}^{\mathrm{\prime }}=\text{ flux of }{S}_1+\text{ flux of charge }2q\\ =\frac{q}{5{\epsilon }_0}+\frac{2q}{{\epsilon }_0}=\frac{q+10q}{5{\epsilon }_0}=\frac{11q}{5{\epsilon }_0}\end{array}$

$\begin{array}{l}\text{ So, ratio }\frac{{\varphi }_{S_1}^{\mathrm{\prime }}}{{\varphi }_{S_2}^{\mathrm{\prime }}}=11\\ \text{ So, }\frac{9K}{11}=9\end{array}$