$S_1,S_2,S_3$ are three concentric spherical shells as shown in figure. $S_1$ and $S_2$ have charge 4Q and 8 Q respactively Whereas $S_3$ is unchanged. Find the ratio of the flux passing through $S_1$ to that through $S_3$

The correct option is A 1 : 3

$\text{Step 1: Draw figure}$ $\text{[Ref. Fig.]}$

Given, Charges on shells:

$Q_{S_1}=4Q,Q_{S_2}=8Q$

$\text{Step2: Applying Gauss law}$

If a charge $q$ is enclosed inside a closed surface then flux through that surface $=\frac{q}{{\in }_0}$

So, ${\varphi }_{S_1}=\frac{Q_{S_1}}{{\in }_0}=\frac{4Q}{{\in }_0}$

From figure, charges $Q_{S_1}$ and $Q_{S_2}$ are enclosed by shell $S_3$

So, ${\varphi }_{S_3}=\frac{Q_{inside{S}_3}}{{\in }_0}=\frac{8Q+4Q}{{\in }_0}$

$\text{Step 3: Ratio of flux}$

$\frac{{\varphi }_{S_1}}{{\varphi }_{S_3}}=\frac{\frac{4Q}{{\in }_0}}{\frac{8Q+4Q}{{\in }_0}}=\frac{4}{12}=\frac{1}{3}$

Hence the ratio of flux passing through $S_1$ to that through $S_3$ is $1:3$

Option $\left(A\right)$ correct.