Consider a closed loop $C$ in a magnetic field as shown in the figure. The flux passing through the loop is defined by choosing a surface whose edge coincides with the loop and using the formula $\phi = \sum \vec{B}_i \cdot d\vec{A}_i$. Now,if we choose two different surfaces $S_1$ and $S_2$ having $C$ as their edge,would we get the same answer for the magnetic flux? Justify your answer.

(N/A) Yes,we would get the same answer for the magnetic flux.
According to Gauss's Law for magnetism,the net magnetic flux through any closed surface is zero $(\oint \vec{B} \cdot d\vec{A} = 0)$.
If we consider the closed surface formed by the union of surfaces $S_1$ and $S_2$,the total flux through this closed surface is zero.
Let the flux through $S_1$ be $\phi_1$ and through $S_2$ be $\phi_2$. Since the magnetic field lines enter through one surface and exit through the other,the flux through $S_1$ must equal the flux through $S_2$ in magnitude.
Therefore,the magnetic flux linked with any surface bounded by the same loop $C$ is the same,as it depends only on the number of magnetic field lines passing through the loop.