Question

Consider a closed loop $C$ in a magnetic field (Figure). The flux passing through the loop is defined by choosing a surface whose edge coincides with the loop and using the formula $\phi =B_1\cdot d{A}_1+B_2\cdot d{A}_2+\dots$ .
Now if we chose two different surfaces $S_1$ and $S_2$ having $C$ as their edge, would we get the same answer for flux. Justify your answer.

Answer 1

One gets the same answer for flux. Flux can be thought of as the number of magnetic field lines passing through the surface (we draw $dN=B\Delta A$ lines in an area $\Delta A\perp$ to $B$). As lines of $B$ cannot end or start in space (they form closed loops) number of lines passing through surface $S_1$ must be the same as the number of lines passing through the
surface $S_2$.