Electric flux through a hemisphere due to these charge configurations

[haruspex]

This is an obviously late reply ( 8yrs to be specific ) but I came across the same problem and the doubt I have is a bit different. I'm wondering why it is that only a line cutting the surface once can make a net contribution to the electric flux and not a line cutting through it twice. I think I might be using the same textbook as the original poster.

Because the direction in which the line passes through matters. Consider a closed manifold. Having passed from outside it to inside it, a second crossing by the flux line must be from inside to outside. That has the opposite sign and the two cancel.

 

[Abhikamya M S]

oh thank you so much that clears my doubt I really wasn't expecting a reply to it.