Flux from Outside Charge - Purcell's Method I understand that the flux from an outside charge of a closed surface is zero...however, I was recently looking through Purcell's book Electricity and Magnetism, and I found this:

Do you understand what he means with this picture?
Thank you!
 A: The flux through the part of the surface to the left of the “pinch” plus the flux through the part of the surface to the right of the pinch is the total flux, which must be $q/\epsilon_0$. In the limit of letting the pinch close, all this flux goes through the left and none through the right.
A: The surface in (b) has three parts: a roughly spherical part surrounding the charge, a tiny bridge, and the surface in (a). Now imagine pinching the bridge to zero thickness.


*

*The flux through the first part is $q/\epsilon_0$. 

*The flux through the second part is zero.

*The flux through the third part is what we want to compute.

*The total flux through the surface in (b) is just $q/\epsilon_0$, as argued earlier in the chapter.


So the flux through the third part, i.e. the surface in (a), is zero.
A: I think the argument is flawed. The 'trick' is a distraction and not a neat proof.
In the first diagram the difficulty is believing that the flux which enters the surface from outside eventually leaves it. To me this is visually obvious. Flux equates to the number of lines of force. Every line of force which enters the closed region (the red lines) eventually leaves it. 

In the second diagram we still have flux (the red lines again) which leaves the new part of the surface (the 'balloon' on the left) and re-enters the original surface (the 'balloon' on the right). We haven't really dealt with the difficulty which we had in the first diagram. All we've done is caused a distraction and ignored the part of the flux (the red lines) which caused the difficulty in the first diagram. We still have to take on trust that flux (re-)entering on the right (having exited on the left) also leaves again on the right. 
In other words we have assumed what we are trying to prove!
To see this more clearly we could shrink the balloon on the left down to zero size at the same time as shrinking the 'pinch' or 'neck' between the two balloons. Then we are back to the first diagram with the same difficulty we had originally. 
