# Gauss's law in closed space

1. Aug 9, 2012

### Khashishi

How does Gauss's divergence law work in a closed finite universe? Let's say the universe were a 4-sphere, with a single electron. How can I work out the field of the electron? If I draw a 3-sphere around the electron, then I split space into two regions. One region contains an electron, so Gauss's law tells me that the flux through the sphere is -e*4pi. But the other region doesn't contain any electron, so Gauss's law tells me that the flux through the sphere going the other way is 0. That's inconsistent.

It seems if I think about it a little more, it makes no sense to have an infinite range force in finite space. An electron's field lines would "wrap around" and overlap itself onto infinity.

2. Aug 9, 2012

### Khashishi

It seems that problems could be avoided by stipulating that the total charge in the universe is 0. But what about gravity then?

3. Aug 9, 2012

### Nabeshin

Well Gauss' law for gravity is Newtonian gravity, which we know is incorrect.

4. Aug 9, 2012

### TheShrike

Why is it inconsistent? Gauss' Law essentially tells you whether or not you have any sources within the volume enclosed by your surface.

For the region without the electron, the flux of electric field through the surface is indeed zero, since any flux entering the volume from outside (where the source is located) is exactly countered by the flux leaving the volume.

For the region with the electron, the only electric fields in the universe (as you defined it) are from the electron, and can only pass outwards through the surface i.e. there is no field coming inwards to counter it, and thus the flux is non-zero.

Are you mixing up flux of the electric field with the actual magnitude of the electric field?

That's not meant to be an insulting question; it's just that I did precisely that when I learned about Gauss' Law.

5. Aug 9, 2012

### TheShrike

Actually, I think I see where you're coming from now. I'm not sure how to answer that.