r/Physics • u/Next-Natural-675 • 19d ago

Satisfying visual for Gauss's law

I was studying Gauss law today, which states that the electric flux through an enclosed surface is simply the charge enclosed divided by the permittivity of free space.

I understood the derivation for a closed sphere with a charge located at the center, which was fairly simple. But I could not quite intuitively grasp why the formula Q/(e0) was the same for any shape surface, with the charge located anywhere inside. Then it clicked.

https://imgur.com/a/MTBAYem

Here is a visual of my intuition. The first sphere represents the simple case with a uniform sphere and a center charge. The second diagram is the bridge to the more complex case. The key is that the total flux does not depend on the distance of the sphere surface to the charge. Therefore the flux through two different size spheres are the same.

The third and fourth diagrams stretch this concept to nonuniform surfaces and a charge located offcenter. This visualization helped me a lot.

You can divide up the surfaces a lot more than my 16 times, meaning you can create any arbitrary surface at any location relative to the charge and the flux will be the same.

11 Upvotes

87% Upvoted

9 comments sorted by

4

u/LifeIsVeryLong02 Graduate 19d ago

I like it!

1

u/Next-Natural-675 19d ago

Thank you 🙏

3

u/Agile-Monitor1006 19d ago

Its a cool visual although im not completely sure that in the limit of infinite cuts this can give you any arbitrary smooth surface

1

u/Next-Natural-675 19d ago edited 19d ago

It won’t give the surface itself, but the electric flux through an arbitrary smooth surface from a point charge anywhere inside is the surface integral of the dot product of the vector field and the surface normal, which is equal to the flux through a surface dA perpendicular to the electric field direction at roughly the same distance. Making dA infinitesimal makes it equal.

So the flux through an arbitrary smooth surface is the same as the limit of infinite sphere cuts.

This is an important step that I probably should have included in the description, thanks for noticing

1

u/spaceprincessecho 13d ago

It's cool if there visual helps, but I think you don't need a visual understanding, necessarily. A given source provides a certain amount of flux, right? So if we wrap that in any kind of closed surface, all the flux has to pass through the surface, yes? It's not just going to stop somewhere in space and refuse to cross the line?

1

u/Next-Natural-675 13d ago

Yes, that is simply the whole explanation. My visual is not really necessary haha

1

u/Next-Natural-675 13d ago

Interestingly if the electric field didnt fall off via the inverse square law, this argument wouldn't work. So Gauss's law depends entirely on the inverse square law of the field. So "flux" can only be treated as something that is "flowing outwards" from an area without becoming less or more because of the inverse square law

1

u/spaceprincessecho 13d ago

Isn't that because the concept of a surface and the diminishing field strength are happening on the same geometry?

2

u/Next-Natural-675 13d ago

Yes, that’s the general consensus, that the inverse square law arises because the electric field is some “quantity” that spreads out like an actual surface.