Find E Field of a Torus: Inside and Outside Surfaces

[Salvador]

Hello , please help me out , I can find the E field of a sphere on google and read that there is no field on the inside of the sphere , but what is the e field of a torus ? Not on the inside but on the outside surfaces also in the inner loop or the middle ?

 

[Simon Bridge]

The problem is underspecified - i.e. is the torus a conductor or an insulator? Charge distribution?
For an insulator with a uniform charge density in the shape of a torus (as pictured) - that should be simplest.
How have you attempted to answer your own questions?
Note: you should be able to figure out the field at the dead center of the torus by symmetry.
Getting an analytical expression for the field everywhere would be a tad tricky but you should be able to find it for subsets - like along the z-axis. Depending on how good your maths is of course ;)

A conducting torus is trickier:
http://link.springer.com/article/10.1134/1.1259067#page-1

 

[Salvador]

Yes , pardon, the torus is a good conductor.It is charged with a positive potential.A closed symetrical torus.

Now my guess is that the E field is the strongest on the outer middle plane of the torus and decreases gradually once we go from upper and lower 90 degrees towards the inside but still its very mind boggling how strong the e field is closer to the inner side of the torus because theoreically i can model it as a number of spheres put in a toroidal shape and electrically connected , still i guess that makes the field weaker on the inside since alike charges tend to push away ?

 

[Simon Bridge]

Do you mean that the torus is maintained at a constant potential everywhere?
You realize that surface charge will not be uniform?

You'd probably do better modelling it as a number of loops of charge.

Since like charges repel, the field dead center will be zero, yes.

You can verify your guesses using maths ;)
BTW: did you look at the link - it treats a conducting torus.

 

[Salvador]

Do you mean that the torus is maintained at a constant potential everywhere?

Yes.


I looked at the springer article , but it doesn't say much that i could understand and the full version is for money.
So if we slice the torus in half horizontally and call that line 0 degrees , then on the 0 degrees plane there is no e field on the inside but as we move away from the inner middle point the e field begins to rise but where in what region does it reach its maximum ?

 

[Simon Bridge]

Yes - that is because the field due to a conducting torus is complicated and hard to calculate in general.

It is a good idea to start with a simple picture - orient a torus with an open center in the x-y pane, centered on the origin. See general geometry for the torus:
http://en.wikipedia.org/wiki/Torus

But if I were you I'd start by working out the field due to a ring of charge in the x-y plane for any point on the z axis, then work from there to a torus. May help to work in cylindrical-polar coordinates.

That will get you used to the maths.
Then you can start looking at different locations.

Note: a conducting torus won't generally have a constant charge density.

 

[Salvador]

Simon I'm not good at maths.
Can I just ask , if we had a negative electron as test particle and we placed the electron in the exact middle position of the torus when the torus is positively charged then the electrion would feel no force but once the electron would move past the middle point it would gradually start to feel the positive charge of the torus unlike in a sphere were every point inside the sphere has zero potential?

 

[Simon Bridge]

The symmetry suggests that the x-y plane inside the torus-hole would have zero potential, but the potential changes in the z direction.

As you've seen, you need very good maths to do this problem - especially for a conductor.

 

[NILOYENDU ROY]

simon sir, I need to calculate the electrostatic energy of a charged conducting torus total charge on which is specified[there are no charges anywhere else]. But elliptic integral is coming into my calculation even if i want to calculate the potential at some simple points on the torus like (a,0,0) [a is the inner radius]. And i can't find any way to calculate the energy without knowing the potential. because integrating (epsilon0/2)E^2 over the whole space is impossible for me..! and other formulas for calculating energy would involve potential.Will knowing E field just out side the surface of the torus be any good for knowing the potential? And i don't think i can do anything with the boundary conditions which will come while solving poisson's equation.. so what can i do? is there any other way to calculate the energy?

 

[Simon Bridge]

Welcome to PF;
Do you know the charge distribution?

 

[NILOYENDU ROY]

thanks.
Simon sir,

I thought that the charge would be distributed over the surface of the torus uniformly.. As because it was mentioned in the problem that the torus is a perfect conductor.. and i planned to calculate the energy by performing the integration (1/2)∫σv(da) and as because over tthe conductor potential is constant and if we consider a surface just enclosing the conductor, 'v' will come out of the integral and the rest would be easy.. i got a hint how to calculate the potential of the conducting torus..​

 

[Simon Bridge]

If you are not told the charge distribution - then you need to work it out or look it up.
http://link.springer.com/article/10.1134/1.1259067#page-1

Note: It's not going to be uniform because charges in the surface of the "hole" in the center of the torus will tend to push each other to the outside.

 

[NILOYENDU ROY]

okey..
thnks for the link..