Calculate Gravitational Field on Torus Using Gauss' Law

[ayae]

I'm a little stuck, how can I go about calculating the gravitational field on the surface of a mass in the shape of a torus using Gauss' Law.

 

[Naty1]

[1/4piG] closed integral (g . dS)

 

[ayae]

I know that but I cannot simplify it to produce a workable result.

 

[Doc Al]

I'm a little stuck, how can I go about calculating the gravitational field on the surface of a mass in the shape of a torus using Gauss' Law.

That shape would not have enough symmetry to make using Gauss' law profitable.

 

[ayae]

That shape would not have enough symmetry to make using Gauss' law profitable.

When you say profitable, do you mean possible or worth while doing?

Because I really had my mind set on using Gauss' law for this example, is there no way of numerically calculating it?

 

[Doc Al]

When you say profitable, do you mean possible or worth while doing?

I mean worth doing. Gauss' law always applies, but you can only use it to find the field in certain cases of high symmetry. (Such as spherical or cylindrical symmetry.)

Because I really had my mind set on using Gauss' law for this example, is there no way of numerically calculating it?

Step one is to find a gaussian surface with a uniform field. Can you do that for a torus?

You'd need to use superposition, not Gauss' law.

 

[ayae]

I mean worth doing. Gauss' law always applies, but you can only use it to find the field in certain cases of high symmetry. (Such as spherical or cylindrical symmetry.)

This is what I feared, I didn't know whether it was symmetrical enough. Thanks for clearing it up.

Step one is to find a gaussian surface with a uniform field. Can you do that for a torus?

You'd need to use superposition, not Gauss' law.

You're going to have to forgive my scientific illilteracy, but I don't understand this.

All I need to know is the gravitational acceleration at the very centre edge of the torus. So is it safe for me to make the assumption that g.n = g at this centre strip? And simplify down to g integral dA = 4 Pi G M, (which I can easily calculate).

 

[Doc Al]

All I need to know is the gravitational acceleration at the very centre edge of the torus. So is it safe for me to make the assumption that g.n = g at this centre strip?

Is the field strength constant over that center strip? Sure. But that's not a gaussian surface.

And simplify down to g integral dA = 4 Pi G M, (which I can easily calculate).

Sounds like you're still insisting on applying Gauss' law. You have to integrate over a closed gaussian surface, not just that center strip.

 

[ayae]

Is the field strength constant over that center strip? Sure. But that's not a gaussian surface.

Sounds like you're still insisting on applying Gauss' law. You have to integrate over a closed gaussian surface, not just that center strip.

If I can't find a gaussian surface for a which passes through the center strip what can I do? How would I do this using superposition?

Can I not utilize the property of the torus being rotationally symetrical around z? :(
Thanks for the help so far.

 

[Doc Al]

If I can't find a gaussian surface for a which passes through the center strip what can I do? How would I do this using superposition?

Find the field contribution from each element of mass and add them up. Not trivial, I'm afraid.

Can I not utilize the property of the torus being rotationally symetrical around z?

I don't see any obvious way. You want the field at the inner edge, not along z.

 

[ayae]

Would it be scientifically sound to conclude with the use of shell theorem that; if the radius is considerablly large compared to the thickness of the torus that the torus can be considered a cross section of a sphere and if we consider that this cross section is the middle of the sphere then the net gravity in the middle area of the torus will be close to zero?

 

[stevenb]

Just to through out a possible idea. Have you tried to use toriodal coordinates.

http://mathworld.wolfram.com/ToroidalCoordinates.html

http://en.wikipedia.org/wiki/Toroidal_coordinates

I haven't tried myself, but perhaps there is enough symmetry to apply Gauss's Law to a torus in this particular general curvilinear coordinate system.

 

[ayae]

Just to through out a possible idea. Have you tried to use toriodal coordinates.

http://mathworld.wolfram.com/ToroidalCoordinates.html

http://en.wikipedia.org/wiki/Toroidal_coordinates

I haven't tried myself, but perhaps there is enough symmetry to apply Gauss's Law to a torus in this particular general curvilinear coordinate system.

Thankyou, that looks very promising.

 

[Vanadium 50]

Yes, but you will have to work out what the divergence operator is in toroidal coordinates. That may be non-trivial.

 

[stevenb]

Yes, but you will have to work out what the divergence operator is in toroidal coordinates. That may be non-trivial.

Actually, it's not too hard to derive that if you have a couple of hours to kill, but one can also find this worked out. The attached pdf gives the formulas for general curvilinear coordinates and the previous references cited above give the h1, h2 and h3 scale factors.

Personally, the issue I'm having is visualizing in this coordinate system. It's not clear to me that the symmetry works out. At first it seems as if the symmetry is not there, but the coordinate system itself resembles the expected field pattern (EDIT: correction, not field but potential function), so there might be a way to make it work. I am curious if it can be done, so if someone is successful, please post the results.

 

[gabbagabbahey]

This is what I feared, I didn't know whether it was symmetrical enough. Thanks for clearing it up.



You're going to have to forgive my scientific illilteracy, but I don't understand this.

All I need to know is the gravitational acceleration at the very centre edge of the torus. So is it safe for me to make the assumption that g.n = g at this centre strip? And simplify down to g integral dA = 4 Pi G M, (which I can easily calculate).

Personally, I wouldn't bother trying to find a Gaussian surface. Instead, just use Cartesian coordinates and find the contribution to the total field (at the field point you are interested in) by an arbitrary infinitesimal piece of mass on the toroid. Integrate over the entire toroid and you're done. You should find that due to symmetry (when you measure the field at a point along the inner middle track of the toroid) two Cartesian components of the field will be zero, and you will only really have to do the integration for one component. The integration itself shouldn't be too bad with a clever substitution.

 

[Doc Al]

Personally, I wouldn't bother trying to find a Gaussian surface. Instead, just use Cartesian coordinates and find the contribution to the total field (at the field point you are interested in) by an arbitrary infinitesimal piece of mass on the toroid. Integrate over the entire toroid and you're done.

Exactly. (Still, non-trivial.)

 

[stevenb]

So, the above advice is definitely good, and as an engineer I'd take that approach to get a practical answer.

Still, the motivation of the OP has not been expressed and it's not clear if an nifty and elegant approach is being sought for some esoteric reason. For this reason, I hope I can take some poetic license to do some (possibly dubious) brainstorming, making clear that these ideas may be deadends, or at best needlessly complicated.

My previous mention of using Gauss's law in toroidal coordinates does not seem fruitful. At least I can't seem to make it work, as I don't see the needed symmetry. Still, there is some gut level feeling that I have that somehow this coordinate system may be useful, if it turns out that a closed form solution can not be found using the above recommendations in rectangular coordinates. Note, that I haven't tried, so I don't know if that works out to find the field at all points in space.

With that preface, note the following. As pointed out in the following link, Laplace's equation is separable in toroidal coordinates. This means that the exterior solution (for potential) can be found from the Laplace equation and boundary conditions. One can then investigate what happens with Poison's equation inside the torus shaped mass. I expect that the mass density is either constant or only dependent on the v-coordinate, in order to have toroidal symmetry. This mass symmetry could (possibly) allow determination of the interior solution for potential and then gravitational field can be found from the gradient of the potential.

http://mathworld.wolfram.com/LaplacesEquationToroidalCoordinates.html