- #1
ayae
- 20
- 0
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 said: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.
Doc Al said:That shape would not have enough symmetry to make using Gauss' law profitable.
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.)ayae said:When you say profitable, do you mean possible or worth while doing?
Step one is to find a gaussian surface with a uniform field. Can you do that for a torus?Because I really had my mind set on using Gauss' law for this example, is there no way of numerically calculating it?
This is what I feared, I didn't know whether it was symmetrical enough. Thanks for clearing it up.Doc Al said: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.)
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.
Is the field strength constant over that center strip? Sure. But that's not a gaussian surface.ayae said: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?
Sounds like you're still insisting on applying Gauss' law. You have to integrate over a closed gaussian surface, not just that center strip.And simplify down to g integral dA = 4 Pi G M, (which I can easily calculate).
Doc Al said: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.
Find the field contribution from each element of mass and add them up. Not trivial, I'm afraid.ayae said: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?
I don't see any obvious way. You want the field at the inner edge, not along z.Can I not utilize the property of the torus being rotationally symetrical around z?
stevenb said: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.
Vanadium 50 said:Yes, but you will have to work out what the divergence operator is in toroidal coordinates. That may be non-trivial.
ayae said: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).
Exactly. (Still, non-trivial.)gabbagabbahey said: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.
Gauss' Law is a fundamental law of physics that relates the electric flux through a closed surface to the enclosed electric charge. In the context of calculating gravitational field on a torus, Gauss' Law can be used to determine the gravitational field strength at any point on the surface of the torus by considering the distribution of mass inside the torus.
The formula for calculating gravitational field on a torus using Gauss' Law is g = G * M / R2, where g is the gravitational field strength, G is the universal gravitational constant, M is the mass enclosed by the surface, and R is the distance from the center of the torus to the point at which the field is being calculated.
Yes, Gauss' Law can be used to calculate gravitational field on any shape as long as the mass distribution inside the shape is known. However, for complex shapes, the calculation may become very difficult and may require advanced mathematical techniques.
The units of measurement for gravitational field strength are typically m/s2 or N/kg, which are equivalent units. These units represent the acceleration experienced by a mass at a given point in the gravitational field.
The calculation of gravitational field on a torus using Gauss' Law is applicable in many real world scenarios, such as in spacecraft design and orbit calculations. It can also be used in studying the gravitational effects of celestial bodies, such as planets and stars, on each other. Additionally, this calculation can be helpful in understanding the behavior of objects on or near the surface of a torus, such as in roller coaster design or in studying the motions of particles in particle accelerators.