Let's look at the torque calculations in some detail.
We start with Ampere's force equation
NOTE: The symbol "r" in Ampere's equation above is the distance between the two current elements. Unfortunately, I have used the symbol "r" in the following diagram and equations as the radius of the VACE torus circle. I hope this will not cause you too much confusion.
Consider the following diagram which defines most of the symbols to be used.
There are 2 Ampere current elements shown. One at P (I1) and one at R (I2).
From this diagram we can see how the following equations have been obtainned.
We next want to express the angle alpha in terms of theta and the distance "d" (the distance from the circle center "O" to I1 at P.) So, consider the following diagram.
Alpha may seem a little harder to get, but looking at the diagram we see it is given by
To rotate I2 to point along the "Force line", I2 is turned counter-clockwise, whereas I1 is rotated clockwise. This is way I chose this equation for alpha. Clockwise rotations will be positive, and counter-clockwise rotations will be negative. So, in the above diagram, alpha is negative and beta is positive.
Epsilon, the angle through which current element #2 has to be rotated to point in the same direction as current element #1 is then given by
Note that in Ampere's force law, all the angles are parameters to cosine functions. So it doesn't matter if we use
or if we use
The cosine of epsilon is the same in either case.
We can now use Ampere's force equation to calculate the magnitude of the force on each VACE in the torus. (Remember the note at the top of this page: "r" in Ampere's equation is the distance between the two current elements and not the radius of the torus.
We can use the usual distance equation to obtain the distance between the two current elements.
Don't worry about the Idl terms in Ampere's equation. They can be set to any constant value provided you use that same value in all calculations. We are only after the "relative" magnitude of the overall torque.
O.K., time for the torque...
Since we now have the magnitude of the force and we know that the force will be along the line joining the two current elements ("Force line" in the above diagram) we see that the magnitude of the torque about "O" is given by
where "r" here is the radius of the circle.Well, that's it. Now just add up each torque about "O" that each VACE in the torus experiences.
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