## Proof That FL = - FR

Consider a VACE (VR) in the torus at some angle theta. There is a "mirror" VACE (VL) on the left side of the torus.

What we are trying to prove is that the force FL on the left current element IL (due to the single external VACE) is equal to the negative of the force FR on the right current element IR.

This is easily seen from the following two illustrations.

In the first illustration we see the current element IR on the right, tangent to the circle, at some angle theta. We construct the mirror of this current element about the right-left dividing line. This gives IM. Note that angles alpha and epsilon are the same as defined by the right current element.

But this mirror current element IM is pointing in the wrong direction. We want the current element on the left IL to point 180 degrees from that of IM.

IL now defines the angles alphaL and epsilonL. In particular, we see that

alphaL = 180 - alphaR
epsilonL = 180 - epsilonR

This changes the angular factor in Ampere's force equation by an overall minus sign.

2cos(epsilonL) - 3cos(alphaL)cos(beta) =
2cos(180 - epsilonR) - 3cos(180 - alphaR)cos(beta) =
- (2cos(epsilonR) - 3cos(alphaR)cos(beta))

So, FL = - FR.

This means that for any force on the right that may be "pushing" on the torus, there is a corresponding force on the left which is "pulling" on the torus. The two torques are then additive and do not sum to zero (unless each torque is zero.)

Therefore, in determining the overall torque on the torus, we need only calculate the torque on the right hand side and then multiply by 2.

If the sum of all the torques on the right hand side do not sum to zero, as seems to be the case, then there is a net torque on the torus and it should rotate.

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Last updated: 06-11-1999