Consider a VACE (V_{R}) in the torus at some angle theta.
There is a "mirror" VACE (V_{L}) on the left side of the
torus.

What we are trying to prove is that the force F_{L}
on the left current element I_{L}
(due to the single external VACE) is equal to the negative of the
force F_{R} on the right current element I_{R}.

This is easily seen from the following two illustrations.

In the first illustration we see the current element I_{R}
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 I_{M}. Note that angles alpha and epsilon are the
same as defined by the right current element.

But this mirror current element I_{M} is pointing in the
wrong direction. We want the current element on the left
I_{L} to point 180 degrees from that of I_{M}.

I_{L} now defines the angles alpha_{L} and
epsilon_{L}. In particular, we see that

alpha_{L} = 180 - alpha_{R}

epsilon_{L} = 180 - epsilon_{R}

epsilon

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

2cos(epsilon_{L}) - 3cos(alpha_{L})cos(beta) =

2cos(180 - epsilon_{R}) - 3cos(180 - alpha_{R})cos(beta) =

- (2cos(epsilon_{R}) - 3cos(alpha_{R})cos(beta))

2cos(180 - epsilon

- (2cos(epsilon

So, F_{L} = - F_{R}.

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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