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Why is it that two power lines carrying anti parallel currents exert force away from each other as oppose to to each other?
I find all that right-hand-rule stuff unconvincing. But you can draw a picture which makes things pretty obvious. Take a constant magnetic field (straight lines) and then superimpose the field of a wire (concentric circles). If you draw it right, you will see than the lines of the external field are bent around the wire in such a way that they seem to push it one way or the other.
The force between the two conductors at constant current is then (dW/db is a partial derivative)
F = dW/db = (u_{0}/8 pi) (a/b) 4 (1/a) I^{2}= (u_{0}/(2 pi b)) I^{2} Newtons per meter
So the force is positive (repulsive), and decreases as 1/b.
I think you are right; the force is attractive, not repulsive. Here is the same problem with a parallel plate capacitor with charge Q, area A, and separation x. I think the force between the plates in this case is attractive:Well that's the whole problem...is the force repulsive?? Are you sure you've got the sign right? Because it seems that as the wires get farther apart, the magnetic energy INCREASES. Doesn't that make the force should be attractive? That's how it works in electrostatics: the forces move so as to minimize the field energy. Yes, I agree with you that the force is in fact repulsive. But then the field energy would INCREASE as the wires move apart. So I don't see how the partial derivative gives you the right answer. Is this not a problem?
Smythe holds all currents constant when he does the differentiation. He specifically states in Section 7.18 that "this [force] is exactly the reverse of the electric case where the force on equal and opposite charges tends to bring them together and destroy the field".Okay. I think the reason its confusing is that the same logic which tells you that capacitor plates attract (decreasing the field energy) gives you the opposite answer when you find that opposite currents repel (INCREASING the total field energy!).
What happens physically to make it different is that once the wires start moving in them, there are voltages induced which create new currents. So it's really not the same as the moving capacitor plates where the charges stay constant.