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Born2bwire

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A similar exercise will show you that the right-hand wire experiences a force to the right.

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Born2bwire

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Wait... what? What are the physics here? The fields don't interact with each other.

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I couldn't find the exact picture I wanted but this website has something similar:

https://rdl.train.army.mil/soldierPortal/atia/adlsc/view/public/8865-1/accp/mm0703/lsn3.htm [Broken]

Ironically in the example shown here you need to use the right hand rule to get the field lines, but if you do the same thing with just two currents instead of a current and a magnet, you don't need to use any special rules. I find the picture to be quite convincing in showing that like currents attract.

https://rdl.train.army.mil/soldierPortal/atia/adlsc/view/public/8865-1/accp/mm0703/lsn3.htm [Broken]

Ironically in the example shown here you need to use the right hand rule to get the field lines, but if you do the same thing with just two currents instead of a current and a magnet, you don't need to use any special rules. I find the picture to be quite convincing in showing that like currents attract.

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Born2bwire

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With just currents, how are you determining the fields? What is interacting with the fields to produce a force? How do you determine the direction of the force?

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In electrostatics its a little easier. You can account for the field energy by calculating E-squared and show that the total field energy is reduced when opposites attract. It's a pretty convincing argument but it doesn't work so well for magnetism. Or gravity for that matter. It's not always that easy to say why things go the way they do. But for the case of the two wires, I still find the picture quite convincing. Have you looked at it by the way?

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So the stored magnetic energy per unit length for both conductors with current I is

The force between the two conductors at constant current is then (

So the force is positive (repulsive), and decreases as

Note: I

For two cables carrying 1000 amps and separated by 10 cm, the force is 2 Newtons per meter. I have observed this in cables for big magnets.

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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?The force between the two conductors at constant current is then (dW/dbis a partial derivative)

F = dW/db = (uNewtons per meter_{0}/8 pi) (a/b) 4 (1/a) I^{2}= (u_{0}/(2 pi b)) I^{2}

So the force is positive (repulsive), and decreases as1/b.

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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?

stored energy = W= (1/2) Q

dW/dx = (1/2) Q

So dW/dx > 0 for attractive force

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This youtube experiment seems to be rather definitive:

Smythe "*Static and Dynamic Electricity*" 3rd Edition, in Section 7.18 proves that the force between two parallel wires with opposing currents is__ repulsive__, using the Biot Savart Law, the Lorentz force, and vector potentials. The inductance of a bifilar wire is given in section 8.12. In Section 8.13, Smythe shows that the force along any coordinate is the __positive__ derivative of the total magnetic energy (e.g., 1/2 LI^{2}) along that coordinate.

Smythe "

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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.

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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".

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.