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I've watched this video on YouTube by Sixty Symbols entitled "Currents and Magnets". In the video, the professor demonstrates the expansion of a wire due to current heating it up and he also demonstrates how the current interacts with a magnet. I want to ask about the magnetic interaction.

At about 2:28 he shows how the orientation of the magnet relative to the wire/current determines whether the wire is repelled or attracted. It appears that, mathematically, whether the wire is pulled or pushed is determined by the cross product between the vector of the flow of the current, $\vec{c}$, and the vector of the magnetic field, $\vec{m}$.

However, the cross product is not commutative (order of the operands is important). What determines whether I should cross $\vec{c}$ with $\vec{m}$ or whether I cross them the other way around?

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There is an expression for the Lorentz force on a charge in a magnetic field. This expression is based on experimental facts, and the order in the vector product of the charge velocity and the magnetic field is fixed.

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It is not based on experimental facts--- it is based on arbitrary human convention. –  Ron Maimon Aug 23 '12 at 5:48
    
@Ron Maimon: I agree that it is based on arbitrary human conventions - how you define the vector product, "how to label magnetic fields by vectors", what order of velocity and magnetic field you choose in the vector product, however I strongly disagree that "it is not based on experimental facts" - it is: if you wish to redefine, e.g., the vector product or the order of the vectors in the vector product, you have to change the sign in the formula for the Lorentz force, otherwise the formula will not agree with experiment, so in this sense the order of the vectors is fixed. –  akhmeteli Aug 23 '12 at 9:18
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The force is given by dF=I dlxB, so there is no ambiguity. I current flow (scalar) dl: element of wire lenght (vector) B: Magnetic field (vector)

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The difference in the two orders is a minus sign, and this is an arbitrary human convention for how to label magnetic fields by vectors.

When you have a current flowing in a loop, the magnetic field produced at the center of the loop is, by convention, the direction of your thumb when you put your fingers around the loop. When you have a current and a field, if you put your thumb in the direction of the current, and the fingers in the direction of the field, you get the force. This resolves the sign ambiguity you asked about.

These definitions require a hand to define them, and so they are not invariant to left-right reflection. But the experimental phenomena are invariant under left-right reflection. If you want to describe electromagnetism with natural quantities which are obviously left-right invariant, you can use the magnetic tensor, which is defined as an antisymmetric tensor with components:

$$ B_{ij} = \epsilon_{ijk} B_k $$

Where summation is implied, and the $\epsilon$ tensor is the totally antisymmetric 3-index tensor. This point of view pictures a magnetic field in the z-direction as a little swish in the x-y plane, with an orientation. When you have a current, it gets turned in the direction of the swish. This is the manifestly left-right invariant description of electromagnetism, which is not presented in elementary books.

It is automatic in relativity, since the B field is in the space-space parts of the Faraday tensor, so it automatically has this correct description.

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