Biot–Savart law: the origin of chirality?

[carrz]

As we know Biot-Savart law follows right-hand rule, like this:

...so the question is why right-hand rule, why not left?

I suppose it must have something to do with electron's magnetic dipole moment or "spin", as that's the only source of asymmetry I am aware of in this scenario. It follows then the orientation of the spin axis could be what defines this magnetic rotational direction, which would imply then the spin axis orientation is defined by the velocity vector. But still, the question remains, how or why could velocity vector have any specific directional impact on either intrinsic electron spin axis orientation or Biot-Savart magnetic field rotation?

 

[Jano L.]

The right-hand rule works because it gives the same direction as the vector product in the Biot-Savart law:
$$
\mathbf j(\mathbf x') \times |\mathbf x - \mathbf x'|
$$
where $\mathbf j$ is current density at point $\mathbf x'$, $\mathbf x$ is point where the magnetic field is sought.

The vector product chooses one direction for $\mathbf B$ from two possibilities. Direction of magnetic field could have been defined to be opposite. Then we would use left hand to get this opposite direction.

 

[Matterwave]

People use the right hand rule mostly because most people are right handed. It's easier to figure out the convention using the dominant hand. One can arbitrarily reverse all the fields and go by the left hand rule. The only physical effect is the motion of particles and forces on these particles inside these fields, so however direction you choose to define the fields is arbitrary as long as you get the motion correct in the end.

 

[Dale]

...so the question is why right-hand rule, why not left?

This is merely a matter of convention. Like why the electron is negative, why not positive? EM could work just fine if we picked different conventions (i.e. electrons positive, left hand rule, different units, etc.) Someone historically picked the right hand and established the convention and people have just used it ever since, there is no physical reason nor effect.

 

[carrz]

This is merely a matter of convention. Like why the electron is negative, why not positive? EM could work just fine if we picked different conventions (i.e. electrons positive, left hand rule, different units, etc.) Someone historically picked the right hand and established the convention and people have just used it ever since, there is no physical reason nor effect.

Convention is arbitrary, but the reason for nature to always stick with the same direction is not.

 

[Dale]

Nature doesn't pick a direction for EM (I.e. there is no chirality observed in EM). Only our conventions do pick one, and we could pick the opposite convention without changing nature or experimental results in any way.

 

[carrz]

Nature doesn't pick a direction for EM (I.e. there is no chirality observed in EM). Only our conventions do pick one, and we could pick the opposite convention without changing nature or experimental results in any way.

Nature does pick always the same direction relative to velocity vector. This has nothing to do with convention or whether you are going to call it right, west, or clockwise. It's always the same.

 

[Dale]

No, it doesn't.

Take any experiment you can think of, and any measurement you can devise. If we picked the opposite convention (left handed coordinates and cross products instead of right handed coordinates and cross products) then you would get all of the B fields reversing direction but none of the physically measurable results being altered in the least.

If you cannot think of an example, then work out the force between two current-carrying wires using both a left handed and a right handed convention. Don't forget to use the same handedness both for determining the direction of the field and the Lorentz force.

In other words, nature does not pick the handedness of the B field (and therefore the direction of the B field), our conventions do. Nature does pick the direction of the force, and that does not exhibit any chirality.

 

[carrz]

Nature does always pick the same direction relative to velocity vector.

 

[Dale]

Work the exercise I suggested and see.