Electromagnetic force between Parallel Currents

[Dyon]

Hi, sorry if this has been asked before.
It is known that two parallel wires carrying electric currents in the same direction attract one another. The force is known as electromagnetic force.

My question is whether this has been checked for two pure currents of charges. Specifically, whether there is any experiment in which two parallel electron beams (or positive ions) have been observed to attract one another in vacuum.

References to any published experimental work will be much appreciated. I have a very old reference with an experiment done by William Crookes in which he says that the electron beams repelled one another and I would like to double check this.

Many thanks.

 

[Drakkith]

I'm not sure it applies to "pure charges". A conductor is generally electrically neutral, whereas beams of charged particles are not.

 

[Dyon]

Thanks Drakkith.
If this doesn't apply to pure charges, then is B = v x E for an electric charge (taken from Jackson's Electrodynamics) correct?

 

[jtbell]

is B = v x E for an electric charge (taken from Jackson's Electrodynamics) correct?

If it were not, would it be in Jackson, as well as in every other E&M textbook out there?

If you have two beams of electrons, there are both electric and magnetic forces. That does not invalidate the magnetic force law. You simply have to combine it with the electric force law:
$$\vec F = q (\vec E + \vec v \times \vec B)$$

 

[Dyon]

So how do we reconcile this? Does a moving charge produce a magnetic field just as an electric current in a wire does?

 

[jtbell]

Does a moving charge produce a magnetic field just as an electric current in a wire does?

Sure it does. See for example

http://farside.ph.utexas.edu/teaching/em/lectures/node125.html

in particular equation 1534.

 

[Dyon]

Do we have an experimental measurement of the B produced by a moving electric charge (as eqn. 1534 says)?
I would appreciate a reference to an experimental work.
Thanks.

 

[Dyon]

So in general we have F ⃗ =q(E ⃗ +v ⃗ ×B ⃗ ).
Then the difference between the behavior of currents in wires and beams of pure charges moving in vacuum is that, in the former case E ⃗ is zero because the wire is neutral and the force is reduced to F ⃗ =q(v ⃗ ×B ⃗ ), while in the latter case the full F ⃗ =q(E ⃗ +v ⃗ ×B ⃗ ) acts between the charges.
Did I get this correct?
Thanks.

 

[jtbell]

Yes.

 

[Dyon]

Ok, so the above is true for the force acting on a charge.

But what about the charges producing the B field?
For charges moving in a wire, their E field is nil since the wire is neutral. Then why does Jackson say that the B field produced by a charge moving in a wire is B = v x E where E is the electric field of the charge moving in the wire?

 

[jartsa]

Ok, so the above is true for the force acting on a charge.

But what about the charges producing the B field?
For charges moving in a wire, their E field is nil since the wire is neutral. Then why does Jackson say that the B field produced by a charge moving in a wire is B = v x E where E is the electric field of the charge moving in the wire?

E is an imaginary electric field. Imagine that the static charges don't exist.

Magnetic field caused by the moving charges: v x E

Magnetic field caused by the static charges: 0 (zero velocity x some electric field )

Total magnetic field is: (v x E) + 0