Relativistic Electromagnetism - The Origin of Magnetic Forces

The Origin of Magnetic Forces

In the simple model of events in a wire stretched out horizontally, a current can be represented by the evenly spaced positive charges, moving to the right, whilst an equal number of negative charges remain at rest. If the wire is electrostatically neutral, the distance between adjacent positive charges must be the same as the distance between adjacent negative charges.

Assume that in our 'lab frame' (Figure 5), we have a positive test charge, Q, outside the wire, traveling parallel to the current, at the speed, v, which is equal to the speed of the moving charges in the wire. It should experience a magnetic force, as can be easily confirmed by experiment.

Considered from a 'test charge frame', where the test charge is at rest (Fig. 6), the only possible force is the electrostatic force F_e = Q * E. In this frame, the negative charge density has Lorentz-contracted with respect to what we had in lab frame because of the gained speed. This means that spacing between charges has reduced by the Lorentz factor with respect to the lab frame spacing, l:

Thereby, positive charges have Lorentz-expanded (because their speed has dropped):

Both of these effects combine to give the wire a net negative charge in the test charge frame. Since the negatively charged wire exerts an attractive force on a positively charged particle, the test charge will therefore be attracted and will move toward the wire.

For, we can concretely compute both, the magnetic force sensed in the lab frame

and electrostatic force, sensed in the test charge frame, where we first compute the charge density with respect to the lab frame length, l:

and, keeping in mind that current, resulting electrostatic force

which comes out exactly equal to the magnetic force sensed in the lab frame, !

If the currents are in opposite directions, consider the charge moving to the left. No charges are now at rest in the reference frame of the test charge. The negative charges are moving with speed v in the test charge frame so their spacing is again:

The distance between positive charges is more difficult to calculate. The relative velocity should be less than 2v due to special relativity. For simplicity, assume it is 2v. The positive charge spacing contraction is then:

relative to its value in their rest frame. Now its value in their rest frame was found to be

So the final spacing of positive charges is:

To determine whether l₍₊₎ or l_(-) is larger we assume that v << c and use the binomial approximation that

After some algebraic calculation it is found that l₍₊₎ < l_(-), and so the wire is positively charged in the frame of the test charge.