MAGNETIC FORCE LAW

From our Coulomb’s law calculation of the electric force in Figure (1a), we were able to obtain the formula for the magnetic force in Figure (1b). The result, Equation (1) repeated here, is
.....(1)

Figure 1: Two views of the same experiment. For the observer moving with the electrons, she sees a positively charged wire exerting an attractive force on the negative charge at rest. We see an electrically neutral wire carrying a negative
current, and a moving negative charge. The spring is still stretched, meaning the attractive force is still there.
where q is the charge on the test particle, i the current in the wire, and r the distance from the wire to the charge as shown in Figure (2). The only thing our derivation does not make clear is whether v in Equation (1) is the speed of the test charge or the speed of the electrons in the wire. We can’t tell because we used the same speed v for both in our thought experiment. A more complex thought experiment will show that the v in Equation (1) is the speed of the test particle.

Figure 2: Force on a charge -q moving at a speed v parallel to a negative current -i a distance r away.