In the field of electromagnetism, every point in space is characterized by two vector quantities, which determine the force on any charge. First, there is the electric force, which gives a force component independent of the motion of the charge, q. We describe it by the electric field, E. Second there is an additional force component, called the magnetic force, which depends on the relative velocity, v, of the charge in relation to reference frame of the magnetic field source. - The Lorentz Force Equation says that the force on an electric charge is dependent not only on where it is, but also on how fast it is moving in relation to something else, as in:
In figure 1, a conducting rod is moving through a magnetic field B. An electron, located in the rod, sees a magnetic force due to motion of the rod through the magnetic field. In the reference frame of the magnetic source (frame S), there is no E, thus the only force acting on the electron, is:
What happens if the rod is at rest with the observer's reference frame, but the magnetic source is moving with velocity -v, as in figure 2? Does the electron stay where it is? Would we see different things happening in the two systems?
Figure 1. A conducting rod is in relative motion with respect to a magnet. An observer S, fixed with respect to the magnet that produces the B-field, sees a rod moving to the right. He also sees a magnetic force acting downward on the electron.
We know from relativity that magnetism and electricity are not independent things - they should always be taken together as one complete electromagnetic field. Although in the static case Maxwell's equations separate into two distinct pairs, with no apparent connection between the two fields, nevertheless, in nature itself there is a very intimate relationship between them arising from the principle of relativity.
In accordance with Special Relativity, we must get the same physical result whether we analyze motion of a particle moving in a coordinate system at rest with respect to the magnetic source or at rest with respect to the particle. In the first instance the force was purely "magnetic", in the second, it was purely "electric". We know that a charge q is an invariant scalar quantity, independent of the frame of reference.
Since the F' equal to F, we can calculate F' as:
For cases where the source of the magnetic field is moving, the relative velocity v becomes the opposite sign. To distinguish this type of motional electric field, we can rewrite the equation, where V is the relative velocity, and B is the magnetic field (seen by S):
since we know that
Figure 2. A conducting rod is in relative motion with respect to a magnet. An observer Sí, fixed with respect to the rod, sees the magnet moving to the left. He also sees an electric force acting downward on the electron.
Mathematically, it can be shown that a purely electric field in one reference frame can be magnetic in another. The separation of these interactions depends on which reference frame is chosen for description. In 1903 - in a now famous experiment - Trouton and Noble showed that two electric charges moving with same constant velocity do not produce a magnetic interaction between themselves. This is consistent with the fundamental postulate of relativity. The force between two electric charges must be the same for an observer at rest with respect to the charges. This is true whether the charges move at constant velocity, or whether they remain fixed with respect with some reference frame.
Since electric and magnetic fields appear in different mixtures if we change our frame of reference, we must be careful about how we look at the fields E and B. We must not attach too much reality to them. The field lines may disappear if we try to observe them from different coordinate systems.
The field lines that we see in our textbooks for electric and magnetic fields are only mathematical constructs to help us understand and clarify the effects more easily. We can say more accurately that there is such a thing as a transformed electromagnetic field with a new magnitude and direction. Einstein's special relativity and Lorentz transformation make this view possible.