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 S’, 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.