Magnetic force
1820  Oersted finds that an electric current produces a magnetic field.
1826  Ampere finds that electric currents attract each other.
      In other words, an electric current produces a magnetic field and the
      magnetic field exerts a force on nearby currents.
1889  Heaviside published the force law for a charge moving in a magnetic field.
1892  Lorentz discovers the "Lorentz transform" for special relativity
      This offered an explanation for the Michelson-Morley experiment
1904  Lorentz finds that the Lorentz transform resolves the paradoxes of
      Maxwell's equations
1905  Einstein and Poincare each publish a complete formulation of the
      theory of special relativity

The magnetic force was the first clue for special relativity. If two like charges are moving in parallel they experience an attractive magnetic force proportional to Velocity^2. If you shift to a frame moving with the electrons the velocity is zero and the magnetic force vanishes, an apparent contradiction. The force shouldn't depend on frame of reference. The contradiction is resolved by special relativity.

We will use CGS units, which are designed to be convenient for electromagnetism. MKS units are awkward for electromagnetism.

Q  =  Charge
V  =  Speed
C  =  Speed of light
Fe =  Electric force between two charged particles
Fm =  Magnetic force between two charged particles moving in parallel at speed V
R  =  Distance between two charged particles
L  =  Lorentz factor for time dilation
   =  (1-V^2/C^2)^(-1/2)

If V << C,

Fe =         Q^2 / R^2
Fm = (V/C)^2 Q^2 / R^2
The magnetic force is weaker than the electric force by a factor of (V/C)^2. Magnetic forces are only noticeable if the charges are moving fast or if the the material is electrically neutral so that the electric forces cancel out.

If two charges are at rest the repulsive force between them is

Fe = Q^2 / R^2
If the charges are moving, then an observer in a stationary frame sees them as slowed down in time by a factor of L, which is equivalent to a reduction in the apparent force.

For a uniformly-accelerating object,

X = .5 A T^2
If T is changed by a factor of L then the apparent acceleration is changed by a factor of L^2.

The electric force adjusted for time dilation is

Fe =     L^-2      Q^2 / R^2
   =  [1-(V/C)^2]  Q^2 / R^2
   =   Q^2 / R^2   -  (V/C)^2  Q^2 / R^2
The term
(V/C)^2  Q^2 / R^2
is the same as the expression for the magnetic force. The warping of spacetime by the Lorentz transform gives rise to the extra term in the electric force, which can equivalently be interpreted as a magnetic force.

For example, two stationary positive charges experience a repulsive electric force. If the charges are moving in parallel at speed V then time dilation makes the repulsion seem smaller. The reduction in the repulsive electric force can be interpreted as the addition of an attractive magnetic force.

The magnetic force and special relativity are equivalent. Without special relativity there would be no magnetic forces.