Moving Magnet and Conductor Problem - Newton's Law of Motion in Modern Notation

Newton's Law of Motion in Modern Notation

The modern approach to obtaining the relativistic version of Newton's law of motion can be obtained by writing Maxwell's equations in covariant form and identifying a covariant form that is a generalization of Newton's law of motion.

Newton's law of motion can be written in modern covariant notation in terms of the field strength tensor as (cgs units):

where m is the particle mass, q is the charge, and

is the 4-velocity of the particle. Here, is c times the proper time of the particle and is the Minkowski metric tensor.

The field strength tensor is written in terms of fields as:

$F^{\alpha \beta} = \left( \begin{matrix} 0 & {E_x} & {E_y} & {E_z} \\ -{E_x} & 0 & cB_z & -cB_y \\ -{E_y} & -cB_z & 0 & cB_x \\ -{E_z} & cB_y & -cB_x & 0 \end{matrix} \right) .$

Alternatively, using the four vector:

related to the electric and magnetic fields by:

the field tensor becomes:

where:

The fields are transformed to a frame moving with constant relative velocity by:

$\acute{F}^{\mu \nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta} ,$

where ${\Lambda^{\mu}}_{\alpha}$ is a Lorentz transformation.

In the magnet/conductor problem this gives

which agrees with the traditional transformation when one takes into account the difference between SI and cgs units. Thus, the relativistic modification to Newton's law of motion using the traditional Lorentz force yields predictions for the motion of particles that are consistent in all frames of reference with Maxwell's equations.

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