Copyright © 1999 David E. Rutherford
All Rights Reserved

1. Forces on a Point Charge Moving Parallel to a Current-Carrying Wire

We would like to find the components of the four-force on a charged particle moving parallel to a current carrying wire. Let there be a long straight current-carrying wire extending along the x-axis. Since the wire lies along the x-axis, the drift velocity four-vector of the charges in the wire is . A point charge with proper magnitude is moving with four-velocity parallel to the wire. We wish to find the components of the four-force on the particle by using the electric field tensor, , in non-relativistic units, where the are the components of the electric field four-vector. We use this, rather than the equivalent , where the are the components of the potential four-vector, because it is simpler in this case. From Gauss' law, the electric field at the position of the charge is in the y-direction, so the electric field four-vector at the position of the charge is .

The four-force on the particle is

or

in non-relativistic units, where the are the components of the four-momentum of the particle.

From Eqs. (1), we find that the 1-component of the force four-vector is

But we see, from the components of the electric field and drift velocity four-vectors, in this case, that the only nonzero components of are , , , and . Therefore, Eq. (3) becomes

Remembering that , we get for the force in the x-direction

Using the same methods, we find

or, for the y-component of the four-force we get

Similarly,

and the z-component is

Finally, we have

or for the t-component we get

Copyright © 1999 David E. Rutherford
All Rights Reserved