The electric and magnetic fields of a particle of charge q moving in a straight line with speed v = ?c, given by (11.152), become more and more concentrated as ? ? 1, as is indicated in Fig. Choose axes so that the charge moves along the z axis in the positive direction, passing the origin at t = 0. Let the spatial coordinates of the observation point be (x, y, z) and define the transverse vector r_?, with components x and y. Consider the fields and the source in the limit of ? = 1.

(a) Show that the fields can be written as

where v is a unit vector in the direction of the particle's velocity.

(b) Show by substitution into the Maxwell equations that these fields are consistent with a 4-vector source density,

J^? = qcv^??⁽²⁾(r_?)?(ct ? z)

where the 4-vector v^? = (1, v).

(c) Show that the fields of part a are derivable from either of the following 4-vector potentials,

where λ is an irrelevant parameter setting the scale of the logarithm. Show that the two potentials differ by a gauge transformation and find the gauge function,x-

Transcribed Image Text:

24 8(et - 2): в 3 2q 8(ct - z) E