Electromagnetic potentials due to a moving electron

We have aheady made implicit use of the fact that an electromagnetic field can be represented as a foin-dimensional vector with components A, Ay, A and i , where 

We now wish to derive the potentials due to an electron with charge —e moving with uniform velocity v along the x axis of the inertial flame S introduced in the previous section, i.e. the electron is at rest in the S frame (see figure 3.1). Suppose that we wish to calculate the potentials at the field point (x, y, z) at time t = 0 in S. By (3.183) this corresponds to the space-time point (x, y, z, t ) in S where 

Now the potentials at this point in S are easily derived since the electron is stationary in this frame  

The expressions are, of course, time independent. If we transform these potentials to 

Note that these expressions refer to a particular instant of time in the S frame. The potentials in this frame are time-dependent, since they follow the charge during its motion. At a general time t, for example. 

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