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Lets say we have a uniform sphere of charges at the origin (at retarded time = 0) with some velocity and we are interested in the field at a point along the x-axis (normal to the surface of the sphere), say (x,0,0).

We could use the Lienard-Weichert field equation to find the electric field of each and every source charge to the observation point and because of symmetry of the system (at the retarded time, of course) the only component of the electric field that would be left would be along the x-direction. The field would be relatively easy to find and simple in nature.

Now, lets say we wanted to find the electric field about the same point but with a very small deviation off the x-axis, ($x,\delta y, \delta z$). Using only the on-axis (x,0,0) field and maxwell's equations is there a way to find the value of this off axis field?

Is there a method(s) one could point me to that would help me with this problem?

Note: It is an important distinction that this is electrodynamics, not statics.

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  • $\begingroup$ Is it critical that you only use maxwell's equations to find the answer? Why aren't you able to use the LW equation? $\endgroup$
    – A Nejati
    Commented Sep 16, 2018 at 5:54
  • $\begingroup$ Mainly, because I want to extend the formalism to more complicated geometries and the LW fields will give a horrendously complicated expression. I'm hoping to use another method but I would like to understand the simple example first before extending it to more complicated examples. $\endgroup$
    – Donkey Kong
    Commented Sep 17, 2018 at 22:17
  • $\begingroup$ Just to be clear, do you want the relativistically correct answer, or are you happy with the classical answer? $\endgroup$
    – A Nejati
    Commented Sep 18, 2018 at 1:57
  • $\begingroup$ Classical is fine, but if you want to give a quick disclaimer of the relativistic case that would be very eye-opening too! $\endgroup$
    – Donkey Kong
    Commented Sep 18, 2018 at 21:52

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