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Imagine a charged ball with surface charges. Let it travel with a constant velocity $v$. Imagine a point outside the ball. Since the charge is moving, there is a electric displacement displacement current $\frac {\delta E}{\delta t} > 0$

There is no current $j$ at this point.

By maxwells law:

$$\nabla \times B = \mu_o \,(j \, + \epsilon_0 \frac{\delta E}{\delta t})$$

there is a rotation field of $B$.

But how does this magnetic field look like?

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  • $\begingroup$ You need to write down what your current density and charge density looks like. Then you can solve Maxwell's equations. If you are comfortable with relativistic electrodynamics, because your velocity is constant, you should be able to solve for static charge density and then simply boost the solution into a frame where the ball is moving. $\endgroup$
    – Cryo
    Commented Mar 18 at 21:32
  • $\begingroup$ Begin by writing down what the charge and current densities are, not with words, but with functions, i.e. $\mathbf{J}\left(t, \mathbf{r}\right)=\dots$. You may find it simpler to solve for infinitesimally small ball, so you need to decide whether finite size is important to you. $\endgroup$
    – Cryo
    Commented Mar 18 at 21:34

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