Imagine a point charge moving along x-axis in empty space at a constant velocity $v$. Since the charge is moving, the electric and magnetic field around it should be surmisable from this image (courtesy: brilliant.org)

[![E/M field around a charge moving at constant velocity](https://i.stack.imgur.com/mQSdJ.gif)](https://i.stack.imgur.com/mQSdJ.gif)

I suppose one can explain the shape of the above magnetic field from the following Maxwell's Equation:

$\nabla\times \boldsymbol{B} = \frac{1}{c^2} \frac{\partial \boldsymbol{E}}{\partial t}$

The current component $\boldsymbol{J}$ is not needed since we have a lone charge moving in space.

If I understand the above equation correctly, the change w.r.t. time in $\boldsymbol{E}$ at any point in space is accompanied by $\boldsymbol{B}$ curling around it. Since the moving charge in our frame of reference concedes $\boldsymbol{E}$ at some points in space, and bolsters $\boldsymbol{E}$ at others, we see a concomitant curling $\boldsymbol{B}$ in proportion to the change in $\boldsymbol{E}$.

However, just like the diverging $\boldsymbol{E}$, the curling $\boldsymbol{B}$ is also changing w.r.t. time. Where is the accompanying $\boldsymbol{E}$ field that should arise due to the equation below?

$\nabla\times \boldsymbol{E} = -\frac{\partial \boldsymbol{B}}{\partial t}$

Why don't the authors usually comment on $\boldsymbol{E}$ accompanying the curling $\boldsymbol{B}$? I am assuming that there is no additional $\boldsymbol{E}$. Maybe, it's just that the curl of the already curling $\boldsymbol{B}$ is the original diverging $\boldsymbol{E}$, although moving in one direction. Am I correct? Where can I learn more about this?