Direction of the magnetic field of an oscillating charge

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The discussion revolves around calculating the magnetic field generated by a harmonically oscillating point charge along the z-axis. The user simplifies the problem by using the instantaneous Coulomb field of the charge, but seeks to understand how to determine the local magnetic field's magnitude and direction. They express difficulty in applying the Ampère-Maxwell equation, as it only provides the curl of the magnetic field, not its specific direction. A suggestion is made to integrate Faraday's law over time to derive the magnetic field from the electric field's curl. The conversation highlights the complexities of superposition and interference of the magnetic field contributions as the electric field changes.
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Homework Statement
A point charge undergoes sinusoidal motion along the z-axis. In the quasi-static, non-relativistic limit—neglecting retardation and radiation—determine, at an arbitrary observation point, the magnitude and direction of the magnetic field produced by this motion.
Relevant Equations
$$\overrightarrow{E}= \frac{q}{4\pi\varepsilon_{0}}\frac{r-r^{,}}{||r-r^{,}||^{3}}$$
Hello everyone,


I’m considering a point charge q that oscillates harmonically about the origin along the z-axis, e.g. $$z_{q}(t)= A\sin(wt)$$
In a strongly simplified / quasi-instantaneous approximation I ignore retardation and take the electric field at the position ##r=(x,y,z)## simply to be the “Coulomb field at the charge’s instantaneous position”:

$$E(r,t)=\frac{q}{4\pi\varepsilon_{0}}\frac{r-r_{q}(t)}{||r-r_{q}(t)||^{3}}$$
with $$r_{q}(t)=(0,0,z_{q}(t))$$
(I’m aware this isn’t completely physically sound—I just want to understand the basic idea.)


My question:
How can I determine locally, for each point,
how large the resulting magnetic field is and in which direction it points?


Where I’m stuck:
Using the “fourth Maxwell equation” (Ampère–Maxwell) I only get ##\nabla \times \overrightarrow{B}##.
That doesn’t directly tell me where ##\overrightarrow{B}##points at my specific point.
I also don’t know how the different contributions to the magnetic field “superpose” or interfere when ##\overrightarrow{E}##changes.


Thank you in advance for any answers and insights!
 
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MadManMax said:
How can I determine locally, for each point,
how large the resulting magnetic field is and in which direction it points?
How about integrating Faraday's law ##\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}## w.r.t. time ##t## to get ##\vec{B}=-\int\nabla\times\vec{E}\,dt\,##?
 
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