- #1
MadManMax
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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!
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!
