electromagnetism - what the explicit formula magnetic field a electron that moving on the curve path?

,hi,we suppose ,we have a single electron that moving on a curve path,the velocity is v (it is variable),the path moving is a curve not direct path.i saw maxwell equation my question is ,is there a explicit( not implicit ) equation for magnetic filed a single electron that moving on curve path.

thanks for reply

Answer

Maxwell alone doesn't specifically attribute an electric or magnetic field due to any specific charge. But an example of a solution to Maxwell can be provided if both the electric and magnetic field are computed as the electric and magnetic parts of the electromagnetic field given by Jefimenko's equations:

$$\vec E(\vec r,t)=\frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec r',t_r)}{|\vec r -\vec r'|}+\frac{\partial \rho(\vec r',t_r)}{c\partial t}\right]\frac{\vec r -\vec r'}{|\vec r -\vec r'|^2}\; \mathrm{d}^3\vec{r}' -\frac{1}{4\pi\epsilon_0c^2}\int\frac{1}{|\vec r-\vec r'|}\frac{\partial \vec J(\vec r',t_r)}{\partial t}\mathbb{d}^3\vec r'$$ and $$\vec B(\vec r,t)=\frac{\mu_0}{4\pi}\int\left[\frac{\vec J(\vec r',t_r)}{|\vec r -\vec r'|^3}+\frac{1}{|\vec r -\vec r'|^2}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\right]\times(\vec r -\vec r')\mathbb{d}^3\vec r'$$ where $t_r$ is actually a function of $\vec r'$, specifically $t_r=t-\frac{|\vec r-\vec r'|}{c}.$

These reduce to Coulomb and Biot-Savart only when those time derivatives are exactly zero, which is statics. So Jefimenko is an example of proper time dependent laws for the electromagnetic field.

If you have a single charge you can consider the situation where the charge is a Delta function and the current is a Delta function times the velocity.

You will find that the term that depends on the change in current requires knowing the acceleration of the charge. And you'll notice that every point in space requires knowing the location and the velocity and acceleration of the charge at different (and earlier) times in the past. And none of them depend on the charge right now.

Simply put. No one can magically see where the charge is now. And no fields are affected by that. Instead, for every place $\vec r$ where the charge isn't located you make a backwards lightcone, and see where, $\vec w,$ it intersects the worldline of the charge. Then you take the acceleration, $\vec a,$ velocity, $\vec v,$ and position $\vec w$ of the charge at that time and compute the electric field and magnetic field at that place $\vec r$ based on those three vectors. To compute the fields you can use $\vec d=\vec r -\vec w$ and $\vec u=c\hat d-\vec v$ and get (adapted from Griffiths' Introduction to electrodynamics):

$$\vec E(\vec r,t)=\frac{q}{4\pi\epsilon_0}\frac{d}{(\vec d\cdot \vec u)^3}\left[(c^2-v^2)\vec u+\vec d\times(\vec u\times \vec a)\right]$$

and $$\vec B(\vec r,t)=\frac{1}{c}\hat d\times \vec E(\vec r,t).$$

Recall that the $t$ is not the time when the charge had the position $\vec w,$ velocity $\vec v,$ and acceleration $\vec a.$ Those are at an earlier time, the time back when the charge broadcasted its position $\vec w,$ velocity $\vec v,$ and acceleration $\vec a$ just in time to arrive at $\vec r$ at time $t.$