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I'm working on an N-body simulator for charged particles. I know that moving charged particles generate a magnetic field, and another moving charged particle could be effected by this magnetic field.

Using the Maxwell's Equations we can derive the magnetic field of a point charge moving at a constant velocity, which is roughly approximated by Biot-Savart's Law if we ignore relativistic effects.

Here's the magnetic field of B at a point due to the motion of a charged particle (ignoring relativity): \begin{equation} \vec{B} = \frac{\mu_0 q \vec{v}}{4 \pi} \times \frac{\hat{r}}{|\vec{r}|^2} \end{equation}

My question is what happens if we stop assuming that the velocity of particle A is constant (for example, two charged particles A and B orbiting and interacting with each other)? Can I accurately assume for a certain time step in my simulation that the velocity is constant and use that to calculate the magnetic field due to that particle's motion? Or are their other terms which would arise/change in the derivation from Maxwell's Equations if velocity is not constant?

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3 Answers 3

If you consider particles with acceleration you would get electromagnetic waves. It would depend on the simulation how much this this is relevant to your situation.

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For argument's sake, how would these affect the n-body simulation, and how would I quantify these? I'm primarily interested in the motion of particles/group of particles. –  helloworld922 May 24 '13 at 23:06
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The equation you quote is only valid in a stationary case. If the particles interact at long distances, and their velocities are changing, there will be some time before the electromagnetic field reaches particles far apart. Basically, you can use your formula only if the velocity changes at a rate much slower than the fields would travel through the simulation box.

In the general case, you cannot simulate N-body interactions like that. One widely-used technique, called "Particle-in-Cell", works by solving the Maxwell equations on a grid with definite cell sizes. The local fields are determined by local charges and currents, as stated in Maxwell equations. In turn, charges are accelerated by the Lorentz force that fields exert on them.

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You may want to refer to chapter 9 of Griffith's book. Using the notations in his book, if $\vec{w}(t)$ denotes the trajectory of the moving charge, $t_r$ is the retarded time and $\vec{r}_1 = \vec{r} - \vec{w}(t_r)$ then the electric field due to the charge is \begin{equation} \vec{E}(\vec{r},t) = \frac{q}{4\pi\epsilon_0}\frac{\vec{r}_1}{(\vec{r}_1\cdot\vec{u})^3}[\vec{u}(c^2 - v^2 + \vec{r}_1\times(\vec{u}\times\vec{a})] \end{equation} where $\vec{v}=\dot{\vec{w}}$, $\vec{u} = c\hat{r}_1 - \vec{v}$, $\hat{r}_1$ is the unit vector along $\vec{r}_1$ and $\vec{a}$ is the acceleration of the charge.

Magnetic field due to the charge is, \begin{equation} \vec{B} = \frac{1}{c}\hat{r}_1 \times \vec{E} \end{equation}

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I'm assuming for the magnetic field equation you mean 1/c^2? –  helloworld922 May 26 '13 at 0:29
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