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I started pondering about the electrostatic approximation in relation to plasmas, where a particle's motion $\mathbf x(t)$ is governed by the Lorentz force:

$$ m\ddot{\mathbf x}=q(\mathbf E+\dot{\mathbf x}\times\mathbf B). $$

It is clear that when the $\mathbf B$-field is constant, or at least slowly varying, Maxwell's equations de-couple such that the $\mathbf E$-field can be solved independently of the $\mathbf B$-field,

$$ \nabla\times\mathbf E =\mathbf 0, \\ \nabla\cdot\mathbf E = \frac{\rho}{\varepsilon_0}, $$

Usually, of course, one introduces the electric potential and solves the Poisson equation. So far all fine. But this does not mean, however, that $\mathbf B=\mathbf 0$. Indeed, the $\mathbf B$-field is given from the remaining Maxwell equations once the $\mathbf E$-field is solved for:

$$ \nabla\times\mathbf B=\mu_0\mathbf J + \mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t} \\ \nabla\cdot\mathbf B=0 $$

Clearly, if there is a non-negligible $\mathbf B$-field, this should be included in the Lorentz force when calculating the trajectory of the particle. However, I usually see that when the electrostatic approximation is applied, the $\mathbf B$-field is usually ignored in the Lorentz force.

Note that I do not talk about external fields, which may come in addition, but fields self-consistently determined from the particles in the plasma, i.e., $\rho$ and $\mathbf J$ given by the collection of particles.

I suppose there must be some condition for neglecting $\mathbf B$ in the Lorentz force, and I would expect this condition to follow from considering magnitudes of the various terms in the equations, but I'm not sure what this condition is. Slow particles such that the current is small? But slow compared to what? Speed of light?

Any help appreciated.

Edit:

Since this is a kinetic plasma the sources can be expressed as a sum of contribution due to each particle in the plasma:

$$ \rho=\sum_p q_p \delta(\mathbf x-\mathbf x_p) \\ \mathbf J=\sum_p q_p \dot{\mathbf x}_p \delta(\mathbf x-\mathbf x_p) $$

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  • $\begingroup$ There are two potential sources for the B field (neglecting a nearby magnet of current carrying device): the one from an incoming radiation, or the one created by another charge moving under the influence of the Lorentz force. In both cases, you can estimate the ratio between E and B and thus see if ignoring B in the Lorentz force is justified. $\endgroup$ – David Jul 15 at 12:02
  • $\begingroup$ Incoming radiation would be an external field not created by the plasma itself, so that is irrelevant. I was initially thinking perhaps I can do some dimensional reasoning on Ampére's law, showing the B-field to be negligible, but if I could show that the magnetic part of the Lorentz force is negligible in comparison to the electric part, that would work too. But how can I do that? And what is the necessary assumption? I suppose slow particles would apply to most of the literature I read. $\endgroup$ – sigvaldm Jul 17 at 7:53

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