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The Lorentz force is

$$ F = q(E + v \times B) $$

where $E$ is denoted the electric field and $B$ is denoted the magnetic field. Further, Maxwell's equations are

$$\begin{array}{rcl} \nabla \cdot D &=& \rho\\ \nabla \times H - \frac{\partial D}{\partial t} &=& J\\ \nabla \times E + \frac{\partial B}{\partial t} &=& 0\\ \nabla \cdot B &=& 0 \end{array}$$

The question now is, there seems to be a difference in that it is $E$ that shows up in the Lorentz force, while the first Maxwell equations say something about $D$, while $B$ is present both in the Lorentz force and in the last Maxwell equation.

What is the reason for this discrepancy, or apparent asymmetry between how the electric and magnetic fields are treated? This question is closely related to the discussion in Asymmetry definitions electric $\chi_e$ and magnetic susceptibility $\chi_m$ but that discussion failed to raise how the field appears in the Lorentz force and there is also no clear answer to the question.

Qmechanic
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Robert
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1 Answers1

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The Lorentz force formula for EM force on charged particle is valid only in vacuum, where $\mathbf D=\epsilon_0\mathbf E$ and $\mathbf B = \mu_0 \mathbf H$. It can be written in different ways:

$$ \mathbf F = q\mathbf E + q\mathbf v\times \mathbf B\,\,\,(*) $$ or $$ \mathbf F = q\mathbf E + q\mathbf v\times \mu_0 \mathbf H $$ or

$$ \mathbf F = q\mathbf D/\epsilon_0 + q\mathbf v\times \mathbf B $$

or

$$ \mathbf F = q\mathbf D/\epsilon_0 + q\mathbf v\times \mu_0 \mathbf H. $$

So the "asymmetry" you observe is due to particular choice of $(*)$ as the convention from the above 4 possibilities.

Roger
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Ján Lalinský
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    Thank you, but I am not sure I believe your explanation. Looking at e.g. Jackson, there is no mentioning that the Lorentz force is only valid in a vacuum. I would think that the Lorentz force equation is rather telling exactly not that, but rather that the magnetic force on a particle is rather due to both external magnetic field and any magnetization in the presence of the particle. – Robert Mar 25 '21 at 22:03
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    Jackson isn't some kind of definitive source on EM theory. You can see the problem with the Lorentz formula easily: it is a function of two macroscopic fields $\mathbf E,\mathbf B$ only, which are too smooth to correctly describe net force highly dependent on position and velocity in the medium due to close presence of big number of charged particles. You can use the Lorentz formula inside material medium to formulate a microscopic theory of charged particles, but the fields have to be the microscopic highly fluctuating fields $\mathbf e,\mathbf b$. – Ján Lalinský Mar 25 '21 at 22:34
  • It is not a given that these microscopic forces sum up to something as simple as the Lorentz formula with two macroscopic fields. The Lorentz formula works with the microscopic fields, but at that level of resolution, there is no point formulating equations in terms of magnetization and polarization. So there is no difference between $\mathbf b$ and $\mathbf \mu_0 \mathbf h$ on that level, we are "in vacuum" back again even inside the material medium. – Ján Lalinský Mar 25 '21 at 22:35
  • For example where net force in medium isn't always simply function of macroscopic fields $\mathbf E,\mathbf B$, see https://feynmanlectures.caltech.edu/II_10.html section 10-5. Force between charges placed in a dielectric is different when the dielectric is liquid than when it is solid of the same dielectric constant. Additional details such as internal stresses in the medium become important. – Ján Lalinský Mar 25 '21 at 22:50
  • Indeed, the stated Lorentz force is only valid in vacuum. The problem of generalizing to a spatially averaged medium, e.g. in which one defines a polarization density, turns out to be unresolved. There are several competing optical force densities, each self-consistently conserving momentum with their own corresponding stress tensors and momentum densities defined in terms of E, D, H, and B. – Gilbert Mar 26 '21 at 04:26
  • @JánLalinský but Jackson is as probably good as a published source. Could you pls cite a source that Lorentz force is not valid in matter (either the macroscopic or microscopic) – lalala Jun 15 '21 at 16:22