Lorentz' Derivation of Lorentz force I have one simple question: Can someone point me to the paper where H.A. Lorentz published Lorentz force? I was digging through the usual literature in electromagnetics (Jackson, Griffiths, Thide, Sommerfeld, Stratton) for the reference on the original Lorentz paper with no success. Any help is greatly appreciated
 A: What is nowadays known as the Lorentz force law was originally due to Maxwell, equation 77 in Part 2 of his 1861 paper On Physical Lines of Force (p. 482 of vol. 1 of his Scientific Papers), which in more modern vector notation looks like:
$$\mathbf{E}=\mu\mathbf{v}\times\mathbf{H}-\frac{\partial\mathbf{A}}{\partial t}-\nabla{\psi},$$
where $\mathbf{E}$ is EMF.
References:• Tombe, F. D. (2012a). Maxwell’s original equations. The General Science Journal. pp. 4-5.• See the references in p. 225 fn. 26:Assis, André Koch Torres; Chaib, J. P. M. C; Ampère, André-Marie (2015). Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience (PDF). Montreal: Apeiron. ISBN 978-1-987980-03-5. p. 225:[It] was first obtained by J. C. Maxwell between 1861 and 1873, and by H. A. Lorentz in 1895.²⁶
A: There is widespread (and false) folklore that the relations that Maxwell posed between the electric displacement field (which we now call $$), the magnetic and electric potentials (which we, respectively, denote $$ and $φ$ today) were early precursors of the Lorentz force. What Maxwell wrote, when expressed in terms of today's notation would be:
$$ = ε , \hspace 1em  = -∇φ - \frac{∂}{∂t} + ×.$$
The equations Maxwell actually wrote are not relativistic, but non-relativistic and Galilean covariant; and he went to great pains to show this covariance.
The velocity $$ is the light speed reference for the non-relativistic theory. In many places in his treatise, he used $$ in place of $$, and sometimes he used both even in the same section.
More precisely, $$ was used to identify the (unique) frame in which the constitutive law becomes isotropic, i.e. the frame in which
$$ = ε \left(-∇φ - \frac{∂}{∂t}\right).$$
The version of the constitutive law with $ ≠ $ is:
$$ = ε \left(-∇φ - \frac{∂}{∂t} + ×\right).$$
This led to a dichotomy between what was known as the "moving" version of Maxwell's equations, where $ ≠ $, versus the "stationary" version, where $ = $.
Today, we would define the $$ in terms of the potential as
$$ = -∇φ - \frac{∂}{∂t},$$
and, instead, write his constitutive relation as:
$$ = ε \left( + ×\right),$$
For the other constitutive law - that linking the magnetic induction $$ to the magnetic field $$, Maxwell initially failed to make any clear distinction between them in his early papers, except to essentially write $$ as the glyph $μ$ - or so it would have appeared, if we used today's names. There was also an extra factor of $4π$ in his version of $$, which further muddied the waters. It was only later in the treatise that he (somewhat) more clearly separated out the two and actually wrote $ = μ$.
But he still failed to make a clean separation between the two, which is a fatal error since they have different transformation properties. For consistency, its constitutive law has to be written as
$$ = μ( - ×),$$
and it wasn't until after Maxwell died that this discrepancy was noted and the correction added. I think it was Thomas who added in the correction.
The additional $-×$ term was actually verified in 1902 by a husband and wife team, I don't remember their names off-hand, it's been over 100 years now.
Maxwell's equations, in full, with these constitutive laws then reads:
$$
 = -∇φ - \frac{∂}{∂t}, \hspace 1em  = ∇×, \\
∇· = ρ, \hspace 1em ∇× - \frac{∂}{∂t} = , \\
 = ε( + ×), \hspace 1em  = μ( - ×),
$$
where $ρ$ and $$ are, respectively, the charge and current density for sources. The other equations following as a consequence, are:
$$∇· = 0, \hspace 1em ∇× + \frac{∂}{∂t} = , \hspace 1em ∇· + \frac{∂ρ}{∂t} = 0,$$
and he also included a field-current law $ = σ $ (the "microscopic Ohm's law"), which isn't any longer counted as being amongst the fundamental equations, but more phenomenological in nature.
In the frame of isotropy, the "stationary" form of the constitutive laws reduces to
$$ = ε, \hspace 1em  = μ.$$
Technically, you also have to have $$-dependent terms for the microscopic Ohm's law, and that wasn't noted until 1908 by Einstein and Laub, and I believe also by Minkowski ... more on that below. But, actually, the frame of the microscopic Ohm's law need not be the same as the frame of isotropy, so it could be a different $$.
In these equations, light propagates in a sphere out from a source, with a center that drifts, if $ ≠ $. When the level of resolution in technology made it possible to look for a reference speed for light, the search for that drift, and for a means to measure $$ to ensue, this is what eventually led to the Michelson-Morley experiment as well as other attempts to find $$ and to find which frame was the frame of isotropy.
That there should be dichotomy between a "moving" and "stationary" form - and that such a dichotomy would exist even in the (near-)vacuum of outer space - did not sit well with Einstein. It was Einstein who showed that the "stationary" form actually applied in all inertial frames and it was the whole point of the term "moving" in the title "On the Electrodynamics of Moving Bodies". Even in the opening section he made circumlocutionary reference to $$, stating that - with his amendment of the Galilean transform to the Lorentz transform - it was no longer needed and became "superfluous".
These constitutive equations are legit - when describing moving media; but they undergo a minor modification to make them consistent with Relativity.
Einstein later returned to this issue, with Laub, in 1908 and addressed the matter more fully. Independently, Minkowski also addressed the issue, and it was in that paper, where he did, that he introduced his four-dimensional geometry.
The relativistic version of the Maxwell-Thomson constitutive relations are, today, known as the Maxwell-Minkowski constitutive relations and would be written as:
$$ + \frac{1}{c^2} × = ε( + ×), \hspace 1em  - \frac{1}{c^2} × = μ( - ×).$$
These relations do hold even in a vacuum! But, in a vacuum, since $εμ = (1/c)^2$, they are (almost) provably equivalent to the "stationary" isotropic version of the constitutive laws $ = ε$ and $ = μ$. So, just as Einstein had asserted: $$ becomes superfluous.
There's is also a $$-dependence that has to be added in the microscopic Ohm's law, which they did; though they failed to note that it doesn't have to be the same reference speed $$.
Thus, in the relativistic version of the theory, in a vacuum, light propagates outward on a sphere that has a fixed center; though in a medium, there will be center-drift - both in the relativistic and non-relativistic versions of the constitutive laws.
Notice, I said "almost" to "provably equivalent". The exception occurs, if the medium itself has $ε μ ||^2 = 1$, even as you approach a vacuum: $ε μ c^2 → 1$, which might be relevant in areas such as plasma physics.
The force law that Maxwell actually tried to write down was a mangled mess and never survived outside of or beyond his treatise. It was most definitely not the Lorentz force law.
