What is Heaviside's version of Maxwell's equations?

Question

I have read, in many places, statements like this:

Heaviside was able to greatly simplify Maxwell's 20 equations in 20 variables, replacing them by four equations in two variables. Today we call these 'Maxwell's equations' forgetting that they are in fact 'Heaviside's equations'.

... but nowhere can I find exactly what these equations look like in the form Heaviside produced. There are many versions of the equations, most with more than just four.

(From this biographical article)

Please can anyone tell me exactly what are the original "four equations in two variables"?

Answer 1

This gives the four equations in the form Heaviside came up with: $$\varepsilon E = \rho$$ $$\nabla \times E = - \mu \frac{\partial H}{\partial t}$$ $$\nabla \cdot \mu H = 0$$ $$\nabla \times H = k E + \varepsilon \frac{\partial E}{\partial t}$$

where $E$ represents the electric field, $H$ represents the magnetic field, $\varepsilon$ is the permittivity, $\mu$ is the permeability, $\rho$ is the charge density, and $k$ is the conductivity.

Here, $\times$ and $\cdot$ denote the cross product and dot product. Also, $H$ is not the same as the perhaps more commonly used $B$.

Note: The first equation should properly be $\varepsilon \nabla \cdot E = \rho$, though the source leaves it out, probably due to a typo.

Answer 2

Are we all talking about the same "Heaviside" who wrote "On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field", received 1891-06-09, read 1891-06-18, that you can find a copy of over at the Royal Society (https://doi.org/10.1098/rsta.1892.0011) or on the Internet Archive (https://archive.org/details/philtrans06858697)?

Absolutely sure? :)

Probably not, because this is what he wrote...

Under the section "The Electromagnetic Equations in a Moving Medium" [Footnote about the term "Moving" pending] $$𝐁 = μ𝐇, \quad 𝐃 = c𝐄, \tag{81}$$ $$U = ½𝐄𝐃, \quad T = ½𝐇𝐁, \tag{82}$$ $$Q_1 = 𝐄𝐂, \quad Q_2 = 𝐇𝐊, \tag{83}$$ $$𝐂 = k𝐄, \quad 𝐊 = g𝐇, \tag{84}$$ $$ρ = \text{div}\ 𝐃, \quad σ = \text{div}\ 𝐁, \tag{85}$$ $$𝐉 = 𝐂 + \dot{𝐃} + 𝐪ρ, \quad 𝐆 = 𝐊 + \dot{𝐁} + 𝐪σ \tag{86,87}$$ $$𝐡 = V𝐃𝐪, \quad 𝐞 = V𝐪𝐁, \tag{88,91}$$ $$\text{curl}(𝐇 - 𝐡_0 - 𝐡) = 𝐉 = 𝐂 + \dot{𝐃} + 𝐪ρ, \label{89}\tag{89}$$ $$-\text{curl}(𝐄 - 𝐞_0 - 𝐞) = 𝐆 = 𝐊 + \dot{𝐁} + 𝐪σ, \label{90}\tag{90}$$ $$\left\{ \begin{matrix} \text{curl}\ 𝐡 = 𝐣, & 𝐉 + 𝐣 = 𝐉_0, \\ -\text{curl}\ 𝐞 = 𝐠, & 𝐆 + 𝐠 = 𝐆_0, \\ \end{matrix}\right\} \tag{92} $$ thus (as he noted), we may also write ($\ref{89}$) and ($\ref{90}$) as $$ \left\{\begin{matrix} \text{curl}(𝐇 - 𝐡_0) = 𝐉_0 = 𝐂 + \dot{𝐃} + 𝐪ρ + 𝐣, \\ -\text{curl}(𝐄 - 𝐞_0) = 𝐆_0 = 𝐊 + \dot{𝐁} + 𝐪σ + 𝐠. \end{matrix}\right\} \tag{93} $$ where

• $𝐃$ & $𝐁$ = electric displacement and magnetic induction,
• $𝐄$ & $𝐇$ = electric and magnetic force,
• $c$ & $m$ = permittivity and inductivity (for isotropic media),
• $U$ & $T$ = stored electrical & magnetic energy (per unit volume),
• $𝐂$ & $𝐊$ = fluxes (read: current densities),
• $k$ & $g$ = electric and magnetic conductivity,
• $ρ$ & $σ$ = the volume densities of electrification and magnetification,
• $σ$ and $𝐊$ are quite likely fictitious and $g$ superfluous,
• $𝐉$ & $𝐆$ = the "true" electric and magnetic currents,
• $𝐪$ = the velocity of electrification (and magnetification), i.e. the velocity relative to the medium,
• $𝐞$ & $𝐡$ = the motional electric and magnetic forces, the latter is a correction to Maxwell, who failed to include it,
• $𝐞_0$ & $𝐡_0$ = the intrinsic force (of electrification) and force of intrinsic magnetization,
• $𝐣$ & $𝐤$ = the current densities associated with the motional electric and magnetic forces,
• $𝐉_0$ & $𝐆_0$ = maybe these are the "true" currents, instead.

Now, I don't know about anyone else, but that doesn't look like four equations! I suspect that you might have been on the receiving end of "telephone tag", running a dozen layers deep, because somebody was definitely asleep at the switch there. It doesn't even resemble the item you cited!

The Footnote: "Moving"
Now ... let's put this in context. All the pre-1905 literature (as well as post-1905 literature) makes a clear distinction between "stationary" and "moving". That's meant in an absolute sense: stationary and moving with respect to The Medium. Everything before 1905 is a medium, including outer space, and has its own reference frame with respect to which "stationary" and "moving" are determined.

This is the meaning and sense of "moving" in Einstein's 1905 "On The Electrodynamics Of Moving Bodies". In that context, you can see clearly that the very title, itself, is a loud wake-up call to those of the time - something that falls on deaf ears today, of those who have no living memory of the earlier context.

But, as I noted: some things after 1905 make the distinction too.

Maxwell's equations, such as we would write them today, and such as we would write them if the magnetic sources were present too, would be: $$ ∇·𝐁 = σ, \quad ∇×𝐄 + \frac{∂𝐁}{∂t} = -𝐊, \\ ∇·𝐃 = ρ, \quad ∇×𝐇 - \frac{∂𝐃}{∂t} = +𝐉. \\ $$ The constitutive relations between the two sets of fields that goes with this, in the non-relativistic case would be: $$𝐃 = ε(𝐄 + 𝐆×𝐁), \quad 𝐁 = μ(𝐇 - 𝐆×𝐃),$$ where, now, I am using $𝐆$ to denote the velocity of the medium, instead of Heaviside's $𝐪$, because that's what Maxwell (sporadically) used.

Heaviside's other letters are scrambled up a bit. I think he's got his $𝐉$ and $𝐂$ swapped, because Maxwell used $𝐂$ for the "total" current. Maxwell's $𝐂$ is our $𝐉 + \dot{𝐃}$. The magnetic letters for currents and sources are his insertions that he called, "probably fictitious", but he's just going to put them there because he likes things to line up symmetrically. Basically, that was actually the reason he gave.

His vector product $V𝐚𝐛$ is our $𝐚×𝐛$. His scalar product $𝐚𝐛$ is our $𝐚·𝐛$. In addition, he also had a generalized inverse $𝐚^{-1}$, which is our $𝐚/(𝐚·𝐚)$. His dot was used as a separator. For instance, his $𝐚·𝐛𝐜$ actually means his $𝐚(𝐛𝐜)$, or our $𝐚𝐛·𝐜$.

I won't even try to line up which combinations of his letters correspond to our letters. They don't exactly match. Maxwell's $𝐄$, for instance, is our $𝐄 + 𝐆×𝐁$, as it was for Hertz (after you convert his letters to our letters), while Lorentz stuck with the bare $𝐄$, and added the $𝐆×𝐁$ as a separate term. Maxwell had no $-𝐆×𝐃$ addition to $𝐇 - 𝐆×𝐃$. Thomson made that correction. I thought it was him first, but maybe Heaviside did it, since he laid claim to the "motional" magnetic force as his idea.

I'm not entirely sure which $𝐄$ Heaviside's is: Maxwell's or ours; the same for $𝐇$. I'll have to look at it more closely and (maybe) add a few remarks on this with a later edit.

That passing over the 1905 divide changes nothing, here. The $𝐆$ is still there - except for the vacuum. The relativistic version of the constitutive relations above is the Maxwell-Minkowski(-Einstein-Laub) relations: $$𝐃 + \frac{𝐆×𝐇}{c^2} = ε(𝐄 + 𝐆×𝐁), \quad 𝐁 - \frac{𝐆×𝐄}{c^2} = μ(𝐇 - 𝐆×𝐃).$$ The inclusions on the left-hand side of each equation are offsets that will cancel the "motional" forces on the right ... if ... $εμ = (1/c)^2$.

Hertz' and Lorentz' constitutive laws are equivalent to the non-relativistic versions, not the Maxwell-Minkowski relations. You'll have to go through the exercise to lay out a "Modern versus Heaviside" Rosetta Stone to verify that his equations line up with the Maxwell-Thomson(-Hertz-Lorentz) relations, as I'm almost certain they do.

But, in any case: there's a whole lot more than four equations there! And, I scanned volumes 1 and 2 of his "Electromagnetic Theory" (1899). Basically, he just lifted his earlier equations and put them in there, with further detail and elaboration.