On Physical Lines of Force/Modern derivations

On Physical Lines of Force → Modern derivations

Every one of the four equations now called Maxwell's equations appears somewhere in this paper, along with the constitutive relations, the vector potential, the field energy densities, the stress tensor, the motional term of the Lorentz force, and the identification of light. This section takes each result in Maxwell's 1861 notation and carries it to the form a reader meets today, making the changes of convention explicit at every step.

The master table

modern equation	Maxwell's number	where in the paper	derivation
$\mathrm{\nabla }\cdot 𝐃=\rho$	(113)	Part III, Prop. XIV	Gauss's law for electricity
$\mathrm{\nabla }\cdot 𝐁=0$	(56)	Part II, Prop. VIII	No magnetic monopoles
$\mathrm{\nabla }\times 𝐄=-\partial 𝐁/\partial t$	(54)	Part II, Prop. VIII	Faraday's law
$\mathrm{\nabla }\times 𝐇=𝐉+\partial 𝐃/\partial t$	(9), corrected at (112)	Part I, Prop. III and Part III, Prop. XIV	Ampère–Maxwell

And the supporting results:

result	Maxwell's number	where	derivation
$𝐁=\mathrm{\nabla }\times 𝐀$	(55)	Part II, Prop. VIII	The vector potential
$𝐄=-\partial 𝐀/\partial t-\mathrm{\nabla }\phi$	(58)	Part II, Prop. VIII	The vector potential
$𝐄=𝐯\times 𝐁-\partial 𝐀/\partial t-\mathrm{\nabla }\phi$	(77)	Part II, Prop. XI	The Lorentz force
$𝐁=\mu 𝐇$, $𝐃=\epsilon 𝐄$	(105) and Part I	Part III, Prop. XIII	Constitutive relations
the magnetic stress tensor	Prop. II, then (12)–(14)	Part I, Props. II and III	The stress tensor
magnetic energy density $𝐁\cdot 𝐇/8\pi$	(46)	Part II, Prop. VI	used in the wave equation
$c=1/\sqrt{{\epsilon }_0{\mu }_0}$	unnumbered	Part III, Prop. XVI	The unit ratio and c
Faraday rotation, results (A)–(F)	—	Part IV	Faraday rotation today

Suggested order

If you are reading this section on its own rather than following links from the Part pages:

1. From components to vectors — read this first; everything else assumes it
2. No magnetic monopoles
3. Ampère's law and its correction
4. Faraday's law of induction
5. Gauss's law for electricity
6. The constitutive relations
7. The displacement current
8. The unit ratio, and why it is a velocity
9. The electromagnetic wave equation
10. The vector potential
11. The motional term and the Lorentz force
12. The Maxwell stress tensor
13. Faraday rotation today

Three things to keep in mind throughout

Maxwell's equations are not Maxwell's

The four equations in the box in every textbook were assembled into that shape by Oliver Heaviside in 1884–85, working independently of and in parallel with Josiah Willard Gibbs, who developed the same vector calculus. Maxwell's own 1865 paper lists twenty equations in twenty variables. He had no del operator, no curl, no divergence as symbols — only components. What is called "Maxwell's equations" is Maxwell's physics in Heaviside's notation.

This is not a quibble. Reading the 1861 paper is much easier once you stop looking for equations that are not there and start looking for the triples that became them. See From components to vectors.

The 4π is a units convention, not physics

Maxwell works in a Gaussian-style system where the factor $4\pi$ — the surface area of a unit sphere — appears explicitly in the field equations. SI absorbs it into ${\epsilon }_0$ and ${\mu }_0$. Neither is more correct; the 4π has to appear somewhere, because the inverse-square law and the sphere are related, and the choice is where to put it.

	Gaussian (Maxwell's style)	SI
Ampère	$\mathrm{\nabla }\times 𝐇=\frac{4\pi }{c}𝐉+\frac{1}{c}\frac{\partial 𝐃}{\partial t}$	$\mathrm{\nabla }\times 𝐇=𝐉+\frac{\partial 𝐃}{\partial t}$
Gauss	$\mathrm{\nabla }\cdot 𝐃=4\pi \rho$	$\mathrm{\nabla }\cdot 𝐃=\rho$
Coulomb	$F=\frac{q_1{q}_2}{r^2}$	$F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$

Maxwell's own system is not quite modern Gaussian either — the explicit factors of $c$ in the table above come from a later tidying. Converting his results is mostly a matter of tracking the 4π, and it is where most transcription errors happen.

The equations here are reconstructions

No equation in this paper could be transcribed from a page scan for this edition. Equation numbers are attested where the pages say so; equation bodies in Maxwell's notation are reconstructions, correct in content but possibly differing from the printing in sign convention, placement of 4π, and whether a relation is written once or three times. The modern equations are, of course, just the modern equations. See Sources and editorial method for the full account.

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