On Physical Lines of Force/Part I/Applications

On Physical Lines of Force → Part I → Applications

Having got equations (12), (13), (14), Maxwell tests them on three cases that were already understood, to show that the stress theory reproduces the known answers — and then notices that in one respect it does not agree with action at a distance, and says how to check.

A single magnetic pole

With no currents present the problem reduces to two conditions: one holding at the pole itself, one holding everywhere else.

The conditions then are, that equation (18) must be fulfilled at the magnetic pole, and (19) everywhere else.

In modern language: the divergence of the induction is fixed by the pole strength at the pole, and vanishes elsewhere. Maxwell solves it and recovers the inverse-square repulsion $m/r^2$ between two poles in a standard medium.

Provenance: the numbers (18) and (19) are attested, but the surviving prose is internally inconsistent about which of the two conditions carries which number. Treat the assignment as unsettled. See Sources.

The prediction Maxwell offers for test

Here the two theories part company. On the stress theory, what passes between two poles is not an influence across empty space but a state of a material medium, and the properties of that medium must matter.

In a medium having a greater value of $\mu$ (such as oxygen, solutions of salts of iron, &c.) the attraction, on our theory, ought to be less than in air, and in diamagnetic media (such as water, melted bismuth, &c.) the attraction between the same magnetic poles ought to be greater than in air.

Then, immediately, the honest qualification:

The experiments necessary to demonstrate the difference of attraction of two magnets according to the magnetic or diamagnetic character of the medium in which they are placed, would require great precision, on account of the limited range of magnetic capacity in the fluid media known to us, and the small amount of the difference sought for as compared with the whole attraction.

Two things are worth noticing. First, Maxwell names real substances — oxygen, iron salts, water, melted bismuth — rather than speaking of "a paramagnetic medium". Second, he raises the objection to his own proposal before anyone else can: the effect is small, the available range of $\mu$ in fluids is narrow, and the measurement would be hard.

The straight cylindrical current

Let us next take the case of an electric current whose quantity is C, flowing through a cylindrical conductor whose radius is R, and whose length is infinite as compared with the size of the field of force considered.

Inside the conductor the current per unit area is $C/\pi R^2$; outside it there is no current at all, and the magnetic force falls off as the reciprocal of the distance. A unit north pole at perpendicular distance $\rho$ from the axis feels a force

$$\frac{2C}{\rho }$$

tending to move it round the conductor in the direction of the hands of a watch, if the observer view it in the direction of the current.

Provenance: the equation numbers for this case could not be established. See Sources.

Observe the phrasing once more. Maxwell has just done the algebra, and he translates the answer straight back into the laboratory — a compass needle, a watch face, an observer looking along the wire. He does this every time. It is the single most imitable habit in his writing.

Two parallel currents

Take a second current, parallel to the axis of $z$, at distance $\rho$, of quantity $C^{\prime }$, length $\ell$ and section $A$, so that its strength per unit of section is $C^{\prime }/A$.

Putting this quantity for [the current density] in equations (12, 13, 14), we find [the force] per unit of volume; and multiplying by [the volume], we find [the total force], showing that the second conductor will be attracted towards the first with a force inversely as the distance.

$$F=\frac{2C{C}^{\prime }\ell }{\rho }$$

The inverse first power, not the inverse square — which is correct for two long parallel wires, and was well established by Ampère.

The second prediction: the same test, with the sign reversed

Maxwell now points out something rather elegant. The force between two poles varies inversely as $\mu$; the force between two currents varies directly as $\mu$.

We find in this case also that the amount of attraction depends on the value of $\mu$, but that it varies directly instead of inversely as $\mu$; so that the attraction between two conducting wires will be greater in oxygen than in air, and greater in air than in water.

So the medium hypothesis makes two predictions that point in opposite directions in the same substances. Immersing the apparatus in oxygen should weaken the force between magnets and strengthen the force between currents. No bookkeeping over distances can produce a pattern like that by accident; only a theory in which the intervening stuff genuinely participates will do it.

This is the strongest empirical claim in Part I, and Maxwell leaves it standing as an invitation.

The closing sentence

We shall next consider the nature of electric currents and electromotive forces in connexion with the theory of molecular vortices.

Part I ends not on a result but on an announcement — a habit of serialised publication, and a useful one. He has established that a stress will do the work. He has not established that anything could sustain such a stress. That is Part II.

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Next: Part II · Previous: Prop. III