On Physical Lines of Force/Part II/Prop. VII

On Physical Lines of Force → Part II → Prop. VII

Maxwell's statement

Prop. VII. — To find the energy spent upon a vortex in unit of time by the layer of particles which surrounds it.

Where P, Q, R come from

This proposition introduces the symbols that will become the electric field. Maxwell defines $P,Q,R$ as the forces acting on unity of the particles — that is, the force per unit of the electric substance, resolved along the three axes.

It is worth pausing on how physical this definition is. $P,Q,R$ are not an abstraction; they are the push exerted on the little rolling spheres by whatever is driving them. Everything the modern electric field does in this theory, it does as a mechanical force on a material particle.

The counting argument

Each particle sits between two vortices, touching each at opposite ends of a diameter. So the reaction of a particle is shared equally between the two vortices it separates — half to each.

Multiply by the superficial density of the particles, which was fixed back in equation (34) of Prop. V:

since the superficial density of the particles is … (see equation (34))

to get the force per unit area of vortex surface. That is equation (47): the work expended on an element of the vortex's surface.

Then integrate over the whole closed surface of the vortex. The result is equation (50): the total work done on the vortex, per unit time, by the layer of particles around it.

$$\frac{dE}{dt}{|}_{\text{supplied}}=-\frac{V}{4\pi }[\mu \alpha (\frac{dR}{dy}-\frac{dQ}{dz})+\mu \beta (\frac{dP}{dz}-\frac{dR}{dx})+\mu \gamma (\frac{dQ}{dx}-\frac{dP}{dy})]\mathrm{(}\mathrm{50}\mathrm{)}$$

Provenance: the equation numbers (47) and (50) are attested by Maxwell's own citations; the printed bodies could not be recovered. The expression above is the form required for consistency with equation (46) and with the result (54) that Prop. VIII extracts from the comparison. In modern notation it is $-\frac{1}{4\pi }𝐁\cdot (\mathrm{\nabla }\times 𝐄)V$. See Sources.

Reading the answer

The structure of the result is worth noticing before the algebra of Prop. VIII arrives.

The rate at which energy flows into the vortex depends on $𝐁$ dotted with the curl of the force on the particles — not on the force itself. A uniform force on the particles does no net work on a vortex: it pushes equally on both sides, and the two contributions cancel around the closed surface. Only a force whose circulation is non-zero can spin a vortex up.

Which is the mechanical statement of Faraday's law before Faraday's law has been written down. A current in a straight line does not induce; a current going round a loop does.

The pairing

Prop. VI and Prop. VII are two accounts of the same quantity:

	what it gives	equation
Prop. VI	the energy a vortex has, as a function of $\mu$ and $\alpha ,\beta ,\gamma$	(46)
Prop. VII	the rate at which the particles supply energy to it, as a function of $P,Q,R$	(50)

Differentiate the first with respect to time, set it equal to the second, and you have an equation connecting the rate of change of the magnetic field to the forces on the electricity. That equation, separated into its three components, is (54).

Maxwell's own phrasing when he gets there:

Comparing this value with that given in equation (50) …

That comparison is Prop. VIII, and it is where induction appears.

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