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

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

Maxwell's statement

Prop. VI. — To determine the actual energy of a portion of a medium due to the motion of the vortices within it.

"Actual energy" is the term of the period for what we now call kinetic energy, as opposed to "potential energy". Maxwell is asking: how much energy is stored in the spinning?

The argument

The energy of a rotating body goes as its mass times the square of its speed. In the medium, the mass is carried by the density $\mu$ and the speed is the circumferential velocity, whose components are $\alpha ,\beta ,\gamma$. So the energy of a volume $V$ must have the form

$$E=C\mu ({\alpha }^2+{\beta }^2+{\gamma }^2)V$$

with $C$ an unknown constant that depends on how the angular velocity and density are distributed inside each vortex — the same unspecified distribution that Prop. I left free.

Maxwell fixes $C$ by conservation of energy, and does it with an argument of real elegance. Move one magnetic system relative to another. The mechanical work done is already known: it is given by equations (12) of Part I, the force per unit volume. And that work must come from somewhere — it comes out of the energy stored in the vortices. Equating the two determines $C$.

The result is equation (46):

$$E=\frac{\mu }{8\pi }({\alpha }^2+{\beta }^2+{\gamma }^2)V\mathrm{(}\mathrm{46}\mathrm{)}$$

Provenance: the equation number (46) is attested — Maxwell cites it by number in Prop. VIII ("Let V be the volume of a vortex, then by (46) its energy is …"). The body is a reconstruction; the coefficient $1/8\pi$ is the value required by consistency with equations (12) and is the standard result. See Sources.

What this equation is

Divide by the volume and it reads: the energy stored per unit volume of the magnetic field is

$$u=\frac{\mu H^2}{8\pi }=\frac{𝐁\cdot 𝐇}{8\pi }$$

which is the magnetic energy density, still written that way in Gaussian units today. It enters physics here, as the kinetic energy of an invisible flywheel.

That reading is not a retrospective gloss. It is the whole point of the proposition, and Maxwell uses it as such immediately: in Prop. VIII he equates the rate of change of this energy with the work supplied by the particles, and Faraday's law falls out. The energy of the magnetic field is treated from the start as mechanical energy of an actual moving medium, with all the consequences that carries — it can be stored, it can be given back, and its rate of change must be accounted for.

Why the mass of the particles does not appear

Because Maxwell said in Prop. V that

the mass of all the particles together is inappreciable compared with that of the vortices

so all the kinetic energy sits in the rotation and none in the translation of the electricity. This is a real physical commitment, and it is what makes the model's inductance purely magnetic. A modern reader may note that the electron's mass does contribute in reality, and that Maxwell's assumption is the reason his theory has no analogue of the inertia of charge carriers.

The bridge to induction

Once you have

• an expression for the energy stored in the field (Prop. VI), and
• an expression for the rate at which the particles do work on the vortices (Prop. VII),

then equating the two is a pure piece of energy bookkeeping, and it must produce a relation between the rate of change of the magnetic field and the forces on the electricity. That relation is Faraday's law. Prop. VIII does exactly this.

This is the structural insight of Part II, and it is worth stating flatly: Maxwell derives electromagnetic induction from conservation of energy in a rotating mechanical system. He does not postulate it. Faraday's law arrives as a bookkeeping identity.

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Next: Prop. VII · Previous: Prop. V