Part II — Prop. IV

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

Maxwell's statement

Prop. IV. — To determine the motion of a layer of particles separating two vortices.

The set-up, in Maxwell's words

Let the circumferential velocity of a vortex, multiplied by the three direction-cosines of its axis respectively, be α, β, γ, as in Prop. II. Let l, m, n be the direction-cosines of the normal to any part of the surface of this vortex, the outside of the surface being regarded positive. Then the components of the velocity of the particles of the vortex at this part of its surface will be …

If this portion of the surface be in contact with another vortex whose velocities are [those of the second vortex], then a layer of very small particles placed between them will have a velocity which will be the mean of the superficial velocities of the vortices which they separate … since the normal to the second vortex is in the opposite direction to that of the first.

The kinematics

A point on the surface of a spinning body moves with the cross product of the angular velocity and the radius. Maxwell has arranged his symbols so that $\alpha ,\beta ,\gamma$ are the circumferential velocity resolved along the axes, so the velocity of a surface point whose outward normal is $(l,m,n)$ has components

$$m\gamma -n\beta ,n\alpha -l\gamma ,l\beta -m\alpha$$

which is the cross product $(l,m,n)\times (\alpha ,\beta ,\gamma )$ written out.

Now put a second vortex on the other side of that surface. Its outward normal points the other way, $(-l,-m,-n)$, so its surface velocity carries the opposite sign in the same construction. The particles wedged between the two take the mean of the two surface velocities — the half-sum rule of the idle wheel.

If $u$ is the velocity of the particles in the direction of $x$, then

$$u=\frac{1}{2}\left[\right(m\gamma -n\beta )-(m{\gamma }^{\prime }-n{\beta }^{\prime }\left)\right]$$

where the primes denote the second vortex. The subtraction is not a sign convention chosen for tidiness — it is the reversal of the normal, and it is the whole physics of the proposition.

Provenance: the printed equation numbers for Prop. IV could not be established, and the expressions above are reconstructions in Maxwell's notation. The proposition statement and the surrounding prose are verbatim. See Sources.

Why the answer is a difference

This is the conceptual heart of the proposition, and it is worth stating twice.

If the two vortices spin at the same rate, the two terms cancel. The particles between them turn on the spot and go nowhere. There is a magnetic field — the vortices are spinning — but no current.

If the two vortices spin at different rates, the terms do not cancel, and the particles drift. A current appears exactly where the rotation varies from place to place.

That is why the relation between current and magnetic force involves derivatives of the field rather than the field itself, and it is why the operation that appears in Prop. V is a curl. Maxwell has built the differential structure of Ampère's law into a gear train.

The half-sum rule for an idle wheel whose centre can move

What Prop. IV is for

By itself it says nothing about electricity. It is a piece of kinematics: given the motion of two neighbouring vortices, find the motion of the layer between them. Its purpose is to be summed. Prop. V takes this expression, integrates it over all the surfaces bounding a volume of the medium, and asks how many particles cross unit area in unit time — and the answer, with one constant fixed, is Ampère's law.

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