On Physical Lines of Force/Modern derivations/The Lorentz force

On Physical Lines of Force → Modern derivations → The motional term

What Maxwell had

Prop. XI of Part II asks for the electromotive forces in a moving body, and answers with equation (77), the last and most complete result of Part II:

$$P=\mu (\gamma \frac{dy}{dt}-\beta \frac{dz}{dt})-\frac{dF}{dt}-\frac{d\psi }{dx}\mathrm{(}\mathrm{77}\mathrm{)}$$

$$Q=\mu (\alpha \frac{dz}{dt}-\gamma \frac{dx}{dt})-\frac{dG}{dt}-\frac{d\psi }{dy}$$

$$R=\mu (\beta \frac{dx}{dt}-\alpha \frac{dy}{dt})-\frac{dH}{dt}-\frac{d\psi }{dz}$$

Provenance: the equation number (77) is attested; the bodies are reconstructions in Maxwell's notation. See Sources.

Writing $(dx/dt,dy/dt,dz/dt)$ as the velocity $𝐯$ of the element, and using the identifications of the concordance, the three lines are one:

$$𝐄=𝐯\times 𝐁-\frac{\partial 𝐀}{\partial t}-\mathrm{\nabla }\phi$$

The last two terms are equation (58), already obtained in Prop. VIII for a body at rest. The first term is new, and it is the magnetic part of what is now called the Lorentz force — arrived at some thirty years before Lorentz gave it as a law about charges in the 1890s.

Where the new term comes from

Maxwell states the plan of the proposition in its opening sentence:

The variation of the velocity of the vortices in a moving element is due to two causes — the action of the electromotive forces, and the change of form and position of the element.

Two causes, two contributions, added.

• The first cause is the one already handled. Prop. VIII relates changes in the vortex motion at a fixed place to the forces on the particles between the vortices, and yields $\mathrm{\nabla }\times 𝐄=-\partial 𝐁/\partial t$.
• The second cause is supplied by Props. IX and X, which work out how a small piece of the medium is stretched, sheared and turned as it travels, and what that does to the vortices it contains.

The modern name for the first half of the second cause is the material derivative. If you sit at a fixed point and watch a quantity change, you measure $\partial /\partial t$. If instead you ride along with the material, you also pick up the change due to arriving somewhere new, and the total rate is

$$\frac{D}{Dt}=\frac{\partial }{\partial t}+(𝐯\cdot \mathrm{\nabla })$$

Maxwell writes $d/dt$ for both, and lets the prose carry the distinction. Prop. XI is exactly where that costs him, and it is worth reading his $d/dt$ there with care.

But $𝐁$ is not an ordinary carried-along scalar. It is a flux density frozen into the medium — the number of lines through a patch of material stays fixed as the patch is carried, stretched and rotated — so the accounting picks up extra terms from the stretching and the rotation, and not merely from the transport. Those are the terms Props. IX and X supply.

The frozen-flux calculation

The cleanest modern route to (77) is through the flux rule, and it needs one piece of vector calculus: Stokes' theorem, which says that the circulation of a vector field round a closed curve equals the flux of its curl through any surface spanning that curve.

Let $C(t)$ be a closed circuit that moves with the material, spanning a surface $S(t)$, and let

$$\Phi ={\int }_{S(t)}𝐁\cdot d𝐒$$

be the flux threading it. Two things make $\Phi$ change: the field may vary in time, and the circuit may move into a region where the field is different. Doing that bookkeeping — and using $\mathrm{\nabla }\cdot 𝐁=0$, so that no lines leak through the sides of the swept volume — gives

$$\frac{d\Phi }{dt}={\int }_S\frac{\partial 𝐁}{\partial t}\cdot d𝐒-{\oint }_C(𝐯\times 𝐁)\cdot d\ell$$

Now impose Faraday's law in the form Prop. VIII gives it, $\mathrm{\nabla }\times 𝐄=-\partial 𝐁/\partial t$, and convert the left-hand side with Stokes:

$$\begin{array}{cc}-\frac{d\Phi }{dt}& =-{\int }_S\frac{\partial 𝐁}{\partial t}\cdot d𝐒+{\oint }_C(𝐯\times 𝐁)\cdot d\ell \\ & ={\oint }_C𝐄\cdot d\ell +{\oint }_C(𝐯\times 𝐁)\cdot d\ell \\ & ={\oint }_C(𝐄+𝐯\times 𝐁)\cdot d\ell \end{array}$$

So the quantity driving current round a moving circuit is not $𝐄$ but $𝐄+𝐯\times 𝐁$. That is equation (77), assembled from the other side.

What it says today

Detach the result from the medium and it becomes a statement about a single charge $q$ moving with velocity $𝐯$:

$$𝐅=q(𝐄+𝐯\times 𝐁)$$

This is the Lorentz force law, and together with the four field equations it is the whole of classical electromagnetism: the fields tell the charges how to move, the charges tell the fields what to be. Maxwell has the second half of the bracket in 1861, in a paper about flywheels.

The cross product deserves a moment. $𝐯\times 𝐁$ is perpendicular to both $𝐯$ and $𝐁$, with magnitude $vB\sin \vartheta$; its direction is given by the right-hand rule. Two consequences follow immediately. A charge moving along a field line feels no magnetic force at all. And a charge moving across one is pushed sideways, which is why beams curve in magnetic fields and why cyclotrons work.

The magnet and the conductor

Consider a magnet and a loop of wire in relative motion, and ask where the current comes from.

• Move the loop, hold the magnet still. The field at any fixed point is constant in time, so $-\partial 𝐀/\partial t$ contributes nothing. The motional term $𝐯\times 𝐁$ supplies the whole electromotive force.
• Move the magnet, hold the loop still. Now $𝐯=0$ and the motional term contributes nothing. The field at the wire is changing, and $-\partial 𝐀/\partial t$ supplies it.

The measured current is identical. The explanation is not. Einstein opens the 1905 relativity paper by complaining about precisely this: his first paragraph is about a magnet and a conductor, and about a theory that offers two accounts of one phenomenon depending on which body is declared to be moving.

Special relativity dissolves the asymmetry. $𝐄$ and $𝐁$ are not two fields but six components of one antisymmetric object, the field tensor $F^{\mu \nu }$; changing frames mixes them, and what one observer calls electric another calls magnetic. There is then only one phenomenon and one explanation, and which term of (77) carries it is a question about coordinates.

Be fair to Maxwell here. His model is numerically correct in both cases — it was never in danger of predicting a different current. And in a medium there genuinely is a fact about what is moving: velocities are measured relative to the ether, so two cases that differ in what moves are two different physical situations, and giving them two accounts is not a defect but a consequence of taking the medium seriously. The trouble was conceptual, and removing it took a change in the concept of space, not a correction to the equations.

The magnetic force does no work

One property of the new term is easy to prove and endlessly confusing. The rate at which a force does work on a particle is $𝐅\cdot 𝐯$, and

$$q(𝐯\times 𝐁)\cdot 𝐯=0$$

because a cross product is perpendicular to both its factors. The magnetic force can bend a trajectory but never speed it up. Magnetic fields do no work — full stop.

How, then, does a motor do work? Because the wire is not a single particle. A charge carrier in a moving wire has two velocities: a drift $𝐮$ along the wire and the wire's own velocity $𝐰$ sideways. The magnetic force is perpendicular to the sum, so it has a component along $𝐰$, which pushes the wire and does positive work, and a component along $-𝐮$, which opposes the drift and does negative work at exactly the same rate. The magnetic force is a broker: it redirects energy from the drift motion into the bulk motion, and the source that keeps the current flowing against the opposition — the battery — is what actually pays. This is why a motor draws more current under load, and it is the same bookkeeping as Lenz's law.

Where the conventions differ

	Maxwell 1861	modern
the motional term	$\mu (\gamma dy/dt-\beta dz/dt)$ and two more	$𝐯\times 𝐁$ in SI; $\frac{1}{c}𝐯\times 𝐁$ in Gaussian
what it acts on	an element of the medium	a charge $q$
$P,Q,R$	force on unit of the particles	$𝐄$, force per unit charge
$d/dt$	one symbol for fixed-point and co-moving rates	$\partial /\partial t$ and $D/Dt$ distinguished
$𝐯$ means	velocity relative to the medium	velocity relative to a chosen frame
status of the law	derived from the mechanics of the vortices	an independent postulate alongside the field equations
the asymmetry	two mechanisms, both correct	one mechanism, seen from two frames

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In the paper: Prop. XI · Props. IX and X · Plain-language version: Why a changing magnet makes a current