On Physical Lines of Force/Modern derivations/Faraday's law of induction

On Physical Lines of Force → Modern derivations → Faraday's law

What Maxwell had

Equation (54), from Part II, Prop. VIII:

$$\frac{d(\mu \alpha )}{dt}=\frac{dQ}{dz}-\frac{dR}{dy},\frac{d(\mu \beta )}{dt}=\frac{dR}{dx}-\frac{dP}{dz},\frac{d(\mu \gamma )}{dt}=\frac{dP}{dy}-\frac{dQ}{dx}\mathrm{(}\mathrm{54}\mathrm{)}$$

Provenance: the equation number is attested, and its content is fixed by Maxwell's own description of it as giving "the relation between changes in the state of the magnetic field and the electromotive forces thereby brought into play." The body is a reconstruction. See Sources.

Reading it

The right-hand sides are a curl again, this time of $(P,Q,R)$ — the forces on the particles, which is the electric field. Note the order of the terms: $dQ/dz-dR/dy$ is the negative of the $x$-component of the curl, since $(\mathrm{\nabla }\times 𝐄{)}_x=\partial R/\partial y-\partial Q/\partial z$. So (54) reads

$$\frac{\partial 𝐁}{\partial t}=-\mathrm{\nabla }\times 𝐄,\text{that is}\mathrm{\nabla }\times 𝐄=-\frac{\partial 𝐁}{\partial t}$$

How Maxwell gets it

This is worth following, because it is not the argument any modern textbook gives, and it is a good one.

1. Prop. VI computes the energy stored in the vortices: $E=\frac{\mu }{8\pi }({\alpha }^2+{\beta }^2+{\gamma }^2)V$, which is equation (46). In modern terms, the magnetic energy density is $𝐁\cdot 𝐇/8\pi$.
2. Prop. VII computes the rate at which the particles do work on a vortex, equation (50). Because each particle touches two vortices at opposite ends of a diameter, and because the integration runs over a closed surface, only the circulation of $(P,Q,R)$ survives: a uniform force does no net work on a vortex.
3. Prop. VIII sets the time derivative of the first equal to the second. The identity must hold for arbitrary and independent $\alpha ,\beta ,\gamma$, so the coefficient of each may be separated. Three equations result, and they are (54).

Faraday's law is derived from conservation of energy in a system of flywheels. Nothing about induction is postulated. The law arrives as a bookkeeping identity: the energy the field gains is the work the electricity does on it.

Where the minus sign comes from

Not from a convention. From inertia.

How an induced current arises

A vortex that is being spun up resists being spun up, exactly as any mass resists acceleration. That resistance is felt by the particles in contact with it, and it pushes them backwards. A backward motion of particles is a current in the reverse direction — one whose own magnetic effect opposes the change that produced it.

That is Lenz's law, and on this model it is not an extra rule. It is Newton's third law applied to a flywheel. Maxwell's narration in the induced-currents passage:

It appears therefore that the phenomena of induced currents are part of the process of communicating the rotatory velocity of the vortices from one part of the field to another.

Once the communication is complete — every vortex turning at its final speed — nothing more moves and the induced current stops. Which is why induction happens only while things are changing: the induced current is the process of change, and it ends when the change does.

The integral form

Stokes's theorem turns (54) into the form Faraday would have recognised. For a fixed loop $C$ spanning a surface $S$,

$${\oint }_C𝐄\cdot d\ell ={\int }_S(\mathrm{\nabla }\times 𝐄)\cdot d𝐒=-\frac{d}{dt}{\int }_S𝐁\cdot d𝐒=-\frac{d\Phi }{dt}$$

The electromotive force round a loop equals minus the rate of change of the magnetic flux through it. This is the 1831 experimental law.

Two things the differential form makes clear that the integral form hides:

• The law is local. It holds at every point of space, whether or not there is a wire there. A changing magnetic field produces an electric field in empty space. That is not an obvious reading of Faraday's experiments, and it is essential to the wave equation.
• The loop in the integral form must be fixed. If the loop moves, there is an extra contribution, and sorting that out is Prop. XI and the Lorentz force page.

The anchor ring, again

Maxwell's toroid experiment is the sharpest illustration of the locality claim, and it points the other way — toward the potentials.

All that is required is that lines of force should pass through the circuit of the conductor, and that these lines of force should vary in quantity during the experiment.

In a perfect toroid, $𝐁$ is genuinely zero outside the ring. A wire encircling the ring nonetheless feels an emf. The integral form explains this without difficulty — the flux through the circuit is what matters, and the circuit encloses the ring. But it means that a wire sitting in a region of strictly zero magnetic field can have a force driving current along it, which is uncomfortable for anyone who wants $𝐁$ to be the whole story locally. See the vector potential.

What it says today

	form
SI, differential	$\mathrm{\nabla }\times 𝐄=-\frac{\partial 𝐁}{\partial t}$
Gaussian	$\mathrm{\nabla }\times 𝐄=-\frac{1}{c}\frac{\partial 𝐁}{\partial t}$
integral, fixed loop	${\oint }_C𝐄\cdot d\ell =-\frac{d\Phi }{dt}$
in terms of potentials	$𝐄=-\frac{\partial 𝐀}{\partial t}-\mathrm{\nabla }\phi$ — Maxwell's (58)

Where the conventions differ

	Maxwell 1861	modern
electromotive force	$P,Q,R$ — a force on the particles at each point	$𝐄$, force per unit charge
magnetic induction	$\mu \alpha ,\mu \beta ,\mu \gamma$	$𝐁$
the time derivative	$d/dt$, also used for the material derivative in Prop. XI	$\partial /\partial t$ here, $D/Dt$ there
the sign	fixed by his arbitrary choice of vortex sense, quoted in the notation page	the same, by convention

Maxwell's "perfectly arbitrary" choice — that looking along a line of force from south to north you see the vortices turning with the hands of a watch — happens to give the modern sign.

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