No magnetic monopoles

On Physical Lines of Force → Modern derivations → No magnetic monopoles

What Maxwell had

In Part II, Prop. VIII, among the five equations that proposition produces, Maxwell writes down the statement that the magnetic induction has no divergence:

$\frac{d (μ α)}{d x} + \frac{d (μ β)}{d y} + \frac{d (μ γ)}{d z} = 0 (56)$

Provenance: the equation number is attested; the body is a reconstruction in Maxwell's notation. See Sources.

From the notation page: $α, β, γ$ are the components of the magnetic force $𝐇$ , $μ$ is the capacity for magnetic induction, and the products $μ α, μ β, μ γ$ are therefore the components of the magnetic induction $𝐁$ . The pattern on the left — each component differentiated along its own axis, all added — is a divergence. So (56) is

$\nabla \cdot 𝐁 = 0$

Where it comes from

Maxwell does not postulate (56). He gets it for nothing, one line after equation (55), which introduced the electrotonic state $F, G, H$ — what we now call the vector potential $𝐀$ :

$𝐁 = \nabla \times 𝐀$

Any field written as a curl automatically has zero divergence. This is an identity of the calculus, true for every smooth vector field whatever, and worth proving once so that you never have to take it on trust.

The divergence of a curl is identically zero

Write out $\nabla \cdot (\nabla \times 𝐀)$ in components. The x-component of the curl is $\partial A_{z} / \partial y - \partial A_{y} / \partial z$ , and the others follow by the cyclic rule, so

$\nabla \cdot (\nabla \times 𝐀) = \frac{\partial}{\partial x} (\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z}) + \frac{\partial}{\partial y} (\frac{\partial A_{x}}{\partial z} - \frac{\partial A_{z}}{\partial x}) + \frac{\partial}{\partial z} (\frac{\partial A_{y}}{\partial x} - \frac{\partial A_{x}}{\partial y})$

Six terms. Now group them in pairs, one from each of two different brackets:

$\begin{matrix} \frac{\partial^{2} A_{z}}{\partial x \partial y} - \frac{\partial^{2} A_{z}}{\partial y \partial x} & = 0 \\ \frac{\partial^{2} A_{x}}{\partial y \partial z} - \frac{\partial^{2} A_{x}}{\partial z \partial y} & = 0 \\ \frac{\partial^{2} A_{y}}{\partial z \partial x} - \frac{\partial^{2} A_{y}}{\partial x \partial z} & = 0 \end{matrix}$

Each pair vanishes because mixed second partial derivatives are equal in either order — differentiating by $x$ then $y$ gives the same result as $y$ then $x$ , for any function with continuous second derivatives. Every term cancels against a partner, and the sum is zero, always, with no assumption about what $𝐀$ is.

So the moment Maxwell writes $𝐁$ as a curl, (56) follows. The two are not independent statements about nature; they are one statement, seen twice.

What it says physically

A divergence measures whether a point is a source: water welling out of a spring has positive divergence there. Electric field lines start on positive charges and stop on negative ones, and $\nabla \cdot 𝐃 = ρ$ says exactly that.

$\nabla \cdot 𝐁 = 0$ says magnetic field lines do neither. They have no beginnings and no ends. Follow a line of magnetic force and it closes on itself, or wanders on without terminating; it never simply stops in empty space, and never emerges from a point the way a field line emerges from a charge.

This is why cutting a bar magnet in half does not leave a north pole in one hand and a south pole in the other. It gives you two magnets. The poles are where the lines enter and leave the iron, not places where they begin.

The integral form

Apply the divergence theorem, which converts the integral of a divergence over a volume into a flux through the surface bounding it:

$\oint_{S} 𝐁 \cdot d 𝐒 = \int_{V} \nabla \cdot 𝐁 d V = 0$

In words: the total magnetic flux through any closed surface is zero. Whatever goes in comes out. Draw any closed bag you like, of any shape, around anything at all — a magnet, a coil, a star — and the flux entering exactly balances the flux leaving. There is nothing you could put inside to unbalance the books, because there is no magnetic charge to put there.

On the vortex model this is not a postulate

Notice how cheaply Maxwell gets this. His magnetic field is a field of spinning vortices in a fluid medium. A vortex is a closed circulation, and circulating fluid does not come from anywhere or go anywhere: it goes round. A field built out of circulation cannot have sources, for the same reason that a wheel has no beginning.

On the modern axiomatic presentation $\nabla \cdot 𝐁 = 0$ is one of four independent postulates, and a student may reasonably ask why nature should obey it. On the vortex hypothesis the question does not arise. That the mechanism explains the law rather than merely accommodating it is a genuine strength of the mechanical picture, and the sort of thing Maxwell built it for. See The vortex sea.

The tension with Part I

There is an apparent inconsistency in the paper, and it is instructive rather than damaging. In Part I, the first term of equations (12) is the force on a magnetic pole, and Maxwell computes it using "imaginary magnetic matter" whose density is $\frac{1}{4 π} \nabla \cdot 𝐁$ . But (56) says that quantity is zero. How can he calculate with a density he later proves to vanish?

The answer is that the pole language is a device, not an ontology. Where $μ$ varies from place to place — at the surface of a piece of iron, say — the quantity $\nabla \cdot 𝐇$ is not zero even though $\nabla \cdot 𝐁$ is, and the bookkeeping of poles is a legitimate way to compute the resulting forces. Maxwell uses it while denying that poles are real sources of anything.

The same distinction is still drawn today: ask a physicist for the field of a bar magnet and there is a fair chance they will put fictitious magnetic charges on the two end faces, because the arithmetic is easier that way. Nobody believes the charges are there, and the field the method produces is the same field a calculation from the currents would give.

What it says today

$\nabla \cdot 𝐁 = 0$

This is the one Maxwell equation that reads identically in SI and in Gaussian units. There is no charge on the right to carry a $4 π$ or an $ε_{0}$ ; zero is zero in every system. It is the simplest of the four, and the only one you never have to convert.

The open question

No magnetic monopole has ever been observed. Searches have been made in cosmic rays, in accelerator collisions and in mineral samples, and none has found one. That is the experimental position, and it is a negative result rather than a proof.

In 1931 Dirac showed that the theoretical situation is more interesting than the experimental one. If even a single magnetic monopole existed anywhere in the universe, consistency of quantum mechanics would require electric charge to come in discrete multiples of a fixed unit — which is exactly what we observe, and for which the standard theory otherwise offers no explanation. One monopole would account for the quantisation of charge everywhere.

This is not a prediction that monopoles exist. It is an argument that their existence would explain something otherwise unexplained, which is a weaker and different claim. Whether any exist remains an open question.

The structural point is the one to take away. If a monopole were found, equation (56) is the one that would have to be amended — its right-hand side would become a magnetic charge density instead of zero, and Faraday's law would acquire a corresponding magnetic-current term. The other three would stand. Maxwell's $\nabla \cdot 𝐁 = 0$ is thus at once the simplest of the four and the only one anybody seriously expects might one day need rewriting.

Where the conventions differ

	Maxwell 1861	Modern
the statement	eq. (56), three terms written out	$\nabla \cdot 𝐁 = 0$
form	one scalar equation (a divergence is a scalar)	one scalar equation — no change
the field	$μ α, μ β, μ γ$ , the magnetic induction	$𝐁$
derivative	$d / d x$	$\partial / \partial x$
factor $4 π$	absent here	absent in SI and in Gaussian alike
sign convention	no sign question arises — the right side is zero	same
logical status	derived from (55), $𝐁 = \nabla \times 𝐀$	usually postulated, with $𝐀$ introduced afterwards as a consequence

That last row is the real difference. The textbook order is: assume $\nabla \cdot 𝐁 = 0$ , observe that a divergence-free field can always be written as a curl, conclude that a vector potential exists. Maxwell's order is the reverse — he reaches the potential first, out of a mechanical argument, and the absence of monopoles falls out as a corollary he barely pauses over.

Next: Gauss's law for electricity · Previous: From components to vectors

In the paper: Prop. VIII · See also: The vector potential · The vortex sea