How this page works

Everything runs in your browser from the equations below; nothing is uploaded. The first and third tabs are established physics (the Biot-Savart law and the field of a magnetic dipole). The second tab is Maxwell's 1861 mechanical model: a picture he used to find his equations, which he himself called "provisional and temporary". The equations it led him to are established; whether space really is an elastic medium of spinning cells (an idea Chantal Roth explores) is a hypothesis, not something experiments have shown.

Biot-Savart law \(\displaystyle d\mathbf B = \frac{\mu_0}{4\pi}\,\frac{I\,d\boldsymbol\ell\times\hat{\mathbf r}}{r^2},\qquad B_{\text{wire}} = \frac{\mu_0 I}{2\pi r}\)

a loop, on its axis \(\displaystyle B(z) = \frac{\mu_0 I R^2}{2\,(R^2+z^2)^{3/2}}\ \xrightarrow{\;z\gg R\;}\ \frac{\mu_0}{4\pi}\,\frac{2\mu}{z^3},\qquad \mu = I\pi R^2\)

a charge going round \(\displaystyle I = \frac{q v}{2\pi r},\qquad \mu = I\pi r^2 = \frac{q v r}{2} = \frac{q}{2m}\,L\)

Maxwell's idle wheel \(\displaystyle V = \frac{a(\omega_1-\omega_2)}{2},\qquad \Omega = -\frac{a(\omega_1+\omega_2)}{2b}\quad(\text{current} \propto \nabla\times\boldsymbol\omega)\)

elastic cells: a wave \(\displaystyle \ddot\theta = c^2\nabla^2\theta,\qquad c = a\sqrt{\frac{k}{4I}}\quad\Longleftrightarrow\quad c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}\)

two dipoles \(\displaystyle \mathbf F = \frac{3\mu_0}{4\pi r^4}\Big[(\mathbf m_1\!\cdot\!\hat{\mathbf r})\mathbf m_2 + (\mathbf m_2\!\cdot\!\hat{\mathbf r})\mathbf m_1 + (\mathbf m_1\!\cdot\!\mathbf m_2)\hat{\mathbf r} - 5(\mathbf m_1\!\cdot\!\hat{\mathbf r})(\mathbf m_2\!\cdot\!\hat{\mathbf r})\hat{\mathbf r}\Big]\)

Maxwell's stress \(\displaystyle T_{ij} = \frac{1}{\mu_0}\Big(B_iB_j - \tfrac12\delta_{ij}B^2\Big),\qquad \mathbf F = \oint T\cdot d\mathbf A\)

A charge going round is a magnet

The wire and the loops are polygons of many straight pieces; each piece's field is the exact Biot-Savart result for a straight segment, and the pieces are added up. The tests check the sum against the textbook results: µ₀I/2πr beside a long wire, µ₀I/2R at the centre of a loop, the on-axis formula, and the dipole field far away. For the electron in the first Bohr orbit (radius 0.0529 nm, speed αc ≈ 2,190 km/s) the moment q v r / 2 comes out as exactly one Bohr magneton, 9.27 × 10⁻²⁴ A·m², and its angular momentum as ħ. The proton and neutron pictures are models: a proton treated as a spinning ball of charge would need its surface to move faster than light to give its measured moment (2.79 nuclear magnetons), which is one reason physicists say spin is not literally a spinning ball. The neutron has no net charge but a magnetic moment of −1.91 nuclear magnetons; measurements show a slightly positive centre and a negative outer region, so turning charge of both signs can still make a magnet. Real particle moments come from quantum mechanics (quarks and their spin), which this page does not model.

Maxwell's spinning cells and idle wheels

In On Physical Lines of Force (1861-62) Maxwell explained magnetic lines of force as the axes of tiny vortices. Spinning cells bulge sideways and shorten along their axis, which gives exactly the pull along field lines and the push across them that Faraday saw. To let neighbouring vortices turn the same way he put a layer of small particles between them, rolling like idle wheels in a gear train; a flow of those particles is an electric current, and the rule that makes them flow is Ampère's law (current = curl of the field). When the current changes, the vortices nearby have to speed up, and the particles in a nearby wire are dragged backwards for a while: that is induction, with Lenz's minus sign built in. In Part III he made the cells elastic: a twist then travels through them as a wave, and from the known electrical constants the speed came out close to the measured speed of light. He concluded that light is a wave in the same medium that carries electric and magnetic effects.

The page uses a square grid of round cells (Maxwell drew hexagons), each turning about an axis out of the screen, so the field shown is the part pointing out of (red) or into (blue) the page, and currents run in the plane. The cells are rigid wheels with an inertia I; each idle wheel sits in the gap between two cells and touches both.
Rolling without slipping fixes a wheel's motion completely: V = a(ω₁ − ω₂)/2 along the gap and spin Ω = −a(ω₁ + ω₂)/2b. In an insulator the wheels are held in place by springs (the elasticity), so they can only shift a little; in a wire they flow, against a friction that plays the part of resistance. A battery is a steady push on the wheels of the wire.
The equations of motion come from energy: spin energy ½Iω² plus spring energy ½ku². They are a discrete wave equation, and the tests check that energy is kept, that a twist travels at a√(k/4I) cells per second, that a steady current settles at push ÷ friction with the two sides spinning opposite ways, and that switching on a current drives a brief opposite current in a neighbouring wire.

Two magnets

Each magnet is a point dipole at its centre, so the numbers are right when the magnets are further apart than their own length and only rough when they nearly touch. The field drawn is the part of the full 3D dipole field lying in the plane of the screen. The force check integrates Maxwell's stress tensor over a sphere around magnet B (100 × 200 points) and compares it with the dipole formula: they agree, which is the mathematical content of Maxwell's "tension along the lines, pressure across them". The force is equal and opposite on the two magnets; the torques are not equal and opposite on their own, because the force also has a turning effect about the other magnet's centre.

Questions

Why does a current loop act like a bar magnet?: Every bit of the wire makes a field circling it. Inside the loop all those circles point the same way, so the field there is strong and along the axis; outside it returns around the loop. Far away the pattern is exactly that of a small bar magnet.
Is the electron really going round in a circle?: Not in modern quantum mechanics, but the Bohr picture gives the right size of the orbital magnetic moment (one Bohr magneton per unit of ħ). The electron also has its own spin moment of almost exactly one Bohr magneton, which is what makes iron magnetic.
Did Maxwell believe in the idle wheels?: He offered them as a mechanical illustration, "not as a mode of connexion existing in nature". He later derived the same equations without any mechanism (1865), and that version is what physics uses.
Why do magnets attract head to tail but repel side by side?: The force depends on how each magnet is lined up with the line joining them. Head to tail, the field lines run straight from one to the other and pull them together; side by side and parallel, the lines between them point opposite ways, are squeezed, and push apart.
How accurate are the numbers?: The wire and loop fields are exact to within the polygon approximation (better than 0.01%). The magnets are point dipoles. Maxwell's lattice is a teaching model with made-up units (cells per second), not SI.

Further

J. C. Maxwell (1861-62). On Physical Lines of Force. Philosophical Magazine 21, 161-175, 281-291, 338-348; 23, 12-24, 85-95. The Open University's The restless universe, section 2.4.1 tells the story with his figure of vortices and idle wheels.
J. C. Maxwell (1865). A Dynamical Theory of the Electromagnetic Field. Philosophical Transactions of the Royal Society 155, 459-512.
Chantal Roth: "What Is Spin-1/2, Really? A Mechanical Model" and the Mechanical Universe demos.

Related: Electromagnetic Fields Simulator (straight wires and Ampère's law in 2D) · Magnetic Isopotentials · Spin Simulator · MES Physics · More MES tools

Maxwell's Spinning Cells and Idle Wheels: A Mechanical Model of Magnetism