Maxwell’s fluid model of magnetism

In 1861, Maxwell derived two of his equations of electromagnetism by modelling a magnetic line of force as a ‘molecular vortex’ in a fluid-like medium. Later, in 1980, Berry and colleagues conducted experiments on a ‘phase vortex’, a wave geometry in a fluid which is analogous to a magnetic line of force and also exhibits behaviour corresponding to the quantisation of magnetic flux. Here we unify these approaches by writing down a solution to the equations of motion for a compressible fluid which behaves in the same way as a magnetic line of force. We then revisit Maxwell’s historical inspiration, namely Faraday’s 1846 model of light as disturbances in lines of force. Using our unified model, we show that such disturbances resemble photons: they are polarised, absorbed discretely, obey Maxwell’s full equations of electromagnetism to first order, and quantitatively reproduce the correlation that is observed in the Bell tests. In 1746 Euler modelled light as waves in a frictionless compressible fluid; a century later in 1846, Faraday modelled it as vibrations in ‘lines of force’ as in figure 1 [1–4]. Figure 1: Faraday’s 1846 model of light as waves in lines of force, and Maxwell’s 1861 figure showing his extension to a magnetic line of force. Fifteen years later Maxwell combined these approaches, proposing that a magnetic line of force is a ‘molecular vortex’ (see the diagram from his 1861 paper in figure 1 [5–7]). A fluidlike medium flows around the line, and centrifugal forces reduce the pressure near the centre, giving a ‘tension’ along the axis which accounts for the forces between the poles of magnets. Maxwell then derived two of his equations of electromagnetism. Suppose the mean momentum per unit volume of fluid is p(x). In modern notation with unit charge, the magnetic field is B = ∇ × p, and it obeys Gauss’s law for magnetism ∇.B = 0, since ∇.(∇ × p) is identically zero. Defining the magnetic flux by φ = ∫ B.ds where ds is a surface element, Stokes’s theorem shows that φ = ∮ p.d` 6= 0 where the path d` encircles the centre. If the fluid exerts a mean force density E on an external system then it must lose momentum, E = −∂p/∂t. Faraday’s law of induction follows immediately: ∇ × E = −∂B/∂t. Feynman later rediscovered a similar derivation [8]. On later interpretations, the momentum density p corresponds to the magnetic vector potential. Maxwell’s magnetic line of force can be almost any axis with mass flow around it ( ∮ p.d` 6= 0). An ordinary vortex in a fluid is not a good exam1 ar X iv :s ub m it/ 11 89 09 8 [ qu an tph ] 2 0 Fe b 20 15 ple, since it is pinned to the fluid and is therefore not symmetric under Lorentz transformation [9]. A better example is suggested by a series of experiments, starting a hundred years later, on quantised magnetic flux.