People already knew that a changing magnetic field makes the electric field curl around. But in 1861, Maxwell proposed that also a changing electric field makes the magnetic field curl around. This means that electric and magnetic fields can form waves, with each one changing the other. Maxwell computed the speed of these waves and got an answer close to the speed of light! The rest is history...
... in a way. Maxwell reached his conclusions using a strange mechanical model involving "molecular vortices". In this model magnetic fields are spinning vortices - the black hexagons here - while electric fields are the displacement of little particles between these vortices - the green circles.
It took him a few years to drop this model and simply write equations governing fields. And it took longer for Heaviside and others to take Maxwell's 20 equations, throw some out, and boil the rest down to 4 vector equations - the ones we now learn in school.
I want to understand Maxwell's molecular vortices because they're the key to how he guessed that a changing electric field makes the magnetic field curl. People already knew a lot about the electric and magnetic fields from experiments. But Maxwell's extra effect completed our theory of electromagnetism!
There are modern ways to see that this extra effect is necessary. Few people go back and try to understand Maxwell's original thinking. Here's one great exception:
@johncarlosbaez Related or not, other discretization of Maxwell equations in terms of a big LC electric circuit mesh is given in this vintage page, https://tinyurl.com/mtfrz869. If the Maxwell and the circuitous views describe the same, there must be some kinship.
@johncarlosbaez@jesuslop
Sorry for the late reply.
Very cool to see an actual diagram by Maxwell!
One aspect that might be confusing about these models, is the idea that “the ether does not exist”. It is true that a mechanical model or a circuit model uses a choice of coordinates, and a choice of discretisation scheme. Other choices will describe the same reality. In the case of rotations and discretisation schemes, this is fairly intuitive. But a change corresponding to a Lorentz transformation will turn electric fields into magnetic fields and vice versa.
But this does not mean that a particular model does not correctly describe reality. We just should be aware that there are other models that do the same.
The laws of nature have rotation, translation and Lorentz symmetry. So an experiment which measures the speed of light gives the same number in all reference frames. We have to remember that our measurement apparatus is also living in the same model, it will transform with the model in such a way that the measurement result is invariant under a change in model.
Although we cannot say electromagnetism is mechanical I think it is not incorrect to make a mechanical model of electromagnetism.
Gerard
@johncarlosbaez “Spinning vortices” makes much more sense than pseudovectors if you ever consider electromagnetism in dimensions other than 3 (4 if you include time). Then the electric field remains a vector field but the magnetic field is a vortical field whose number of components is only the same as the electric field for our special case of 3 (spatial) dimensions. In 2D, the electric field vector has 2 components but the magnetic field has only 1. In 4D (Kaluza-Klein), it’s 4 and 6.
@johncarlosbaez Technical details: This follows from Lorentz invariance and the electromagnetic field tensor F_ab, which has N(N-1)/2 independent components in N spacetime dimensions (F_ab is anti symmetric). The electric field is the inner product of the unit timelike basis vector with F, which has N-1 components, and the magnetic field is “the rest” (simplifiable as the inner product of the dual of the unit timelike basis vector with F), which has (N-1)(N-2)/2 components: a spinning vortex.
@jzsimon - Yes, you're right. I said "pseudovectors" because I figured more people would understand that than "2-forms" or "bivectors" or "antisymmetric rank-2 tensors". But maybe I should have said "axial vectors" - I think that term may be a bit more common than "pseudovectors"?
Anyway, whatever you call them, it's clear from his writings that Maxwell understood the difference between axial and polar vectors in 3d, but had no understanding of the n-dimensional generalization, even though Grassmann wrote his "Ausdehnungslehre" in 1844 so theoretically Maxwell could have read that.
By the way, Maxwell was the guy who invented the word "curl"!
@BashStKid - you can get them free in pdf or djvu formats, pirated, on LibGen. It's illegal - but predictable, given how much the publishers charge for their books.
@johncarlosbaez I like knowledge to be freely available, and authors being fairly paid. (Never mind the publishers, they've already extracted a ton of rent).
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