Comment by adrian_b | Hacker News Reader

Comment by adrian_b | Hacker News Reader

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The originally published equations were "20 or so" because one equation was written for each scalar component.

Rewriting the equations in vector form reduces the number to the modern number.

Moreover, the original equations are the complete system.

The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum.

The complete system of equations has around 6 or 7 or even more equations, depending on whether one chooses to have distinct notations for various physical quantities, such as electric polarization, magnetization, current of free electricity carriers etc., or not.

The variants with less equations are simplifications that are valid only in linear media, because only there you have proportionality relationships between quantities like electric field and electric polarization.

Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell.

The many simplified variants, for media without discontinuities where differential forms are valid, for stationary media, for linear media, for vacuum and so on, can be easily derived from the general form, while the reverse is not true.

> The originally published equations were "20 or so" because one equation was written for each scalar component.

> Rewriting the equations in vector form reduces the number to the modern number.

And if you use the differential form or 4d tensor notation they get reduced to 1 equation. Of course, for a lot of practical problems this is not very useful and it's better to work with the 3d vector form.

> The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum.

> Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell.

Here I have to strongly disagree. The version of Maxwell's equations that is fundamental and exactly correct [1] is the vacuum version. The ones with magnetization and displacement vectors are only approximations where you assume continuous materials that respond to fields in simple way. In truth, materials are made of atoms and are mostly vacuum: there is no actual displacement vector if you look close enough.

Also the vacuum Maxwell equations are useful in many scenarios. For instance, that's how you compute the energy levels of Hydrogen atom or how you derive QED. Also, you have to start from them to derive the macroscopic versions with magnetization and displacement that you seem to like.

[1] Well, up to non-linear quantum mechanic effects.

Even the vacuum version is incomplete without adding an equation for force or energy, because no meaning can be assigned to the electromagnetic field or potential otherwise than by its relationship with the force or energy.

Even today, there exists no consensus about which is the correct expression for the electromagnetic force. Most people are happy to use approximate expressions that are known to be valid only in restricted circumstances (like when the forces are caused by interactions with closed currents, or the forces are between stationary charges).

Moreover, when the vacuum equations are written in the simplified form present in most manuals, it is impossible to deduce how they should be applied to systems in motion, without adding extra assumptions, which usually are not listed together with the simple form of the equations (e.g. the curl and the divergence are written as depending on a system of coordinates, so it is not obvious how these coordinates can be defined, i.e. to which bodies they are attached).

While the vacuum equations are fundamental, they may be used as such only in few applications like quantum mechanics, where much more is needed beyond them.

In all practical applications of the Maxwell equations you must use the approximation of continuous media that can be characterized by averaged physical quantities that describe the free and bound carriers of electric charge. The useful form of the Maxwell equations is that complete with electric polarization, magnetization, electric current of the free carriers and electric charge of the free carriers. It is trivial to set all those quantities to zero, to retrieve the vacuum form of the equations.

I agree that to fully specify electromagnetism you also need to include how the fields affect charged matter. So EM = Maxwell's equations + Lorentz force equation (not sure why you say there is no consensus about what this is, that is new to me).

This is just a matter of taste, but OTOH I would not include descriptions of how some materials respond to the fields in the continuous limit as part of a definition of EM.

It is true that for most terrestrial applications you do need those to do anything useful with EM. But if you want to study plasmas you need to add Navier-Stokes to EM, doesn't mean hydrodynamics is part of EM. To study charged black holes you need EM + GR, but it still makes sense to treat them as mostly separate theories.

You also need to include how charged matter affects the forcing fields in Maxwell's equations (i.e. moving charges depositing a current field).

I actually basically agree with your viewpoint, I studied Plasma Physics in graduate school in a regime where we did _not_ use Navier-Stokes or constitutive relations and everything was in fact just little smeared-out packets of charge moving according to the Lorentz Force Law and radiating.

The fact that you can write it in one equation shows that the theory is very simple because it is an expression of symmetry. E and B are not these two different things related by an inscrutable cross product but just two aspects of the same thing.

You could write all physics in a single simple equation. deltaW=0 Where deltaW is deviation of the universe from the relevant math.

Writing Maxwell's as 1 equation or 4 or more is just esthetic choice where you decide what to accentuate.

20 might be too much because three dimensions are not really different from each other so the notation that maps over them wholesale is probably a good idea.

4 equations seem perfect if you want to differentiate between classical effects of the electric field and relativistic effects (magnetism).

I don't know if single equation really shows that they really have the same source and the relativity is involved or is it just a matrix mashup of the 4 separate equations that doesn't really provide any insights.

It's true that you can always define notation to combine all equations you want into one. This means that, by itself, the observation that you can write Maxwell's equations as a single equation doesn't say anything very meaningful.

However, the notation that lets you do this in this specific case is very natural and not specific to Maxwell's equations. Differential forms are very natural objects in differential geometry, mathematicians would have likely introduced them and studied without inspiration from physics. The fact that Maxwell's equations are very simple in this natural geometrical language does say something meaningful about their nature and elegance, I think.

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