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What is Maxwell's theory of Wave propagation and what is its physical interpretation?

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closed as too broad by Chris White, Nathaniel, Brandon Enright, joshphysics, John Rennie Feb 27 at 6:38

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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Have you tried using the Internet? –  DumpsterDoofus Feb 27 at 4:27
    
More specifically, as well as my answer, try searching for "displacement current". There is also a good section on the Wiki page for "Ampère's Circuital Law". To the close voters: in all fairness, it sure helps to know a little history to know exactly what to search here and to find out what it was exactly that Maxwell did: the four Maxwell equations are a far cry from his work alone. –  WetSavannaAnimal aka Rod Vance Feb 27 at 5:21
    
I answered a similar question here physics.stackexchange.com/q/95912 –  user288447 Feb 27 at 14:02

1 Answer 1

up vote 3 down vote accepted

Before Maxwell, the known laws of electromagnetism were thought to be:

$$\begin{array}{ll} \nabla \cdot \vec{D} = \rho& \text{Gauss's Law for Electricty}\\ \nabla \cdot \vec{B} = 0& \text{Gauss's Law for Magnetism}\\ \nabla \wedge \vec{E} = -\partial_t \vec{B}& \text{Faraday's Law}\\ \nabla \wedge \vec{H} = \vec{J}& \text{Amp}\grave{\text{e}}\text{re's Law}\\ \end{array}\tag{1}$$

where $\vec{E},\,\vec{H},\,\vec{D},\,\vec{B}$ are the electric and magnetic field, electric displacement and magnetic induction, respectively and $\rho,\,\vec{J}$ the electric charge and current density vectors, respectively.

The great problem was that Ampère's law as it stood was inconsistent with the principle of conservation of charge. If you take the vector divergence of both sides of the before-Maxwell Ampère law, you get $\nabla\cdot\vec{J} = 0$, whereas the general statement of charge conservation is $\nabla\cdot\vec{J} = -\partial_t \rho$. The latter form expresses the fact, for example, that charges can build up on capacitor plates as long as the nett charge of the system doesn't change, whereas $\nabla\cdot\vec{J} = 0$ means that current flow lines cannot end anywhere (as they do on capacitor plates in time-varying circuits). I say more about the intuitive physical interpretation of this discrepancy in my answer here. To cut a long story short, one way (not the only one) to get rid of these inconsistencies is to postulate that the RHS of Ampère's law should instead be $J+\partial_t \vec{D}$; then if you take the divergence of both sides of Ampère's law you get the continuity equation $\nabla\cdot\vec{J} = -\partial_t \rho$. This is exactly what Maxwell did.

Once this is done, it can be shown that all the six Cartesian components of the electric and magnetic field vectors in freespace fulfill the D'Alembert wave equation:

$$c^2 \nabla^2 \psi = \partial_t^2 \psi$$

which is the equation for a dispersionless wave. To see this, in one spatial dimension the equation is:

$$c^2 \partial_x^2 \psi = \partial_t^2 \psi$$

and its general solution is:

$$\psi = f(x-c\,t) + g(x + c\,t)$$

where $f$ and $g$ are arbitrary, continuously twice differentiable ($C^2$) functions. They are equations for pulses with invariant shape (defined by $f$ and $g$) running at speed $c$.

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I noticed you use the $\wedge$ notation from geometric algebra; do you find geometric algebra useful or was that just a notation convention you use? GA seems to have some rather vocal proponents, but I hardly ever see it used anywhere and I don't really know much about it. –  DumpsterDoofus Feb 27 at 4:51
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Two reasons: reason 1: Check my age on my user page! When I was doing my PhD, my supervisor used this notation and Born and Wolf was a heavily read text in the field. Reason 2: as a bivector, the outer product is indeed the antisymmetrised geometric product. However, whilst I do like the presentation of geometric algebra a great deal, I don't believe it's quite worth evangelising as vigorously as some do. It's simply an excellent way to present already living and well understood ideas in Clifford algebras and general tensor analysis. So I'd call something like Doran and Lasenby .... –  WetSavannaAnimal aka Rod Vance Feb 27 at 5:13
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....a great piece of technical writing, but not new science. –  WetSavannaAnimal aka Rod Vance Feb 27 at 5:13
    
Okay, makes sense. I started learning about it, but I kept noticing that most GA texts seemed to spend roughly 1/3 of the text preaching about how revolutionary GA was, how it will replace vector algebra in a couple years, how it's superior to vector algebra, how "Dirac and Heisenberg would have used it if they had known about it", and other such grandiose claims. I came away from the reading with a distinct impression that it had been written by a used car salesman, but figured a second opinion might be helpful. –  DumpsterDoofus Feb 27 at 5:23
    
@DumpsterDoofus LOL! Yes, there is certainly the used car salesman quality to it. But I believe if you don't already know about tensors and clifford algebras in particular that this kind of thinking is a good way to learn known concepts. It is certainly true that the "standard" vector calculus notation (essentially Heaviside's /Gibbs's notation) is quite restricted in the geometric concepts that it can describe and it did displace alternative notations. The "evangelism" reminds me very much of proponents of different computer languages, and the debate underlying it is very like that in ... –  WetSavannaAnimal aka Rod Vance Feb 27 at 5:35

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