I'm trying to learn more about Maxwell equations and stumbled upon an essay by professor Freeman J. Dyson from Princeton. He explained Maxwell theory in a very interesting way.

The modem view of the world that emerged from Maxwell's theory is a world with two layers. The first layer, the layer of the fundamental constituents of the world, consists of fields satisfying simple linear equations. The second layer, the layer of the things that we can directly touch and measure, consists of mechanical stresses and energies and forces. The two layers are connected, because the quantities in the second layer are quadratic or bilinear combinations of the quantities in the first layer. To calculate energies or stresses, you take the square of the electric field-strength or multiply one component of the field by another. The two-layer structure of the world is the basic reason why Maxwell's theory seemed mysterious and difficult. The objects on the first layer, the objects that are truly fundamental, are abstractions not directly accessible to our senses. The objects that we can feel and touch are on the second layer, and their behaviour is only determined indirectly by the equations that operate on the first layer. The two-layer structure of the world implies that the basic processes of nature are hidden from our view.

Another part that I found really interesting was this:

The ultimate importance of the Maxwell theory is far greater than its immediate achievement in explaining and unifying the phenomena of electricity and magnetism. Its ultimate importance is to be the prototype for all the great triumphs oftwentieth-century physics. It is the prototype for Einstein's theories of relativity, for quantum mechanics, for the Yang-Mills theory of generalised gauge invariance, and for the unified theory of fields and particles that is known as the Standard Model of particle physics. All these theories are based on the concept of dynamical fields, introduced by Maxwell in 1865. All of them have the same two-layer structure, separating the world of simple dynamical equations from the world of human observation. All of them embody the same quality of mathematical abstraction that made Maxwell's theory difficult for his contemporaries to grasp. We may hope that a deep understanding of Maxwell's theory will result in dispersal ofthe fog of misunderstanding that still surrounds the interpretation of quantum mechanics. And we may hope that a deep understanding of Maxwell's theory will help to lead the way toward further triumphs of physics in the twenty-first century.

Here is the link for the essay: Why is Maxwell's theory so hard to understand?

Can someone please give me a further explanation of this? Which parts of the equations are the first layer and which one are the second? What does he mean by the mechanical stresses and energies? And how are the layers connected (I didn't get what he said)?

  • $\begingroup$ Maxwell's equations are the first layer. The second layer are the things affected by the Lorentz force such as a balloon charged by rubbing against your hair. Btw. you cite the same text two times. $\endgroup$ – Void Sep 24 '14 at 20:00
  • $\begingroup$ In addition to @Void's comment, you should take a look at this. I think the author is talking about the Maxwell stress tensor when mentioning that energies etc. are quadratic/bilinear in the fields. $\endgroup$ – Danu Sep 24 '14 at 20:05
  • $\begingroup$ I fixed it. Thank you. How are Maxwell's equation connected to Lorentz force? $\endgroup$ – user1123975 Sep 24 '14 at 20:07
  • $\begingroup$ @user1123975 clearly, the force is completely determined by the fields & charge distributions. $\endgroup$ – Danu Sep 24 '14 at 20:08

The first layer are Maxwell's equations. They look like this: $$\nabla \cdot \vec{E} = \rho, \; \nabla \times \vec {E} = -\frac{\partial \vec{B}}{\partial t}$$ $$\nabla \cdot \vec{B}=0,\; \nabla \times \vec{B} = \mu_0( \vec{j} + \epsilon_0 \frac{\partial \vec{E}}{\partial t})$$ Where $\vec{B}$ is the magnetic field, $\vec{E}$ is the electric field, $\rho$ the charge density and $\vec{j}$ the current. It is not very important what the equations exactly mean, but it is important to understand that they determine $\vec{E}$ and $\vec{B}$ for given $\rho, \vec{j}$. For example, they tell you that for a static point charge $q$ at origin, the electric field at position $\vec{r}$ will be $$\vec{E}(\vec{r}) = \frac{q}{4 \pi \epsilon_0} \frac{\vec{r}}{r^3} $$ If you do not completely understand this expression, no worries, it is just to demonstrate that the Maxwell equations give us precise formulas for the fields.

This is all nice and beautiful, but how do we know this mysterious $\vec{E}$ is somewhere? That is because it acts on matter by a force called the Lorentz force. Even though we cannot ever see the field $\vec{E}$, we say it is there because two opposite charges start to move towards each other. The field is the invisible first layer and we conclude it's existence only through seeing it's effect on the second layer - moving charged objects. We say "opposite charges attract" and "negative charges repel", but the thing that carries this Lorentz force is actually the electromagnetic field.

Consider now a magnetic field - when you have a bar magnet, you are accustomed to drawing the magnetic field-lines around the magnet. They really do exist, but do we actually ever see them? You could say that yes, by putting the magnet under a piece of paper and put magnetized dust on the paper to see lines exactly as in the pictures? But do we see the field itself? Not directly, once again we see it only through the second layer of specks of magnetized dust in certain patterns. You remove the layer of the dust specks and the image disappears - but the field is still there.

But you could correctly object that it could just be "bodies acting on bodies" - no extra handles in the form of fields needed. But it turns out that the field can carry information on it's own. When you wiggle a charge, it creates an electromagnetic wave which can affect other objects long after the wiggle is finished. You hear your radio reproducing a speech taking place kilometers away from you, so you conclude that something must have carried the recording towards you. The second layer is your radio, but the carrier, the first invisible layer is once again the electromagnetic field.

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Maxwell's equations, in their microscopic form as formulated by Lorentz, are the standard postulates of electrodynamics. From them all electromagnetic formulas and properties can be derived. The symmetry of these equations relative to spatial translation implies, as follows from Noether's theorem, the law of conservation of linear momentum of the charges and their associated EM fields. From this balance equation (which is, in fact, a generalisation of Newton's second law) follows that the force on a (charged) particle, in the non-relativistic approximation, is precisely the Lorentz force. In other words, the Lorentz force follows from Maxwell's equations.

For more on this, see Chapter 4 of my book "Electromagnetic Field Theory", freely downloadable from http://www.physics.irfu.se/CED/Book/index.html where this is discussed at some length.

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