Dear Omer, it seems that you are a bit confused about electromagnetism:
What are axioms and what are theorems?
(short answer: by convention, Maxwell equations are axioms and all the rest is deduced from them).
On an electromagnetic wave, are there charges or currents?
(short answer: no).
Now I will try to explain this with more detail. First, let me make an analogy with Newton gravitation, that I’m sure it’s more familiar to you. The analogy is the following:
| Newton gravitation | Electromagnetism |
|---|---|
| mass | charges and currents |
| gravitational field | electromagnetic field |
| Newton equation | Maxwell equations |
I do not know what Tipler does (I only browsed through this book once, and I did not enjoy it). But, according to what you say, I suppose it does the following: there are many equivalent formulations of the whole of electromagnetism, for instance, these three:
These three formulations are matematically equivalent, so it is fairly arbitrary which one is taken as an “axiom”, and which ones are to be deduced. Historically, the particular laws of the third formulation were discovered first, and later came Maxwell and wrote his equations, which turned out to be a neat way to summarize everything. I suppose that Tipler follows the historic order, and you are confused by this. Now I will try to do the same thing using Newton gravitation (the three formulations of it); thus, you can see how it works. This is much simpler than electromagnetism, because the third formulation contains a single law relating masses to forces. I will start with this basic law, which we were taught in high school, and then I will rewrite this same law as an integral equation, and then as a differential equation.
The force F that attracts two masses \(m\) and \(n\) located at points \(x\) and \(y\) is: \[ F = \frac{y-x}{|y-x|^3}\cdot m\cdot n \] were units have been chosen so that the gravitational constant is \(1\). This formula is called the inverse square law, because it says that the force attracting two masses is inversely proportional to the square of the distance. Usually, we make an abstraction here and we talk about the gravitational field \(G\) created by a particle of mass \(m\) located at point \(x\): \[ G(y) = \frac{y-x}{|y-x|^3}\cdot m \] this describes the force that would act upon a test particle of unit mass located at each point \(y\). From here, by the principle of superposition (or linearity), we can compute the gravitational field produced by masses \(m_i\) located at points \(x_i\): \[ G(y) = \sum_i\frac{y-x_i}{|y-x_i|^3}\cdot m_i \]
Now, let’s move to the continuous case. If we have a planet, for example, we do not want to sum the gravitational fields produced by each atom of the planet. It is more confortable to consider that we have a mass density distribution \(m(\cdot)\) defined on the whole space, instead of a finite set of masses \(m_i\). Then, the gravitational field created by a mass distribution is: \[ G(y) = \int\frac{y-x}{|y-x|^3}m(x)\mathrm{d} x \] were the domain of integration can be any set that contains the support of \(m\), for example the whole \(\mathbf{R}^3\).
This is the integral form of Newton gravitation, analogous to the integral form of Maxwell equations. See how it works: you have a scalar field \(m\), which is the mass distribution, and an vector field \(G\), which is the gravitational field. This equation allows to compute the gravitational field \(G\) from the mass distribution \(m\). Similarly, Maxwell equations in integral form allow us to compute the electromagnetic field \((E,B)\) from the charge and current distributions \((\rho,J)\).
Notice that the integral formulation contains the discrete formulation as a particular case: you only have to set \(m\) equal to a sum of diracs of weight \(m_i\) at the points \(x_i\), and you recover the expression with the sum.
Now let us move on to the differential formulation. I don’t make all the computations because I’m lazy, but it is a standard exercice in vector analysis to show that the integral expression above is equivalent to the following two differential equations \[\mathrm{rot}\ G = 0\] \[\mathrm{div}\ G = m\] well, the second equation is missing a factor \(4\pi\) somewhere, but I don’t remember where (to obtain nicer differential equations, we could have put the factor \(4\pi\) on the formula for the attraction of two masses, and here it would disappear). When we meet I can show you the detailed computation on a blackboard. The computation is beautiful, but not exactly fun to write on a computer. So, voilà!, we already have the theory of gravitation written in differential form! Notice that you can deduce many properties of gravitation from this equation (well, all of them, in fact!). The first equation says that, if the topology of the space is not too complicated, the gravitational field is conservative: \(G=\mathrm{grad}\ V\). The second equation is simply the potential or Poisson equation \(\Delta V = m\), which is probably the most well known and easy to solve PDE in the world. Another observation is that \(m=0\) implies \(G=0\), thus, if there are no masses then there is no gravitational field. This seems trivial, but it is important to notice. More about this later.
Now let’s move to electromagnetism. Your book may have done a similar deduction as my explanation above, but for electromagnetism instead of gravitation. The problem is that instead of a single law describing the force between two charged particles, you have a lot of laws depending on the speed of the particles, etc. After many complicated deductions (but ultimately similar to what we have done above), they arrive to Maxwell equations in integral form, and later, in differential form. These equations relate the charge and current densities that you have (\(\rho\) and \(J\)) with the electric and magnetic fields that they create (\(E\) and \(B\)). They are like that (with units selected so that the speed of light is \(1\)): \[\mathrm{div}\ E = \rho\] \[\mathrm{div}\ B = 0\] \[\mathrm{rot}\ E = -B_t\] \[\mathrm{rot}\ B = J + E_t\] where the subindices denote partial derivatives (with respect to time, in this case). This is the first novelty, that the Maxwell equations depend on time, not like gravitation which is “static”.
In gravitation, if we did not have masses, the gravitational field was zero. In electromagnetism, if we do not have masses nor currents, there are still things to do. In fact, many things. See what happens: \[\mathrm{div}\ E = 0\] \[\mathrm{div}\ B = 0\] \[\mathrm{rot}\ E = -B_t\] \[\mathrm{rot}\ B = E_t\] it turns out that these equations are very easy to solve: take the last two ones and apply \(\mathrm{rot}\) to each of them: \[\mathrm{rot}\ \mathrm{rot}\ E = -E_{tt}\] \[\mathrm{rot}\ \mathrm{rot}\ B = -B_{tt}\] now, and you can check that using coordinates, for any vector field \(K\) you have \[\mathrm{rot}\ \mathrm{rot}\ K = \mathrm{div}\ \mathrm{grad} K-\mathrm{grad}\ \mathrm{div}\ K\] (where, on the first term the operators are “vectorial”, that is, they act on each component: \(\mathrm{grad}\ K\) is a matrix and \(\mathrm{div}\ \mathrm{grad}\ K\) is again a vector). Applying this little formula to the two equations above, and using the fact that the divergences vanish (as said by the first pair of Maxwell equations), we obtain the following equivalent system of six equations and six variables: \[\Delta E = E_{tt}\] \[\Delta B = B_{tt}\] Does it say something to you? this is the wave equation in each component of the electromagnetic field! The solutions of this system are called electromagnetic waves.
This is a beautiful property of the electormagnetic field, that the Newtonian gravitational field lacks. In classical gravitation, if you don’t have masses, there is nothing to do. On the contrary, in electromagnetism, the electromagnetic field can exist on its own, without need for charges nor currents. When this happens, we have light rays, radio waves, or electromagnetic radation in general.
This has important consequences. You have to think that electromagnetic waves carry energy. Imagine the following situation. Imagine what happens if you put two equal charges of opposite sign orbiting around each other. The force that attracts them is exactly the same as in the case of gravitation (obeying the ivnerse square law), however, the behaviour is very different. A moving mass does not dissipate energy. On the contrary, a moving charge creates an electromagnetic field, that transports energy away from the charge. A system of two masses orbiting around each other does not lose energy and continues to orbit forever. On the other hand, two charged particles orbiting around each other lose energy very fast (radiated away as electromagnetic waves), and they aproach each other until they collide. This observation made Bohr very nervous because it was incompatible with his orbital model for the atom: indeed, Maxwell equations predict that atoms would radiate energy and the electrons would collapse immediately towards the nucleus. They had to make some tricks to avoid this collapse, and quantum mechanics is all they could got with.
And well, just a final comment: in Einstein gravitational theory, there are gravitational waves! It turns out, that they are so weak that they have not been detected yet. To try to detect them, people look at pairs of super-massive black holes orbiting around each other, and look if the system loses energy or not. For now, the predicted energy loss is so small that the current instruments are not precise enough to detect it.
Soothly we live in mighty years!