Where does the 3rd and the 4th Maxwell's equations lead us in the end? Take the 3rd and the 4th equation from this table.
The first tells us that an electric field can be generated by a magnetic field.
The second, says that a magnetic field can be generated from an electric field (or more specifically, from its variation in time).
Well, at least, I've been told so.
My question are two and are rather simple.


*

*If the two fields can generate reciprocally, won't a single tiny variation in time of the first eventually always finish in two infinite fields? Maybe they change in different directions? If this is the case consider question number 2.

*How can we study the effects separately? I mean, let's take for example an experiment, where the magnetic field changes in time. Something will happen, and we conclude that the electric field generated by the variation of the magnetic field (we can calculate it using the second equation), caused this thing. Can we really say that? (in my Physics book there are some exercises that are told this way). But shouldn't this variation be the point of origin of other infinite variations? Why do we make conclusions here?
 A: To answer your first question, take a look at their form:
$$\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$$
$$\nabla\times\mathbf{B}=\frac{\partial\mathbf{E}}{\partial t}$$
(clearly I am ignoring constants and $\mathbf{J}$). What happens if over a period time $t$, $\mathbf{E}$ increases? The curl (how much a vector rotates) of the magnetic field changes (increases). Conversely, increasing $\mathbf{B}$ over a time $t$ decreases the curl of $\mathbf{E}$. This sort of balance (due to the signs) will not introduce infinite fields.

If you take the curl of Maxwell 3, you get
$$\nabla\times\nabla\times\mathbf{E}=-\nabla\times\frac{\partial\mathbf{B}}{\partial t}$$
Since $\partial_t\nabla=\nabla\partial_t$ (due to equivalence of 2nd-partial derivatives), the above can be written as
$$\nabla\times\nabla\times\mathbf{E}=-\frac{\partial}{\partial t}\left(\nabla\times\mathbf{B}\right)$$
Using a vector calculus identity and Maxwell 4, you get
$$\nabla\left(\nabla\cdot\mathbf{E}\right)-\nabla^2\mathbf{E}=-\frac{\partial}{\partial t}\left(\mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}\right)$$
Not much can be done here unless we assume free-space (i.e., $\rho=\mathbf{J}=0$) and use the relation $c=1/\sqrt{\varepsilon_0\mu_0}$; in this case we get
$$\nabla^2\mathbf{E}+\frac{1}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2}=0$$
which is a wave equation. This can also be solved for $\mathbf{B}$, giving you the same equation. If we make a perturbation in $\mathbf{B}$ or in $\mathbf{E}$, we can follow the propagation with only that field.
A: You can have coupling between equations that leads to stable motion -- see the harmonic oscillator
\begin{equation}
\dot{p} = - x
\end{equation}
\begin{equation}
\dot{x} = p
\end{equation}
in this case, the sign difference buys you stable modes. There's the same sign difference between the curl equations so you can start to suspect oscillatory motion (although this is very rough and not a terribly dependable rule). Feedback does not imply instability.
As for the specific physics textbooks, it would be helpful to see an example of where this comes up, although for example if you have a current, current causes a time varying $\vec{B}$ field, and then having finite $\vec{B}$ gives you $\vec{E}$. Usually how $\vec{E}$ interacts with $\vec{B}$ has to be cast as to whether your source is a current or a charge density. But again, a specific example would be helpful.
