# Electromagnetic waves according to Maxwell

If a variable Electric field creates a variable magnetic field and VICE VERSA (according to Maxwell's equations), then why don't we enter a loop where E vector and B vector keep creating one another until they reach infinite magnitudes?

• Because it does not work that way. Have you had a look at any solutions to Maxwell's equations? – Cryo Jul 9 at 15:22
• by the way, your feeling is based on the fact that an infinite sum is necesseraly infinite. Which is not true: $\sum_n \frac{1}[n^2} = \pi^2/6$ for example. – StarBucK Jul 9 at 15:32
• More often than not, verbose descriptions of physics are not precise enough to be relied upon to base such new arguments in them that which would rely on the precise details of the description. – Dvij Mankad Jul 9 at 16:56
• It is a misunderstanding that electric and magnetic fields generate each other. They do not. – my2cts Jul 9 at 18:19
• Care to elaborate, @my2cts? I'm down with the idea that the statement is loose, but there are Maxwell's equations and an array of practical devices that work off this basic idea. Like motors and generators. – Brick Jul 9 at 19:12

The fields are vectors with (signed) direction. In a wave, the $$\mathbf{B}$$ field "creates" $$\mathbf{E}$$ field components, but they are, at some times at least, opposite to the currently present $$\mathbf{E}$$ field and therefore reduce the total field. And vice versa. This manifests via the relative minus sign between Faraday's law $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$ and Ampere's law $$\nabla \times \mathbf{B} = \frac{\partial \mathbf{E}}{\partial t}$$ (shown here in units where $$c=1$$ and in vacuum).
Ah, this was actually the great insight of Maxwell. What you are referring to is electromagnetic waves (i.e. light). These waves are just the electric and magnetic field continuously generating each other. Unlike what you may intuit though, looking at the actual mathematical solutions that yield such behavior shows that the magnitude of the fields never actually grow but either stay the same (e.g. plane waves) or shrink (e.g. spherical waves). This can easily be seen by Poynting's theorem which shows that Maxwell's equations conserve energy or, more specifically, the quantity $$\frac{\epsilon_0E^2}{2}+\frac{B^2}{2\mu_0}$$ is conserved in vacuums.