# Does $E$ cause $B$ or does $B$ cause $E$ in Maxwell's equations?

From the Maxwell's equations we get

$$\frac{\partial E}{\partial x} = -\frac{\partial B}{\partial t}$$

and

$$\frac{\partial B}{\partial x} = -\mu_0\epsilon_0\frac{\partial E}{\partial t}$$

My question is: A change in the electric field causes a change in the magnetic field, while a change in magnetic field is causing a change in electric field. Is this situation not similar to sitting inside a bucket and lifting yourself up?

-
You should use equations which apply to the same context . –  Dimensio1n0 Aug 9 '13 at 15:00
You need some context to define what you mean by your equations. They're not true in general. Is this for an electromagnetic wave propagating in a certain direction? –  Ben Crowell Aug 9 '13 at 17:29

I think it would be more accurate to say that $\nabla\times E$ causes a change in $B$, and $\nabla\times B$ causes a change in $E$ –  Larry Harson Aug 12 '13 at 16:20
Therefore, such separation is somewhat arbitrary and should not be taken too seriously. If both electric AND magnetic fields are defined at some initial moment $t_0$ in the entire 3-space, you can calculate their first derivatives with respect to time in the entire 3-space using the Maxwell equations, so you can pose the so-called (initial) Cauchy problem (http://en.wikipedia.org/wiki/Cauchy_problem ). Mathematicians proved that this problem is well-posed under some natural conditions, and the Maxwell equations can be integrated, so no problems of "sitting inside a bucket and lifting yourself up" arise. I should say though that in general your question is not trivial, it just so happens that in this particular case integration of the equations of motion is not problematic (in principle).