# 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?

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@Qmechanic: That was a quick suggested accept, thanks : ) I was ssurprised to see why there was a "Thanks for your edit! This edit will be visible..." on top and yet an "edited 53 seconds ago..." below . –  Dimensio1n0 Aug 9 '13 at 14:59
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
Evil Communirty –  Dimensio1n0 Sep 29 '13 at 12:00

That's right, electric fields can cause magnetic fields and vice-versa. This is what allows electromagnetic waves (light, radio, etc.) to travel through empty space. Shine a laser pointer into the sky, and the light from it can travel through space for a billion years. The electric field of the light will be a source for the magnetic field, and the magnetic field of the light will be a source for the electric field -- on and on it goes through the vacuum of space.

All waves are a little bit like this. A wave on a string: The motion of the string makes the tension change, and the tension causes the string to move. A sound wave: The pressure causes the air to move, and the motion of the air causes pressure to build up.

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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).