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?    
 A: 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.
A: Components of the vectors of electric field and magnetic field are in the same time components of a single Lorentz-covariant tensor of electromagnetic field (http://en.wikipedia.org/wiki/Electromagnetic_tensor ). Separation of electromagnetic field into electric field and magnetic field is not Lorentz-invariant: something that looks like electric field in one frame of reference is a combination of electric and magnetic fields in another frame of reference moving with respect to the first frame of reference.
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).  
