Why is a coil behaving different in DC and AC? (the formula Z=omega*L shows that it is like a short in DC and has a frequency growing resistance in AC) But how about the curent through wire (supposing there's no core in the coil)? Why is it proportional to the derivative of the voltage applied? If I place a coil at the output of a DC source do i shortcircuit the outputs and probably destroy the source?

It is the magnetic field created by the current flowing in the wire of the coil. In your formula for impeadance you forgot the j or i representing the imaginary unit. The correct fomula is: Code ( (Unknown Language)): Z = j*ω*L It is a complex number. A real inductor will have a small DC resistance from the properties of the wire and this is modeled as a small resistance in series with the inductance. The complex impeadance of this non-ideal inductor will be Code ( (Unknown Language)): Z = R + j*ω*L In a steady state where the current is not changing, which implies that di/dt = 0, the voltage accross an ideal inductor will be zero. In a non-ideal inductor let us say there is a constant current of 100 milliamperes flowing through a DC resistance of 0.2 Ohms. The voltage drop would be 20 millivolts. You ask why the voltage is a function of the time rate of change of current and the circular answer is that inductance is a property that resists changes in current. It resists the change in current by developing a voltage across itself. A DC resistance resists the magnitude of the current , while the inductance resists the "velocity" of the current. A mechanical example may help. A spring is a device for which Force is proportional to displacement. A shock absorber or dashpot, or damper is a device for which Force is proportional to the velocity of displacement. If you consider the 2nd order, ordinary differential equation for a mechanical spring-mass-damper system you quickly discover that it is exactly the same equation as the one that describes a series R-L-C circuit. You seem to be more than a little bit lost, is this of any help to you at all?

It is helpful! I have many gaps on "how it really works" and i try to fill them. You're explaining very well, thanks a lot! [I know that there is a j in the formula, I only meant the inductive reactance, not impedance] When I was studying these at school we were mostly explained mathematically and at that time I didn't ask to much why (my fault). Now I want and need to know why (I'll work with it). Now, while reading books, working with (simple!) circuits, I find out the lack of knowledge that I have. That's it! :|

The first derivative shows the 90 degree phase difference between the current flowing through the coil and the voltage "induced" across the coil. Why is there a phase difference? - An induced voltage requires a changing magnetic field. The greater the rate at which the field changes, the greater the magnitude of the voltage. If you look at the graph of a sinusoidal current, the slope (rate of change) is greatest at the zero point and least (zero) at the positive and negative peaks. Therefore, the current is 90 degrees out of phase with the induced voltage (which is not the same as the voltage of the source). The induced voltage opposes the current (Lenz's Law), thus the reactance. A DC current through the coil has a steady current, therefore no opposing voltage. The result is nearly a dead short. Formulas sometimes mix a person up more than they help.

One of the really surprising things about inductors is their ability to produce large positive and negative voltages when switched on and off. A large di/dt will produce a large voltage across the inductor. This behavior is the principle behind switching regulators in the boost configuration, which produce a higher voltage from a lower voltage.