I want to find $\Delta V_{ab}$ for the circuit on the figure

I have tried to use the superposition theorem, and then

Here I find my first question, because if there is a coil between A and B, then $\Delta V_{ab}=0$ and it not make sense to me in this case... Isn't it? But looking at the solution in some moment the authors write this two circuit

and I don't understand why they open the circuit in the capacitor branch when there should be a short circuit (I think). Maybe there is some special rule for capacitors and AC source? Maybe in this special case We have to open the circuit in order to don't have a direct coil between the two terminals? or maybe I have missed something...