I have a question about charging a capacitator in an alternating current. If the process starts the current charges the capacitator, therefore the voltage on the capacitator will be increased. Now the current negates itself, the capacitator gets discharged.
But the overall Charge should be:
$$Q_{max}=\int_{0}^{T/2} I_0 \cdot sin(2t/T)\,dt$$ $$Q_{min}=\int_{0}^{T} I_0 \cdot sin(2t/T)\,dt = 0$$
So how the charge $Q$ can be smaller as 0? How does the charge of a capacitator behave in an alternating current, so that the voltage can be negated?
EDIT:
Basically, I cannot imagine how exactly the phaseshift ($\alpha \neq \pi/2$) comes. If the voltage of the voltage-source is high, basically many electrons should be pushed into the capacitator, this decreases the current. If the voltage is high, there should be current pushing out of the capacitator. This does not explain the phase shift.
I have seen this equation $i = C\frac{dU_{current}}{dt}$ (which seems obvious to me). This would mean that if $i$ is $0$, $U$ would have to be maximal. This is not the case in every (capacitator) circuit, is it?
$$i = C\frac{dU_{current}}{dt} \implies Q = \int_{U_0}^{U_{T}} CdU_{current} \implies Q_{min} = 0 \implies U_{min,capacitator} = 0$$
But this conclusion is a contradiction to obvious observations.