Why does the Voltage of a Capacitor change in an alternating current (AC)?

Question

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.

Answer 1

First of all, the driving voltage is going to be a sinusoidal one as you are picking an AC source.

$$V=V_0sin(\omega t)$$

The current as a function of time is given by: $$I=I_0cos(\omega t)$$

This is because: $Q=CV$ $\implies$ $I=\frac{CdV}{dt}$ (by taking the derivative of the LHS and RHS with respect to time.

Now you asked why there is a delay which I am assuming to be the phase shift between the voltage and the current - which is $\pi /2$ in this case. This is why the voltage was assumed to be a sinusoidal one (because the capacitor plates were not charged and hence no potential difference could exist between two neutral plates). and the current turned out to be a cosine wave, much to your surprise.

But this makes sense of course because the current through the circuit will decrease when the voltage increases throughout the charging process in the capacitor. But what you should realize is that the AC source can only help in generating $Q$ amount of charge in the capacitor (after charging one cycle). So this means there would be increased repulsion with time when the electrons are moving across the wire. So, the rate of charging would reduce although charge would build up anyway with time.

Rate of charging is literally the current $I$ in this scenario.

EDIT: As to the scenarios where there is a phase shit other than $\pi /2$, you can use: $tan \phi =\frac{X_L-X_C}{R}$.

In other words, you get such phase shifts when you add different components (like resistors and inductors) into the circuit in addition.