I see in this picture a beam of electrons leaving the cathode. But I read before, that the electrons can flow out of a copper wire only, if they are replaced by other electrons. I guess, that in the battery we can replace it with another one, but in the AC current we can't power companies do not create or destroy electrons.
It's called Thermionic Emission. All of the old vacuum tubes used it. Wikipedia it.
Let us first examine what happens to the electrons in a conductor of length $l$ that is connected to an AC voltage source. The electrons oscillate to and fro with the oscillating voltage. Conservation of charge says that in a length $dl$ of the conductor in a time interval $dt$, whatever number of electrons flow out of $dl$ must be gained by the remaining part of the system, which in our case is the rest of the conductor. Remember that the conservation must hold globally.
Now however, if we have a mechanism by which an electron goes out of our system (via emission), conservation of charge says that the surrounding has gained an electron. That is fine. But the question is what happens to the lost electron in our system? Well, our system will gain one electron back from the surrounding, just elsewhere (for example from the table it is on). In case of a closed circuit where there is no pathway to gain electrons from the surrounding, there will be a buildup of positive charge until there is enough potential build up such that it can rip electrons off from the surrounding. This is why it is always recommended to have a grounding. Which basically is a pathway between your system and the surrounding to avoid charge buildup.