Let we suppose to have a HV Tube or CRT . The filament is connected to the secondary of the transformer used to supply the filament.
At time $t=0$ the wires and cathode are neutrally charged. Then when we switch the power on, the filament begins to heat and free electrons are being evaporated from the cathode. Now I would come to conclusion that if the electron has left the metal there should remain a hole and the cathode would begin to be more and more positively charged, but this is not happening.
Are those electrons replaced? If yes, then from where?
Or if they aren't emitted at all, are they generated?
It might be clear from the picture that, electrons emmited by cathode never strikes the anode. The anode is used for accelerating the electron, not to recombine the electron.
Further question is: Where the electrons in anode come from? For accelerating the electron beam, energy is used proportional to the kinetic energy of electrons, but how the current flows if the electons never strike the anode?
Best Answer
Here is an image with the conducting layer all around the glass from the inside:
and here is the circuitry (from a different link):
As you see the circuit closes with the conducting layer. The power supply provides the energy to keep the cathode negative and the anodes positive. In this diagram the heating of the cathode comes from a different power supply. The electron beam part of the circuit current
Here is another paper
and of course, close the circuit with the power supply.
With the above in mind,
because the power supply through the circuit replaces the charge
Of course they are thermally emitted. If there were no thermally emitted beam of electrons the circuit would represent an elaborate capacitor circuit, no current would be flowing.