In a Coolidge tube, electrons strike the target and stop inside it. Does the target get more and more negatively charged as time passes?

A Coolidge tube is a complete circuit, with filament as the cathode, metal plate as the anode and a power supply to drive the circuit (as shown in the figure above). The electrons ejected from the cathode by thermionic emissions are received by the metal where they give off X-rays and knock off electrons from the target atoms till they finally come to rest. These newly generated electrons are then driven by the electric field of the circuit towards the terminals of the power supply. There are no excess electrons in the metal and thus the target does not get more negative.

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