Final answer:
The relationship between filament current and tube current is influenced by the filament resistance which changes with temperature, according to Ohm's law. Filament current affects the quantity of electrons striking the anode in a CRT, which in turn influences X-ray production.
Step-by-step explanation:
The relationship between filament current and tube current depends on the physical properties of the filament, primarily its resistance, which changes with temperature. Notably, the resistance of the filament increases significantly as it heats up to a high temperature; hence, the current through the filament depends on the filament's resistance and the voltage applied.
When addressing a filament such as that in an incandescent light bulb or a cathode ray tube (CRT), we must also consider Ohm's law, which states a proportional relationship between voltage (V), current (I), and resistance (R). This relationship can be mathematically represented as I = V/R. Therefore, when the filament is cold (at lower temperatures), its resistance is lower, leading to a higher initial current when the voltage is applied. As the filament heats up and its resistance increases, the current stabilizes to a lower value at the operating temperature.
In the specific context of X-ray production in a CRT, the filament current is the electron flow that heats up the filament, causing electrons to be emitted towards the anode. When these energetic electrons strike the anode material (often made of copper), they produce X-rays through their interactions.