Final answer:
Doubling the current through the filament (the cathode) would not change the electron charge to mass ratio as it is a fundamental and constant property of electrons.
Step-by-step explanation:
If you were to double the current through the filament (the cathode), the electron charge to mass ratio would remain the same. This is because the charge-to-mass ratio of the electron is a fundamental property of the electron and does not depend on the current through the circuit or the intensity of the cathode rays. According to J.J. Thomson's experiments, this ratio is a constant for electrons regardless of changes in external conditions such as the type of gas in the cathode ray tube or the metal used for the electrodes.