The force is calculated with the help of the second principle of the mechanics, variation of propulsion in unity of time:
F=delta p/delta t.
We consider that electrons enter in anode without displacing others and each electron will transmit it's impulse to anode, because the electron will move in anode with drift speed.
So,variation of propulsion in unity of time of anode is the impulse "m*v" , brought by the electron multiplied with the numbers of electrons which hit the anode in unity of time.
So if the voltage increases n times, the force will increase n^2 times.