Suppose there’s a long, positively charged beam oriented along the x-axis. Two test masses, each with the same positive charge, are placed at an equal distance from this beam. In one reference frame, the beam and test mass B are stationary, while test mass A is moving to the left:
Since the beam is stationary, there are no magnetic forces. Since the test masses have equal charge, the electric force is the same for both of them. If both test masses begin accelerating downward at the same rate, this tells us they must have the same mass. But if they have the same mass while mass A is moving, special relativity tells us that mass B must have a larger rest mass, by a factor of:
$$
m_{B}=m_{A}(1+v^{2}/2c^{2})
$$
(assuming v<<c)
If we switch to another reference frame, where mass A is stationary while the beam and mass B are moving to the right, the two masses should still have the same vertical acceleration:
But in this frame, mass B is moving and therefore should have a higher relativistic mass, by a factor of:
$$
m_{B}=m_{A}(1+v^{2}/c^{2})
$$
(again disregarding terms of higher order than 2)
In order for both masses to accelerate at the same rate, a greater repulsive force must be applied to mass B. Since they have the same charge, any discrepancy in force must be due to a magnetic force on mass B. However, since mass B is moving in the same direction as the beam, the magnetic force on it will be attractive, thus reducing the net repulsive force. This leads to the conclusion that mass B will accelerate slower than mass A in this reference frame, which is impossible. Where is the error in my reasoning?
I tried finding an answer to this online, and found that the ratio between the electric and magnetic forces for moving charges will go as (v/c)^2:
https://academic.mu.edu/phys/matthysd/web004/l0220.htm
Which is consistent with my calculation, but the force is still in the wrong direction.
