How Does Charge Movement Affect Electric Field Perception in Different Frames?

[vin300]

Charge 1=+e
Charge 2=-e
Charge 3=+e




If chg3 moves along the positive x in a magnetic field directed in +ve y, it sees an electric field in the +ve z, then in this frame a chg 1 and 2 both in rest in the initial frame experience an electric field in the -ve z, with the positive charge accelerated in the -ve z and negative charge in the +ve z, but this is not possible because both + and - charges are at rest in the initial frame, so in chg 3's frame both must be accelerated in -ve z.

 

[vin300]

If r is the lorentz factor,in chg1 and 2's comoving rest frames, for chg3's velocity v along + x and B along + y, the induced electric field is
E(z) =r[vxB/c]
In chg3's frame, charge 1 which is positive and charge 2 which is negative both move with a velocity -v, the induced electric field for each is
E'(z)=r[-vxB/c]
This is exactly equal and opposite to the former, but only charge 1 is accelerated in - z because the force also depends on charge. Charge2 is forced along +z, but this defies logic since in chg2's frame chg3 is accelerated along +z and in chg3's frame chg2 is accelerated along + z