I have studied that the net force on a charged object moving with velocity $v$ under both electric and magnetic fields in given as $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$. I have also been told by my teachers that the net force given by this equation is frame independent for inertial frames.
Regarding this I have two doubts:
- Is the force frame independent in only magnitude or both magnitude and direction?
- Why is either magnitude or direction frame independent at all, since the force is dependent on velocity which is not frame independent. For example, if we assume a constant electric field $E$ and a constant magnetic field $B$ and two observers $O_1$ and $O_2$ where $O_1$ is at rest and $O_2$ is moving with constant velocity $v$. Now if we project a charged particle also with velocity $v$, $O_1$ will observe the particle's velocity as $v$ and hence the force on the particle observed by $O_1$ is $q(E+v\times B)$. $O_2$, however, observes the particle to be at rest as there is no relative velocity between O2 and particle. So shouldn't $O_2$ observe the force as only $qE$? The net direction of the forces observed by both observers should also be different.
How do we resolve this?