Is the magnitude of the Lorentz force always the same in different inertial frames?

[silmaril89]

Sorry if I should be posting as homework, but it's not homework anymore and I'm just curious about the answer. My question is simple and doesn't require answering the actual homework question I had. If there is a force on a charge in one inertial frame of reference, will there also be a force of equal magnitude and direction in any other inertial frame? I'll put the inspiring homework question below in case anyone is interested in it. Also, the reason I'm asking is because I found in the homework problem, that the magnitude of the force was different, by a factor of gamma. This was not what I was expecting. I thought they would be equal.

The Homework question:

In one frame F, there is no electric field, but there is a magnetic field pointing in the y direction of magnitude b. There is a charge q moving in this frame perpendicular to the magnetic field (lets say parallel to the x axis), and is moving with a velocity u. Therefore in this frame there will be a magnetic force on the charge in the z direction with magnitude qub.

Let the rest frame of the charge q be F'. Find the electric and magnetic fields in the frame F'. Then find the force on the charge in the frame F'.

Here is how I solved it. From looking in a table. I found the electric and magnetic field transformations. I found there to be an electric field equal to $$\gamma$$ub in the z direction. There is also a magnetic field, but that doesn't matter, because the charge is at rest. The force on this charge will then be q$$\gamma$$ub in the z direction. The forces are in the same direction in both frames, but they are different in magnitude by the factor $$\gamma$$. Is this right? I was thinking they would end up being the same magnitude. But then I thought, maybe it's stronger due to length contraction

 

[bcrowell]

Thanks for being clear about the homework-related aspects of this. IMO there's nothing inappropriate in asking this here.

The nice way to think about this kind of thing is that every four-vector transforms the same way under a Lorentz transformation. Therefore the final result of this calculation shouldn't depend on the fact that it's an electromagnetic force. Here's how the force four-vector is defined: http://en.wikipedia.org/wiki/Four-vector#Four-force Note that its spatial part differs from the force three-vector by a factor of gamma. Under a Lorentz boost, the components of a vector perpendicular to the boost don't change. That means that the *four*-fource is the same in both frames. But since the spatial part of the four-force differs from the three-force by a factor of gamma, I think that confirms the factor of gamma that you found suspect.

 

[silmaril89]

Thanks for the reply. So it seems that my answer was in fact correct. I think I understand why as well.