Magnetic Field due to a Point Charge

[Electric to be]

Hi. I'm currently learning about magnetism, however the course I'm in doesn't combine special relativity with E&M so I just wanted to do some personal exploring.

The magnetic field for a point charge is proportional to V1 x R of the charge. Then, the magnetic force is proportional to V2 x B where V2 is the velocity of some test charge. This makes sense to me because length contraction only occurs along the direction of motion. So, as a result if V of a test charge is parallel to B, that means it is perpendicular to V of the charge creating magnetic field and should experience no magnetic force.But, my main question is for the case when V x R = 0 because R is parallel to V. If some test charge is moving parallel to the field creating charge, then in it's frame the distance between the two charges should increase, and this "difference" in electric forces seen becomes the magnetic force in the frame where both are moving.

So how is the magnetic field, and therefore force 0 just because a charge is on the same line that the field creating charge is moving on? Shouldn't length contraction/dilation work just the same, and therefore create a discrepancy in the electric forces, which becomes the magnetic force?Sorry if this isn't clear enough, I can elaborate more if needed.

 

[Electric to be]

This question focuses on the fact that the sum of EM forces should be the same in all reference frames. Is this a true assumption to begin with? Is force invariant?

 

[Electric to be]

In fact, the R x V makes even less sense to me now. For example, say the angle between the magnetic field creating charge and the distance was 90 degrees. Say also the charge the magnetic field was acting on had a velocity parallel to the the magnetic field creating charge.

This means that there would be no length contraction, since the distance along their axis of travel is already zero. Yet, at this position the magnetic force is the strongest??

 

[Orodruin]

This question focuses on the fact that the sum of EM forces should be the same in all reference frames. Is this a true assumption to begin with? Is force invariant?

No, force is not invariant. Just like momentum, it becomes part of a 4-vector and is not invariant by itself.

 

[Electric to be]

No, force is not invariant. Just like momentum, it becomes part of a 4-vector and is not invariant by itself.

Yep got it. In either case, could you explain why there is no magnetic field made along the line of movement of a charge? A different moving charge would see a length contraction/expansion in this frame, and I feel that this could lead to contradictions. (For example if there is a neutral situation in one frame and no magnetic field) and then a non neutral and still no magnetic field since it is along the line in a different frame?

 

[Orodruin]

could you explain why there is no magnetic field made along the line of movement of a charge?

The magnetic field is the cross product between the current and the separation. Both of these are in the same direction in the case you describe and therefore the cross product is zero.

You should not be thinking of "length contraction" when it comes to the EM field, it is a rank-2 tensor field and arguing in terms of length contraction is not going to help you in general.

 

[Electric to be]

The magnetic field is the cross product between the current and the separation. Both of these are in the same direction in the case you describe and therefore the cross product is zero.

You should not be thinking of "length contraction" when it comes to the EM field, it is a rank-2 tensor field and arguing in terms of length contraction is not going to help you in general.

Well there was a different example that helped me that used length contraction. It at least explained to me why V x B is in the direction that it is. That used a length contraction explanation. (toward the bottom)http://physics.stackexchange.com/qu...or-why-lorentz-force-is-perpendicular-to-a-pa
Could something similar be applied to explain the this situation as well?