Problem with magnetic field due to relative motion We know that, moving charge produces magnetic field in the surrounding space.
Consider this scenario :
A charge 'q' is moving with a constant speed 'v' in the direction of positive x axis of a coordinate frame 'A'.
As a result, there exists magnetic field everywhere in the space.
Now, consider another frame 'B' moving with speed 'v' in the direction of positive x axis (same velocity as that of the charge). Thus the velocity of the charge as seen in 'B' frame is zero. Thus there should be no magnetic field produced. Could someone explain this to me?
 A: Electric and magnetic fields are not relativistically invariant. What you measure will depend on the frame of reference you are in.
In your example, the moving charge in frame A will be responsible for both an electric field and a magnetic field.
In frame B where the charge is stationary, then an observer would only see a static electric field.
Exactly the situation you propose is used as an example on the relevant wikipedia page https://en.m.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity
A: This is correct. The electric and magnetic fields can morph into eachother if we change the reference frame we're observing it from. This is the reason there is only one force called electromagnetic force.
When you're travelling with the proton, you only see electrostatic field.
When you're standing still, you observe both electrostatic and magnetic field. Mind that electrostatic field actually get's bigger when the proton is moving according to the observer. But so does the magnetic field, but magnetic field increases slightly more, than the elctrostatic.  This increase is not linear, but it is acting according to special relativity.
The total net force another proton would feel traveling in the same direction as the original proton Will be decreasing (because magnetic force Will cause them to attract, while electrostatic to repel). The magnetic force is increasing more with speed than electrostatic one is. Electrostatic force is increasing due to the space conrtaction.
Now you might still be confused because the protons Will repell at different rates depending wether observer is stationary or moving with the charges. This is correct and the reason for this is time dilation which says that moving objects experience time slower.
If the protons would be moving away from observer at the speed of light, the magnetic force would become equal to electrostatic force, causing them to freeze in time relative to eachother. Which is exactly what would happen if it was possible.
Hope this helps clarrify some things without using maths.
