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?

• The moving charge will produce a changing electric field. Although it may not appear to move in the second frame, it will change in time, and from Ampere's law we know that a time-changing E-field requires a curling magnetic field to be present: – M Barbosa Jun 10 '16 at 17:38
• Please note that even if a charge is in rest there is a magnetic field. This is because a electron as well as a proton have the intrinsic property of a magnetic dipole moment. – HolgerFiedler Jun 10 '16 at 22:37
• You just found a great paradox in relativity, and a proof for multiple universes:). It might mean that the frame of reference as seen in B will diverge to another universe different from the universe of the frame observed in laboratory.. not sure though. I was also puzzled by this.. – philip_0008 Jun 11 '16 at 7:58

2 Answers

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

• "In frame B where the charge is stationary, then an observer would only see an electric field." This is not what in a experiment will be observed. Since a charge like a electron has a permanent magnetic dipole moment a observer will observe a magnetic field. – HolgerFiedler Jun 10 '16 at 22:39
• In frame B would the observer see a changing E field or the normal electrostatic field – Shashaank Mar 8 '17 at 16:18
• @Shashaank see edit – Rob Jeffries Mar 8 '17 at 17:26
• @RobJeffries Ok that means M barosa is wrong then – Shashaank Mar 8 '17 at 17:45

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.