# Relativity, and Lenz's law with respect to the Lorentz force acting on a wire?

For this system: It's simple for me to predict the induced EMF, and if the wire were connected to a complete circuit(for current $I$ to flow), the opposing Lorentz force($-F$): However, if the uniform magnetic field was produced by a solenoid, and we now fixed the wire and moved the solenoid away from the wire with the same $\| \mathbf{v} \|$ such as this diagram: From the principle of relativity, I know that the wire would produce the same EMF as the second diagram,and possibly the same Lorentz force? Yet, how can that be true? Wouldn't the wire try to resist the change? Like so: Related question.

• Why would the wire resist? Is it glued to something? – CuriousOne Jul 20 '16 at 5:55
• From 1st diagram wouldn't the induced current resist the change(hence lenze law)? Why wouldn't the final diagram be true? Is it true...? For the polarity of $\epsilon$? Or should it be the same as the second diagram? As you can a see I'm extremely confused due to Lenz's law and Relativity. The wire is fixed in anyway you imagine it to the ground wall etc... – Pupil Jul 20 '16 at 6:05
• When you move the magnetic field, you will see an electric field that starts accelerating the charges. A local current will flow until the electric field is balanced. There will, of course be an electromagnetic field generated that will impact some momentum on the wire, but I don't think you have captured that with any of the diagrams. For that you need a full analysis with Maxwell's equations and the boundary conditions imposed by the wire's charge carriers. Because of the Lorentz invariance of the equations it is irrelevant in which observer system that calculation is carried out. – CuriousOne Jul 20 '16 at 6:15
• For both frames(moving the wire or moving the magnetic field) the electric field direction is the same? Due to the Lorentz force acting on separating the charges in the same direction? My intuition makes me believe that both cases are the same regardless, but somehow when considering Lenz law I lose myself. – Pupil Jul 20 '16 at 6:30
• In one frame it's a constant magnetic field, in a different frame it's a magnetic and an electric field. The field components can not be the same in all frames and they aren't. The Lorentz transform mixes them. The physics can not change, though. How would the system know which observer to "obey"? – CuriousOne Jul 20 '16 at 7:32

## 1 Answer

I think the resolution to the op is quiet simple and the entire paradox is because the solenoid in the third figure, should be moving to the left and not to the right! (If you are in a car moving to the right, then according to you the trees are moving in the opposite direction, ie to the left!) In other words, the system: in the reference frame of the wire looks like so: and if you close the circuit it will be: Note that in both reference frames the force acts in a way that resists the change as expected.

• When I study more about the "Moving magnet and conductor problem" It seems that they would both produce the same result regardless of what is moving, whether it's the magnet or the conductor(assuming both moved in the same direction) yet what you posted makes sense, however, doesn't it contradict things? – Pupil Jul 20 '16 at 20:37
• Your diagram and your analysis when the magnet moves to the right is correct. (It's just that the magnet moving to the right is not the same thing as the first diagram. It's not just changing frames.) Nevertheless, your analysis is correct and the force will be to the right. Once more, the force tries to resist the change, ie it tries to maintain the same relative position between the magnetic field and the conductor (remember that absolute positions do not mean anything. Relative positions count.) – Heterotic Jul 20 '16 at 21:03
• In that sense, I still fail to see a paradox. Maybe if you could rephrase which statements exactly you find contradictory, I could elaborate further, – Heterotic Jul 20 '16 at 21:03
• No you are right, I'm just confused and things are clearer now. Thank you! – Pupil Jul 20 '16 at 21:47