# Induced EMF in open loop

Take a wire that doesn't form a closed loop, moving at velocity $$v$$ perpendicular to a constant magnetic field. In the frame of reference to a stationary person the induced emf caused by the magnetic force is the emf just the non closed line integral of ($$v \times \vec B$$) still? I guess so.

But in the reference frame of the moving wire you have a stationary wire with a moving magnetic field. Now, although this moving magnetic field is moving, at each point in space (because it's a constant field) is the vector same? So my main question is would the value of $$\frac{\mathrm{d} \vec{B}}{\mathrm{d} t}$$ be zero same even though the field is technically moving? If so, how can there be an induced emf in this frame as there is no curl of the electric field (Is the case similar to Faraday Disk Paradox, or is this Special Relativity?)

Also, lets say that there is a curl of the electric field in this frame, Stokes Theorem is only for surfaces and a line is not a surface so how can it be said that there is the curl of the electric field anyway? (I guess generalisation from one scenario to another)

Consider another scenario where in a static magnetic field a rotating loop of wire exists. Here emf is caused by magnetic Lorentz force, but in moving the charge a changing magnetic field is created around the charges causing a curl in the electric field. So is this scenario not just magnetic Lorentz causing emf, but the electric as well?

• as if there is curl at all points in the field how do u calculate emf as a line has no.surface attached to it Oct 23 '20 at 10:09

So my main question is would the value of $$\frac{dB}{dt}$$ be zero same even though the field is technically moving? If so, how can there be an induced emf in this frame as there is no curl of the electric field
Yes $$\frac{dB}{dt}$$ is 0 since magnetic field is not changing. In the frame of reference where rod is at rest, the observer observed an electric field along the rod. The end result would be same as that in the original frame of reference. An EMF of magnitude $$Bvl$$ is set up in the rod in the same direction in either of the frames. The electric field comes into existence as a consequence of Relativity. Please refer chapter "Electromagnetic induction" of "Electricity and Magnetism" by Edward M. Purcell for more details.