Question

A circular loop of radius r is moved with a velocity v the force needed to maintain its velocity constant is

Answer 1

Well if we take B and v as constants here then it is pretty easy, because all you need to do is calculate the change of the magnetic flux through the surface enclosed by either loops, and you know that the induced EMF (ϵ) is given by:

ϵ=−NdΦdt

where Φ=B⋅A, since we started off with B constant then,

dΦdt∝dAdt

In the case of the rectangular loop, the amount by which the surface changes as the loop moves out, is constant, so:

dArectdt=constant↔ϵ=constant

Whereas the change of magnetic flux in the ciruclar loop isn't constant since the surface of the circle doesn't change by equal amounts as the loop moves out:

dAcircdt≠constant