Question

A conducting rectangular loop of mass, M, resistance R, and dimensions w x l falls from rest into a magnetic field B. During the time interval before the top edge of the loop reaches the field, the loop approaches a terminal speed v_t ​. Show that v_t=MgR/B²w²

Answer 1

As the rectangular loop is moving from rest in to the magnetic field  

the area of the loop , changing in the magnetic field so that there will be change in magnetic flux through the loop so that from Faraday's law the induced emf , associated magnetic field develops in the loop

let the magnetic field (original) out of the page then the induced field is into the page and  

when ever the loop attains the terminal velocity the force acting on the loop is zero

that is the magnetic force and gravitational forces are equal

given mass M , dimensions are W X l  

field B

terminal velocity vt

flux is phi _B = B*A cos theta = B*l*W cos theta

from Faraday's law e = -d(phi)/dt

e = - d(B*l*W cos theta)/dt

e = B*W dl/dt

e = B*W vt

from Ohm's law V = I*R

I = V/R

I = B*W*vt /R

the force is F = IWB sin theta

F = (B*W*vt /R )(W*B)

F = B^2*W^2*vt /R

the gravitational force is F = mg

mg = B^2*W^2*vt /R

vt = mg*R /(B^2*W^2)

so the terminal velocity is vt = mg*R /(B^2*W^2)