Problem 23.15

23.15. a)
The magnetic force is given by
F = B I d
Where d is the length of the wire that is perpendicular to both the direction of motion and the magnetic field. So here,
F = BIw
But the magnetic force is equal to the gravitational force when the system is in equillibrium.
IwB = Mg
Now, we can also find two equations for the voltage in the problem. From Ohm's law, we have,
$\epsilon = IR$
and from our equation for motional emf, we have that
$\epsilon = Bwv_{z}$
Combining the two, we have
IR = Bwv_z
When the system is in equillibrium, we have
$v_{t} = \frac{MgR}{B^{2} w^{2}}$

b)
The EMF is directly proportional to v_t but the current is inversely proportional to R. A large R means a small current at a given speed, so the loop must travel faster to get to the equillibrium condition where F_magnetic = F_{gravitational}.

c)
At a given speed, the current is directly proportional to the magnetic field. But the force is proportional to the product of the current and the field. For a small magnetic field, the speed must increase to compensate for both the small B and also the current, so the terminal velocity is proportional to the inverse square of the magnetic field.