Jenny Physics
- 111
- 4
- Homework Statement
- The rectangular loop of wire in the figure is placed so that all but its bottom segment are in a uniform magnetic field ##B_{0}## perpendicular to the plane of the loop. The wire falls under the influence of gravity. The loop has resistance ##R## and mass ##M##.
(a) As the wire falls will the induced current flow clockwise or counterclockwise?
(b) What is the induced current when the velocity of the loop is ##v##?
(c) What are the forces on each segment of the loop when it has fallen a distance ##y## from the initial position?
(d) If the initial vertical velocity of the loop is ##v_{0}## the total force on the loop vanishes as long as ##y<h##. Find ##v_{0}##.
(e) If the wire begins with velocity ##v_{0}## after falling a distance ##h## the loop gravitational energy decreases by ##Mgh##. Where did this energy go? Show that that source gets exactly ##Mgh## so that energy is conserved.
- Relevant Equations
- Use Faradays' law and Lorentz force on a current
(a) Let's say the loop has fallen ##y## from its initial position. Then the magnetic flux is ##B_{0}w(h-y)## and the induced voltage is ##\mathcal{E}=B_{0}wdy/dt##. Since this voltage is positive, the current flows clockwise.
(b) ##I=\frac{\mathcal{E}}{R}=\frac{B_{0}wv}{R}##
(c) The force on the bottom segment is zero since it is outside the magnetic field.
The force on the top segment is ##IB_{0}w\hat{x}\times \hat{z}=-IB_{0}\hat{y}## (downwards)
The force on the left segment is ##IB_{0}(h-y)\hat{y}\times\hat{z}=IB_{0}(h-y)\hat{x}## (to the right)
The force on the right segment is ##-IB_{0}(h-y)\hat{x}## (to the left)
(d) It appears that I have both the gravitational force and the magnetic force downwards?
(e) Not sure where the potential energy goes since it is constant speed motion.
Comments?