# Loop falling in a magnetic field

#### Jenny Physics

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.
Homework Equations
Use Faradays' law and Lorentz force on a current

(a) Lets 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.

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#### TSny

Homework Helper
Gold Member
(a) Lets 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.
Why does positive $\mathcal{E}$ imply clockwise current? I think a better approach for this question is to use Lenz's law.

(d) It appears that I have both the gravitational force and the magnetic force downwards?
This might indicate that you got the wrong direction for the current.

(e) Not sure where the potential energy goes since it is constant speed motion.
Does current in a circuit with resistance produce any form of energy?