Electromagnetism with Lentz's Law

[Taciturn Type]

Homework Statement

A rectangular loop of wire resistance R and dimensions h and w moves with a constant speed v toward and through a region containing a uniform magnetic field of strength B directed into the plane of the page. The region has a width of 2w as shown above. (I think our teacher told us this was a free response question from a Physics C exam, although I don't know what year it's from. Also I didn't include part a because I knew how to solve it.)

(b) On the axes below, plot the following as functions of position x of the right edge of the loop shown above.

i) The induced current I in the loop
ii) The applied force F required to keep the loop moving at a constant speed.

Let the counterclockwise current be positive, clockwise current be negative, forces to the right be positive, and forces to the left be negative. The graphs should begin with the loop in the position shown (x = 0) and continue until the right edge of the loop is a distance 2w to the right of the region containing the field (x = 5w). Sorry I don't have a picture of the graph, I would link a website that has a pdf of the question, but I'm pretty sure that's not allowed.

Homework Equations

The question only asks for relative values of current and force versus time, so it's mostly right-hand rule, but I = V/R = E/R = Blv/R and F_p = F_b = BIlsinθ, and E_avg = ΔBAcosθ/Δt

The Attempt at a Solution

First of all, for both graphs the current and force will be zero from 0w to w and from 4w to 5w because the loop is outside of the field.
For current: From w to 2w, the current is counterclockwise and constant because there's a constant increase in flux (Lentz's law, field points into page). From 2w to 3w, the current is zero again because there's no change in magnetic flux (the entire loop is inside the magnetic field). From 3w to 4w, the current is clockwise and constant (Lentz's Law, field points into the page but magnetic flux is decreasing).
For applied force: From w to 2w it's constant to the right (right-hand rule, field points into the page, current in the right side of the loop travels upward, magnetic force pushes left). From 2w to 3w, there's no applied force because F=BIL and I=0 from 2w to 3w. From 3w to 4w, the force is constant to the right (right-hand rule, field points into the page, current travels upward because the left side of the loop is in the magnetic field).

I've discussed with some classmates and this is the best we could figure it out, but I'm still not sure if what I've written is correct or if the current increases from w to 2w and then remains constant and counterclockwise from 2w to 3w because the current is caused by an average induced emf over the given time interval. Anyways if you made it through the whole post, thanks and I appreciate your help.

 

[anathema]

Dear student,

Your approach to solving this problem is mostly correct. Here are a few suggestions to improve your solution:

1. In the homework equations section, you have mentioned the equations for induced current and applied force in terms of magnetic flux and magnetic field. However, you have not mentioned the equation for magnetic flux, which is Φ = BAcosθ. This equation is essential for solving this problem.

2. In your attempt at a solution, you have correctly identified that the current is zero from 2w to 3w because there is no change in magnetic flux. However, you have not mentioned the reason for the constant counterclockwise current from w to 2w and from 3w to 4w. This is because the magnetic flux is increasing from w to 2w and decreasing from 3w to 4w, as you have correctly stated in the previous sentence.

3. In your attempt at a solution, you have mentioned that the current is constant from w to 2w and from 3w to 4w. However, you have not mentioned the magnitude of this current. The magnitude of the current will depend on the dimensions of the loop, the strength of the magnetic field, and the speed of the loop.

4. In your attempt at a solution, you have mentioned that the applied force is constant to the right from w to 2w and from 3w to 4w. However, you have not mentioned the magnitude of this force. The magnitude of the force will depend on the dimensions of the loop, the strength of the magnetic field, and the speed of the loop.

5. In your attempt at a solution, you have mentioned that the applied force is zero from 2w to 3w. However, this is not entirely correct. The applied force will be zero only if the loop is moving at a constant speed during this time interval. If the loop is accelerating or decelerating during this time interval, there will be a non-zero applied force.

Overall, your understanding of the concepts involved in this problem is good. However, you need to be more thorough and precise in your solution. I would suggest that you revise your solution and include all the necessary equations and explanations to make it more complete. Also, it would be helpful if you could include a graph to show the variation of current and force with position, as this would make it easier to visualize the solution.

I hope this