Magnetic Field Induced Current Problem

[isthisadream]

Homework Statement

http://tinypic.com/r/rlidg7/5

The picture above shows four different cases of a conducting square loop (in orange), traveling at speed v and passing through a rectangular region. In the left half of the rectangular region, the magnetic field is uniform and directed into the screen; in the right half, the magnetic field is uniform and directed out of the screen. The field has the same magnitude in each half of the rectangular region, and there is no magnetic field outside the rectangular region.

Rank the four cases based on the magnitude of the total magnetic flux passing through the loop, from largest to smallest, at the instant shown in the picture.

Rank the four cases based on the magnitude of their induced current, from largest to smallest, at the instant shown in the picture.

What is the direction of the induced current for each case: counter-, clockwise, or none.

Homework Equations

I don't think I will be needing any equations for this problem.

The Attempt at a Solution

I think I got part a down. Cases A and C have zero total magnetic flux due to opposing directions of magnetic field. And Case B has the highest magnitude due to most field lines.

I am really confused for the other two parts. How can there be an induced current at an instant? If it is moving for an instant afterwards, how can I tell what the magnitude of the induced current is?

Additionally for the last part, the direction, I think Case C would have no direction. Case A seems to be getting more out of the page direction, so the opposing field of the induced current should be into the page. Using the right hand rule, would the direction of the induced current be clockwise? Using the same method, I feel that B would also be clockwise and D would be counterclockwise.

Thanks for taking your time to read this and helping me out!

 

[NascentOxygen]

http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif

I think I got part a down. Cases A and C have zero total magnetic flux due to opposing directions of magnetic field. And Case B has the highest magnitude due to most field lines.

I agree.

I am really confused for the other two parts. How can there be an induced current at an instant?

At the moment the photo is taken, what is the current doing.

Additionally for the last part, the direction, I think Case C would have no direction.

I agree.

Case A seems to be getting more out of the page direction, so the opposing field of the induced current should be into the page. Using the right hand rule, would the direction of the induced current be clockwise?

I would say current is CCW, but your explanation is too vague, seems like a guess. :uhh:

 

[isthisadream]

Thanks NascentOxygen for the feedback and welcome.

For the 2nd part, you asked what the current was doing at the moment the picture is taken.

I first thought that if there was no movement, there would be no induced current, so all would be equal to zero. However, since there is a velocity at the instant, I thought that would be false.

For case A, the loop is moving into the out of the page magnetic field. For case B, it is moving into the out of the page field. For case C, the the loop does not change direction of fields or strength. For case D, it is moving out of the out of the page field.

Using the above, case A's induced current is opposing what will happen, so it will be CW. B will also be CW, C will have no direction, and D will be opposing the flux to becoming zero and be CCW.

I feel that I am doing all of this incorrectly. And I have no idea on how to figure out the magnitude ranking of the induced current. Is it looking at how many field lines are through the loop in each case?

The "instant that the picture is taken" is really throwing me off! Any further clarification would be appreciated.

 

[NascentOxygen]

I think this question is tricky. Think of each loop as comprising 4 wires. Each wire is moving through a magnetic field. Some of the wires will have no induced voltage, such as where that wire is not cutting any field lines. Only if the sum of the 4 voltages comes to a non-zero value can we say there is a voltage induced in the loop.

You need to apply one of the right/left hand rules. For example, this right hand generator rule: http://www.diracdelta.co.uk/science/source/f/l/flemings right hand rule/source.html

Remember the rule applies to a straight piece of wire *
and we have 4 straight pieces that make up each loop.

 

[asteriatic]

Hello,

Thank you for your detailed explanation and attempt at solving this problem. Let me provide some clarification and further guidance for you.

Firstly, you are correct in ranking the four cases based on the magnitude of total magnetic flux passing through the loop. Case B has the highest magnitude, followed by Case D, Case A, and then Case C with zero flux.

Moving on to the magnitude of induced current, it is important to note that the induced current is caused by a changing magnetic flux. In this problem, the loop is moving through the magnetic field, causing a change in the magnetic flux passing through it. Therefore, at the instant shown in the picture, we can determine the magnitude of the induced current by considering the rate of change of magnetic flux at that instant. This can be calculated by using the equation: Emf = -N(dΦ/dt), where Emf is the induced electromotive force, N is the number of turns in the loop, and dΦ/dt is the rate of change of magnetic flux. Since the number of turns and the rate of change of magnetic flux are the same for all four cases, we can simply compare the magnitude of the induced electromotive force to determine the magnitude of the induced current. Therefore, the ranking for the magnitude of induced current would be the same as the ranking for the magnitude of total magnetic flux, with Case B having the highest magnitude of induced current.

For the direction of the induced current, you are correct in using the right hand rule. For Case A, the induced current would be in a clockwise direction, as it opposes the increasing magnetic field into the screen. For Case B, the induced current would also be in a clockwise direction, as it opposes the decreasing magnetic field out of the screen. For Case C, there would be no induced current due to the opposing magnetic fields canceling each other out. And for Case D, the induced current would be in a counterclockwise direction, as it opposes the decreasing magnetic field into the screen.

I hope this helps to clarify and guide you in solving this problem. Keep up the good work!