I have a question about force exerted by magnetic field

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Homework Statement


W.college physics problem.jpg


Homework Equations




The Attempt at a Solution


I uploaded the problem which I want to solve. Getting a general expression for magnetic field created by plane is difficult, I solve it another method.
X is still larger than W, we can think palne current as just a infinite wire.
Furthermore, because of large X, magnetic field of z-direction around z=0 is considered as a constant. (magnetic field has x,y dependence and x is still larger than w. So, of course, magnetic field affected by x coordinate is dominant.)
Consequently, z-dependence will be vanish and gradient potential has no z dependence.
So, a force has a just x direction component.
Is it right? [/B]
 

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  • #2
Charles Link
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I think the problem wants you to quantify the result. You need to determine the magnitude and direction of the magnetic field ## B ## as a function of distance from the wire to the loop. You also need to determine the magnitude of the magnetic moment ## m ## of the loop as well as the direction. Can you write an expression of the energy ## U ## of the magnetic moment in the magnetic field? ## \vec{F}=- \nabla U ##.
 
  • #3
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I think the problem wants you to quantify the result. You need to determine the magnitude and direction of the magnetic field ## B ## as a function of distance from the wire to the loop. You also need to determine the magnitude of the magnetic moment ## m ## of the loop as well as the direction. Can you write an expression of the energy ## U ## of the magnetic moment in the magnetic field? ## \vec{F}=- \nabla U ##.

First, thank you for your reply.
As you have said, problem wants quantify solution. But, I want to just know my logic is correct.
For getting more help I`ll calculate based on my logic.
If X is much larger than W, as I mentioned before, z dependence of the magnetic field and gradient about z can be negligible because of large X( dominant effect).
And there is no y dependence because of current distribution.
So, If I want to caculate force, we have to know magnetic field only about x coordinate.


upload_2016-10-22_11-18-23.gif
(magnetic moment)






upload_2016-10-22_11-37-13.gif


Hence, U



upload_2016-10-22_11-36-42.gif



F

upload_2016-10-22_11-39-0.gif



Is it correct?
 

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  • #4
Charles Link
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Looks somewhat good. The magnetic field with the natural log could then be expanded in a Taylor series. Qualitatively, you can work out the direction of the result. Does ## m ## point in the same direction as ## B ## ? ## U=-m \cdot B ##. Suggest you check the sign on ## U ##. The system will tend to go to a state of lower energy. If ## U ## is positive, this will mean the force will push the loop to a direction where ## B ## is lower in amplitude. Also, I might point out that this one has a coordinate system with the x coordinate set up in such a manner that it makes it difficult to use formulas like ## F=- \nabla U ## and get the sign correct. For this formula to work, x needs to be the position of the loop. That is essentially given by ## X ## and not ## x ##.
 
  • #5
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Looks somewhat good. The magnetic field with the natural log could then be expanded in a Taylor series. Qualitatively, you can work out the direction of the result. Does ## m ## point in the same direction as ## B ## ? ## U=-m \cdot B ##. Suggest you check the sign on ## U ##. The system will tend to go to a state of lower energy. If ## U ## is positive, this will mean the force will push the loop to a direction where ## B ## is lower. Also, I might point out that this one has a coordinate system with the x coordinate set up in such a manner that it makes it difficult to use formulas like ## F=- \nabla U ## and get the sign correct. For this formula to work, x needs to be the position of the loop. That is essentially given by ## X ## and not ## x ##.


Thank you for your help. But I have a question.
You said that "Looks somewhat good". Is it mean that not only my calculation but also assumption(neglect z coordinate) is good?
 
  • #6
Charles Link
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Thank you for your help. But I have a question.
You said that "Looks somewhat good". Is it mean that not only my calculation but also assumption(neglect z coordinate) is good?
Most of it, including neglecting the z-coordinate is correct. The ## B ## and ## m ## point in opposite directions though, if I'm not mistaken. Please verify this. This means ## U ## is positive. editing... Also expand ## ln(1+u)=u ## for small ## u ## will make your algebra/gradient operation simpler.
 
  • #7
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Most of it, including neglecting the z-coordinate is correct. The ## B ## and ## m ## point in opposite directions though, if I'm not mistaken. Please verify this. This means ## U ## is positive. editing... Also expand ## ln(1+u)=u ## for small ## u ## will make your algebra/gradient operation simpler.

Thank you for your help.
I missed out - sigh
 
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