I have a question about force exerted by magnetic field

[gaus12777]

Homework Statement

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]

 

[Charles Link]

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? $\vec{F}=- \nabla U$.

 

[gaus12777]

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.

(magnetic moment)

Hence, U

F

Is it correct?

 

[Charles Link]

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 $x$.

 

[gaus12777]

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?

 

[Charles Link]

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.

 

[gaus12777]

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