What is the Magnetic Force on the Loop in this Rigid Contour System?

[Ivan Antunovic]

Homework Statement

Rigid contour of an isosceles triangle and a long line cable lie in a plane and are located in vacuum . Determine the intensity of the magnetic force F acting on the loop. Current I = 100 A.

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Correct answer:
F = 386 mN.
I am getting 2 mN.
Notice that I found dS by splitting while triangle in two triangles.

Homework Equations

The Attempt at a Solution

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[BiGyElLoWhAt]

In your force relation, you have Newtons on the left (Kg*m*s^-2) and (Amper*Wb*t^-1) on the right. Check that out, and see what's up.
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html

 

[Simon Bridge]

How is the derivative in the last line not zero?
Does the magnetic flux through the loop change with time?

In the problem description, the wire loop is not carrying a current.
ds is a line element and dS is a surface element... not always clear which you intend.
ie. The area element between x and x+dx, inside the triangle, is dS=2y(x)dx : y=(x-a)/2
So dS=(x-a)dx ... that what you mean?
Then: $\vec B = \frac{\mu_0 I}{2\pi x}\hat k$ and $d\Phi = \vec B\cdot d\vec S$
... means the flux density $$\Phi = \frac{\mu_0 I}{2\pi}\int_a^{2a}\frac{x-a}{x}\text{d}x$$ ... which is where you ended up.

What about $\vec F= I\oint (d\vec s\times \vec B)$ ?

 

[Ivan Antunovic]

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You mean something like this , still I don't understand how there can be force on the loop if there is no current I flowing through loop.

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By the way I think I messed something up because a didn't cancle out in F.Guess because I didnt do the loop integral the right way

 

[Ivan Antunovic]

In your force relation, you have Newtons on the left (Kg*m*s^-2) and (Amper*Wb*t^-1) on the right. Check that out, and see what's up.
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html

I wrote it incorrectly it should be F= I dFi/ dx

 

[Ivan Antunovic]

Please guys help me I am trying to figure out the whole day how my solution is incorrect, in the solution there is mi zero * I^2 / (2*pi) [1/2 - ln2] and I am having 1 - ln2 in the brackers , where did he get that one half from.

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