Torque from straight wire segment

[DrIxn]

Homework Statement

A straight wire segment of length L makes an angle of 21.0 degrees with respect to the x axis. The wire carries a current of 4.10 A in the direction shown in the figure. There is a magnetic field in the vicinity of the wire which points in the negative z direction with a magnitude given by B=B0(x/L)^3, where B0=3.80 T and L=4.91 m.

What is the force in the y direction from the magnetic field?

If the wire can rotate freely about the z axis what is the magnitude of the torque?

Homework Equations

dF=Idl x B

Torque= r x F

The Attempt at a Solution

I found the magnetic force using the first equation

dF=IdL x B and since B is with respect to x i simplified the cross product to dx

dF=Icos(theta)dx B

And plugged in the equation for B and integrated from 0 to L*cos(theta)

Now for the torque I thought back and if you have the wire laying across the x-axis or at least the xz plane and applied a force in the positive y direction you would be rotating about the z axis so i did r x F with the F from the previous answer and r= L*cos(theta). Is that incorrect?

 

[rude man]

You didn't show the drawing so I'll assume the wire runs from, and pivots at, the origin.

You haven't shown enough math detail for me to trace your steps so I'll just give you my way of approaching the problem.

If you write out the vectors in vector notation it makes things easier (at least for me):

dF = I dl x B
dl = dx i + dy j
B = -B0 (x/L)^3 k
dτ = dl x F,

all vectors in bold.

Can you write dy = (constant) dx to eliminate dy?

Then you can formally use vector math from then on. All integrations will be with respect to x only.

 

[DrIxn]

I tried it your way(dont know how to make the vectors bold):

tan(theta)=dy/dx

I used dl= dx i + tan(theta) dx j
So when I took the cross product i got dF=-(I*B0*tan(theta)/(4*L^3))*x^3 dx i + (I*B0/(4*L^3))*x^3 dx j

And Integrated that from 0 to L*cos(theta) so
F= -(I*B0*tan(theta)*L*cos^4(theta)/4) i + (I*B0*L*cos^4(theta)/4) j

this is the same answer that I got (though I only calculated the y component)
To simplify writing it the Fi(magnitude)=5.576 N for the i component and Fj=14.53 N for the j component
Now for the torque:
dl= dx i + tan(theta) dx j

F=Fi i + Fj j

So crossing those two you end up with just the k vector

(Fj dx + Fi tan(theta) dx) k
Went ahead and used the fact that Fi is negative here

So again integrate (Fj+Fi*tan(theta) dx from 0 to L*cos(theta)
= Fj+Fi*tan(theta)*L*cos(theta)= L*(Fj*cos(theta)+Fi*sin(theta))

Numerically I got 76.41 N*m, still not the right answer

 

[rude man]

I tried it your way(dont know how to make the vectors bold):

Select "Go Advanced" & use the "B" on the toolbar.

I need to leave for about 1 hr but will try to follow your work which basically looks right.

 

[rude man]

Your F is the same as mine.
Let a = tan(dy/dx) = tan(theta) where theta = 21 deg.

Define F = F_x i + F_y j. Note that F_x < 0, we both got that.

I question how you obtained torque τ. Did you perform

dτ = (dx i + a dx j) x (F_y j + F_x i) correctly? Did you wind up with

dτ = (F_ydx - aF_xdy) k ?

You had a sin(theta) term in there & I didn't.

 

[DrIxn]

Thanks for your help rude man, I ended up crossing r x dF to get the answer, not dr x F

 

[rude man]

Thanks for your help rude man, I ended up crossing r x dF to get the answer, not dr x F

You are right. I wish I could say that was a typo on my part but it wasn't.

Differential amount of force along the wire is dF = I dl x B and
dτ = l x dF.

Good catch!