Net Force on a circular current carrying wire, from an infinite wire.

[bluecadetthree]

Homework Statement

-I've attached a picture of the problem-

An infinitely long straight wire of steady current I₁ is placed to the left of a circular wire of current I₂ and radius a as shown. The center of the circular wire is distance d(≥ a) away from the straight wire. Let’s find the net magnetic (Lorentz) force acting on the entire circular wire as follows:

(A) Express the magnetic field B_p at point P (due to the
current I₁) in given quantities (including its direction).

(B) The Lorentz force due to the magnetic field Bp acting on a small current segment I₂dℓ at P is given by

dF = I₂dℓ × B_p​

Express dF = (dF_x, dF_y, dF_z) in component representation in given quantities.

(C) By integrating your results from (B) show explicitly that the net Lorentz force for the
entire circular wire is given by

Fnet = µ_oI₁I₂[itex]\left(1-[itex]\frac{d}{\sqrt{d²-a²}}[/itex]\right)[itex][itex]\hat{y}[/itex]​

Homework Equations

Most given in question.

B_p=[itex]\frac{-μ₀I₁}{2π(d+x)}[/itex]​

The Attempt at a Solution

The first two parts I got through pretty easily. Part (A) was just giving the equation of the B-field for an infinite wire. For part (B) I ended up with:

dF=[itex]\left(\frac{-μ₀I₁I₂acosødø}{2(d+acosø)}\right)[/itex]$\hat{x}$+[itex]\left(\frac{-μ₀I₁I₂asinødø}{2(d+acosø)}\right)[/itex]$\hat{y}$​

which I got from taking the cross product of I₂dl and B_p, and am fairly certain is correct.

Now, for part (C) I keep getting stuck, I see what I'm supposed to get for F_net, but my answer always has a natural log, or becomes zero, and I don't know how else to approach the problem.

 

[TSny]

Hello bluecadeetthree. Welcome to PF!

For part (B) I ended up with:

dF=$\left(\frac{-μ_0I_1I_2a\cos\phi d\phi}{2(d+a\cos\phi)}\right)$$\hat{x}$+$\left(\frac{-μ_0I_1I_2a\sin\phi d\phi}{2(d+a\cos\phi)}\right)$$\hat{y}$​

which I got from taking the cross product of I₂dl and B_p, and am fairly certain is correct.

Yes, I think that's correct. (I tried to fix the latex, hope I got it the way you intended.)

Now, for part (C) I keep getting stuck, I see what I'm supposed to get for F_net, but my answer always has a natural log, or becomes zero, and I don't know how else to approach the problem.

The integration of the y-component is straightforward. (The answer for this component should also be evident by symmetry.) The x-integration seems more difficult. I threw it into Mathematica and it popped out the answer. I also managed to do it by converting it to a contour integral in the complex plane. I'm not seeing an elementary way to do it, but I'm probably just not seeing it. I spent a few minutes searching integral tables on the internet but did not find it.

The answer for the net force that you stated in part (C) appears to have the wrong direction.

 

[bluecadetthree]

I was thinking the given net force had the wrong direction. I'll have to email my professor about that. Thanks for your help!