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**1. Homework Statement**

A long straight wire carrying a constant current I

_{1}and a circular wire loop carrying a constant current I

_{2}lie in a plane. The radius of the loop is R, and its center is located at distance D from the straight wire. What is the magnetic force exerted on the loop by the straight wire?

**2. Homework Equations**

The currents are constant, so ##\overrightarrow{F}_{m}=I \int(d\overrightarrow{l}\times\overrightarrow{B})##

For an infinite wire, ##\overrightarrow{B}=\frac{\mu_{0}I}{2\pi s}\hat\phi##

(cylindrical coordinates)

**3. The Attempt at a Solution**

So I set up the coordinate system like this.

##I=I_{2}## and ##B=B_{1}##, since the magnetic force due to the line's field should be on the current loop.

I thought that the spherical coordinate system would be the easiest to use for this problem. In that case:

##\theta=\frac{\pi}{2}##

##z=0##

##s=y=R sin\phi## (by symmetry)

##dr=0## (R is constant for the loop)

##d\overrightarrow{l}=Rd\theta\hat\theta+Rd\phi\hat\phi##

Therefore, above the wire ##B_{1}## is in the ##\hat\phi##→##\hat z=-\hat\theta## direction. In addition, the origin is displaced by length D, so the equations become:

##\overrightarrow{F}_{m}=I_{2}\int(d\overrightarrow{l}\times\overrightarrow{B}_{1})##

##\overrightarrow{B}_{1}=-\frac{\mu_{0}I_{1}}{2\pi(R sin\phi +D)}\hat\theta##

So the cross product says that ##\overrightarrow{F}_{m}## is only in the ##\hat r## direction.

##\overrightarrow{F}_{m}=\frac{\mu_{0}I_{1}I_{2}R}{2\pi}\int_0^{2\pi} \frac{d\phi}{R sin\phi +D}\hat r=\frac{\mu_{0}I_{1}I_{2}R}{\sqrt{D^{2}-R^{2}}}\hat r##

Is this the correct approach and result? I'm not sure if my replacement for s in the equation for ##\overrightarrow{B}_{1}## is valid, or if I set up the coordinate system in the appropriate way. I believe my answer has the correct dimensionality, I just want to verify the approach. Thanks for your help!