Showing forces between 2 current loops are in accord with Newtons 3rd law

[EngageEngage]

Homework Statement

show that the forces between two current loops obey Newtons 3rd law.

Homework Equations

$$\textbf{F}_{12}=\frac{\mu I_{1}I_{2}}{4 \pi s^{2}} d\textbf{r}_{1}\times(d\textbf{r}_{2}\times \textbf{s})$$
s is the distance between the elements of current

The Attempt at a Solution

So, i have to show that:
$$\textbf{F}_{12}=-\textbf{F}_{21}$$
I will set all the scalar stuff equal to a constant k. i use the BAC CAB rule to get:
$$\textbf{F}_{12}=k(d\textbf{r}_{2}(d\textbf{r}_{1}\cdot\textbf{s})-\textbf{s}(d\textbf{r}_{1}\cdot d\textbf{r}_{2}))$$

I set up my coordinate system so that my first loop lies at x = 0, and the other at x = 1, with equal radii, r = 1, which gives me the following equations:
$$d\textbf{r}_{1}=(-sin(\theta),cos(\theta),0)d\theta$$
$$d\textbf{r}_{2}=(-sin(\phi),cos(phi),0)d\phi$$
$$\textbf{s}=\frac{\textbf{r}_1-\textbf{r}_{2}}{||\textbf{r}_1-\textbf{r}_{2}||}=\frac{(cos(\theta)-cos(\phi),sin(\theta)-sin(\phi),-1)}{||(cos(\theta)-cos(\phi),sin(\theta)-sin(\phi),-1)||}$$
Is this right? This will allow me to first integrate around the first loop and then the second. When I do the resulting integral i end up getting 0. Could someone please tell me why I am doing this incorrectly? Any help is greatly appreciated

 

[Dick]

I don't think you have to integrate anything. Can't you just show that if you interchange 1<->2 (which means also changing s->-s) that the sign of the integrand reverses?