Can anyone explain Jackson's circular current loop formula?

[dingo_d]

Homework Statement

Every time I pick up Jackson I feel dumb :(

So in 5.5 chapter about Circular Current Loop, he says:

$$|\vec{r}-\vec{r}'|=[r^2+r'^2-2rr'(\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos\phi')]^{1/2}$$

So it's cosine rule, but how did he got to that sine/cosine thing? It should be the angle between r and r'. I tried drawing but to no avail :\

help...

 

[sgd37]

in 3d space the vector r has coordinates

$$\bold r = (x,y,z) = (rsin(\theta)cos(\phi),rsin(\theta)sin(\phi),rcos(\theta))$$

and vector r' has coordinates

$$\bold r' = (x',y',z') = (r'sin(\theta ')cos(\phi '),r'sin(\theta ')sin(\phi '),r'cos(\theta '))$$

so basically you use that and the fact that

$$\left|\bold r - \bold r' \right| = \sqrt{ (x-x')^2 +(y-y')^2 +(z-z')^2}$$

and that should be equivalent to what you have

 

[dingo_d]

I thought of sth like that right now in the morning, only I thought of cylindrical coordinate system.

Altho that would get complicated in the z part.

I'll try this. Thank you :)