Magnetic Moment Definition Verification/ Proof

[cubejunkies]

I saw the equation here http://en.wikipedia.org/wiki/Magnetic_moment#Current_loop_definition for the definition of the magnetic moment for a non-planar loop. Can someone tell me if there's a name for this equation $m= \frac { I }{ 2 } \int { \overrightarrow { r } } \times d\overrightarrow { r }$, if it's even right, and where I could find a derivation and/or proof of the equation? I've been fumbling over this for several hours now and I've gotten absolutely nowhere. I don't have any access to any texts on the matter, all I have is google and wikipedia at the moment.

Thanks

Anthony

 

[Jano L.]

It is really just a convenient definition. It assigns to some portion space (where the currents are) a quantity called magnetic moment. The motivation is, we want to find some approximate expression for the magnetic field due to these currents, and it turns out that in large distances, the magnetic moment is sufficient to find it. Try to get and look into D.J. Griffiths, Introduction to electrodynamics, sec. 5.4.3 or J.D. Jackson: Classical Electrodynamics, you should find the details there.

 

[Meir Achuz]

It follows from
$${\bf m}=\frac{1}{2}\int{\bf r\times j}d^3r$$
with the substitution $${\bf j}d^3r\rightarrow I{\bf dr}.$$
The derivation of the j equation takes about one page in an EM textbook.
It is on J. Franklin, "Classical Electromagnetism" on page 212.
Your equation is derived directly on page 210.
You should also know that $$\frac{1}{2}\int{\bf r\times dr}=$$
the area of the loop.

 

[andrien]

see the page 185 from here,I do not want to copy anything from that.so you can see here
http://books.google.co.in/books?id=8qHCZjJHRUgC&pg=PA794&dq=jackson+electrodynamics+page+187&hl=en#v=onepage&q=jackson%20electrodynamics%20page%20187&f=false