- #1

bryanso

- 28

- 7

https://books.google.com/books?id=k7XCAgAAQBAJ&pg=PA129&lpg=PA129#v=onepage&q&f=false

Briefly, starting from Biot-Savart Law, the circulation Integrate(B . dr

_{t}) is calculated. At one point (dr

_{s}x R

_{st}. dr

_{t}) / R

^{2}is replaced with the equivalent (dr

_{t}x dr

_{s}. R

_{st}) / R

^{2}. This the author turns first into (dS . R

_{ts}) / R

^{2}then finally he equates this with dOmega the solid angle (bottom of p. 131). I am in agreement with this algebra.

(R

_{st}and R

_{ts}represent unit vectors)

Then comes a really interesting change of point-of-view. He explains if we hold P fixed (P is a point on the test circuit T) instead of moving P around T, then effectively it is the entire circuit S moving in space. And the complete movement is a torus. Figure 3-16.

If S and T are linked (kind of like the magician's linking rings :) then the point P lies within the "torus".

He then equates Integrate(dOmega) to 4pi.

Now I am not able to convince myself or prove a solid angle inside a torus sum to 4pi. All online help I can find are integration of solid angles inside a sphere or a potato :) A torus is not the same because a cone from P will eventually go out of the torus and comes back into the opposite arm of the torus.

It's funny the author writes "This is also left for the student to prove..."

Thanks