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- In Shadowitz's book The Electromagetic Field, Section 3-4, there is a very interesting derivation of Ampere's Law. Need a bit of elaboration for my understanding of the use of Solid Angle in Torus.

In Shadowitz's book The Electromagetic Field, Section 3-4, p. 129 to 134, there is a very interesting derivation of Ampere's Law. It is a general derivation with a circular source circuit S, and a circular test circuit T. The pages can be seen here, at least from where I post (USA):

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

(R

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

### The Electromagnetic Field

This outstanding volume, designed for junior and senior undergraduates in physics or electrical engineering, is an unusually comprehensive treatment of the subject. The book begins with the basis of electric and magnetic fields and builds up to electromagnetic theory, followed by a number of...

books.google.com

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