Simple Nature - Light and Matter

self-check EBased on how l entered into the derivation in example 13, how shouldit be interpreted? Is it the total length of the wire? ⊲ Answer, p. 927self-check FSurprisingly, we never needed to know the radius of the solenoid inexample 13. Why is it physically plausible that the answer would beindependent of the radius? ⊲ Answer, p. 927Example 13 shows how much easier it can sometimes be to calculatea field using Ampère’s law rather than the approaches developedpreviously in this chapter. However, if we hadn’t alreadyknown something about the field, we wouldn’t have been able toget started. In situations that lack symmetry, Ampère’s law maymake things harder, not easier. Anyhow, we will have no choice innonstatic cases, where Ampère’s law is true, and static equationslike the Biot-Savart law are false.11.3.2 A quick and dirty proofHere’s an informal sketch for a proof of Ampère’s law, with nopretensions to rigor. Even if you don’t care much for proofs, it wouldbe a good idea to read it, because it will help to build your abilityto visualize how Ampère’s law works.First we establish by a direct computation (homework problem26) that Ampère’s law holds for the geometry shown in figure f/1,a circular Ampèrian surface with a wire passing perpendicularlythrough its center. If we then alter the surface as in figure f/2,Ampère’s law still works, because the straight segments, being perpendicularto the field, don’t contribute to the circulation, and thenew arc makes the same contribution to the circulation as the oldone it replaced, because the weaker field is compensated for by thegreater length of the arc. It is clear that by a series of such modifications,we could mold the surface into any shape, f/3.Next we prove Ampère’s law in the case shown in figure f/4:a small, square Ampèrian surface subject to the field of a distantsquare dipole. This part of the proof can be most easily accomplishedby the methods of section 11.4. It should, for example, beplausible in the case illustrated here. The field on the left edge isstronger than the field on the right, so the overall contribution ofthese two edges to the circulation is slightly counterclockwise. However,the field is not quite perpendicular to the top and bottomsedges, so they both make small clockwise contributions. The clockwiseand counterclockwise parts of the circulation end up cancelingeach other out. Once Ampère’s law is established for a square surfacelike f/4, it follows that it is true for an irregular surface like f/5,since we can build such a shape out of squares, and the circulationsare additive when we paste the surfaces together this way.By pasting a square dipole onto the wire, f/6, like a flag attachedf / A proof of Ampère’s law.Section 11.3 Magnetic Fields by Ampère’s Law 675