f/A proof of Ampere’s law.`
and DA. Ampere’s law gives`Γ=4 πk
c^2
It hr ough(B)(length of AB) =4 πk
c^2
(η)(length of AB)B=4 πkη
c^2
=4 πk NI
c^2 `self-check D
What direction is the current in figure e? .Answer, p. 1060
self-check E
Based on how`entered into the derivation in example 13, how should
it be interpreted? Is it the total length of the wire? .Answer, p. 1060
self-check F
Surprisingly, we never needed to know the radius of the solenoid in
example 13. Why is it physically plausible that the answer would be
independent of the radius? .Answer, p. 1061
Example 13 shows how much easier it can sometimes be to cal-
culate a field using Amp`ere’s law rather than the approaches de-
veloped previously in this chapter. However, if we hadn’t already
known something about the field, we wouldn’t have been able to
get started. In situations that lack symmetry, Amp`ere’s law may
make things harder, not easier. Anyhow, we will have no choice in
nonstatic cases, where Amp`ere’s law is true, and static equations
like the Biot-Savart law are false.11.3.2 A quick and dirty proof
Here’s an informal sketch for a proof of Amp`ere’s law, with no
pretensions to rigor. Even if you don’t care much for proofs, it would
be a good idea to read it, because it will help to build your ability
to visualize how Amp`ere’s law works.
First we establish by a direct computation (homework problem
26) that Amp`ere’s law holds for the geometry shown in figure f/1,
a circular Amp`erian surface with a wire passing perpendicularly
through its center. If we then alter the surface as in figure f/2,
Amp`ere’s law still works, because the straight segments, being per-
pendicular to the field, don’t contribute to the circulation, and the
new arc makes the same contribution to the circulation as the old
one it replaced, because the weaker field is compensated for by the
greater length of the arc. It is clear that by a series of such modifi-
cations, we could mold the surface into any shape, f/3.
Next we prove Amp`ere’s law in the case shown in figure f/4:
a small, square Amp`erian surface subject to the field of a distant702 Chapter 11 Electromagnetism