Simple Nature - Light and Matter

Field at the center of a circular loop example 10

.What is the magnetic field at the center of a circular current loop

of radiusa, which carries a currentI?

.This is like example 9, but with the outer loop being very large,

and therefore too distant to make a significant field at the center.

Taking the limit of that result asbapproaches infinity, we have

Bz=

2 πk I

c^2 a

Comparing the results of examples 9 and 10, we see that the

directions of the fields are both out of the page. In example 9, the

outer loop has a current in the opposite direction, so it contributes

a field that is into the page. This, however, is weaker than the field

due to the inner loop, which dominates because it is less distant.

11.2.4 The g factor (optional)
In section 11.2.3 we exploited a particular trick for superimpos-
ing dipoles consisting of small square current loops. Let’s now turn
to a somewhat different way of superimposing dipoles. The idea is
that matter is made out of atoms, which may act like little magnetic
dipoles, but atoms are themselves made out of subatomic particles
such as electrons, neutrons and protons — and there is no obvious
way that we can ever know whether we have taken this process of re-
ductionism (p. 18) to its conclusion. We can, however, look for clues
in the electrical and mechanical properties of matter. Suppose that
a particle of chargeqand massmis whizzing around and around
some closed path. We don’t even care whether the trajectory is a
square or a circle, an orbit or a random wiggle. But let’s say for
convenience that it’s a planar shape. The magnetic dipole moment
(averaged over time) ism=IA. But the angular momentum of a
unit mass can also be interpreted (sec. 4.1.2, p. 256) as twice the
area it sweeps out per unit time. Aside from the factor of two,
which is just a historical glitch in the definitions, this mathematical
analogy is exact: mass is to charge as angular momentumLis to
magnetic dipole momentm. Therefore we have the identity
q
m

·

|L|

|m|

= 2

(wheremis the dipole moment, whilemis the mass). The left-hand
side is called thegfactor. We expectg= 2 for a single orbiting
particle.
Now suppose that we have a collection of particles with identical
values ofq/m(or a continuous distribution of charge and mass in
which the ratio of the charge and mass densities is constant). Then
vector addition of theLandmvalues gives the sameg= 2 for the
system as a whole. On the other hand, if the different members of

Section 11.2 Magnetic fields by superposition 695