0.1. POTENTIAL ENERGY OF A MAGNETIC DIPOLE 217
So fine. We can now see that our classical model proton should havea magnetic moment that is
related to its angular momentum by the simple relation:
~mp=μpL~p (472)wheremup= 2 mep, the magnetic moment of a classical electron should be the same withμe= 2 mee
and so on. This result works adequately in the quantum case as well, as long as we remember to
use the intrinsic spin of the particles in question.
Why do we care? It is because we canusethis result in a clever way by taking advantage of
themotionthat results when we place a proton in a strong magnetic field. The motion, as we shall
see, is aprecessionof the magnetic moment of the proton in a conearoundthe applied magnetic
field that has a precession frequencyωp=μBindependent of the relative angle between the angular
momentum or spin of the proton and the magnetic field.
While precessing in this way, we can easily trick the magnetic dipole moments of the charged
protons toabsorb or emit electromagnetic radiation of the same angular frequency asωp. By
detecting the signal produced by the protons in various clever ways (beyond the scope of this course
to detail, butwithin your capabilities of understandingif you master the next section) we can
measure thedensityof bare protons in almost any substance and create athree dimensional mapof
that density at a remarkably fine resolution.
Protons, of course, are the nuclei ofhydrogen atomsand water is dihydrogen oxide, with two
protons just waiting to be mapped. And what are we? Well, mostly water! The precession of
magnetic moments of protons around strong applied fields is the basis ofmagnetic resonance imaging
(MRI), one of the most important technologies in use in hospitals around the world today. With
MRI one can safely map out soft tissue densities of the human body ina lovely complement to x-rays
(that map out dense tissues but that go right through soft tissuewithout much differentiation). My
wife is a physician, and she orders MRIs on patients on at least a weekly basis, if not a daily one.
Spin resonance is also a very important experimental probe for physicists, as this trick works
for more than “just protons”. Whether you are a potential physics major or engineering student
or a premedical student, you reallymustmaster the next section, then, as it is actually directly
important to your future planned career. To encourage this mastery, I typically tell my students
that a problem on magnetic resonance and precessionwillbe on at least one quiz, hour exam, or on
the final. This is usually enough incentive to motivate them to take thetime to plow through the
complexities of torque as the time rate of change of thevectorangular momentum.
I present this result two distinct ways below – the first suitable for any student, the second perhaps
better for students that have mastered the concept of the cross product in cartesian coordinates. I
strongly suggest thatallstudents at leasttryto master both, but at the very least get to where you
fully understand the first one.
Example 6.3.4: The Precession of Magnetic Moments: Magnetic Reso-
nance
In figure 75 above, you can see a cartoon classical proton in astrongexternal magnetic fieldB~ 0.
The proton (we imagine) is spinning like a little planet – very little indeed given that its radius is
order of 10−^15 meters – and hence has anangular momentumvLpointing in the direction up and
to the right along its axis of rotation. Because its charge ispositive, it has a magnetic moment that
is parallel to its angular momentum, and in the previous section we argued strongly (leaving actual
proof to the student) that its magnetic moment can generally enough be written:
~m=e
2 mp~L=μpL~ (473)wheree= 1. 6 × 10 −^19 Coulombs is its charge,mp= 1. 67 × 10 −^27 kilograms is its mass (in SI units).