0.1. POTENTIAL ENERGY OF A MAGNETIC DIPOLE 219
x
y
∆φL
∆L
z(out)
Figure 76: The torque causesL⊥to precess around thez-axis (out of the page as drawn).L⊥moves
in a circle, and in a short time ∆tit moves through an angle ∆φand hence changes the (vector)
angular momentum by ∆L~as shown.
or (dividing both sides by ∆t):
∆L
∆t
=L⊥
∆φ
∆t(478)
This mustalsoequal the torque in terms of the field, in the limit that we let ∆t→dt:
τ=dL
dt=μpL⊥B 0 =L⊥dφ
dt=L⊥ωp (479)where we usedL⊥=Lsin(θ) andωp=dφdt(the angular precession frequency). Solving for the latter,
we find:
ωp=μB 0 (480)
independent of the angleθbetween~mandB~ 0!
Note well that this derivation, while correct enough for the moment, doesn’t directly result in
equations of motion for the individual components of the angular momentum. It is easy enough,
however, to write down the three (coupled) equations of motion forLx, Ly, Lzusing the cartesian
form for the cross product. One of these is trivial as there is no torque in the direction ofB~=B 0 zˆ.
The other two first order coupled differential equations becomesecond order equationsfor the
oscillatorymotion ofLxandLyseparately. The solutions, however, are not independent, as the
phaseof one is determined by the phase of the other. Indeed, the solution describes~L⊥tracing out
an explicit circle at the precession frequency.
Instead of covering this solution in the text, this is left as an optional exercise for the interested
(non-major) student or a required problem for physics/engineering/math majors in the homework.
The math for this, note well, is very similar to the math used to derive the wave equation for
the electric and magnetic field components from Maxwell’s equations ina few chapters, so it isn’t
completely crazy to give this a try now even if you don’t “have” to to make it easier on yourself
then!
6.4: Spin Echoes and Magnetic Resonance Imaging
One of the primary reasons for many students to take a course in electricity and magnetism is to
learn enough about how magnetic fields and moments work that theycan understandMagnetic
Resonance Imaging (MRI). MRI is one of the most important non-invasive diagnostic tools