Larmor Precession
Larmor Precession
Larmor Precession
When a pure static magnetic field is present and acting on a system of charges, the
Lorentz force of the field on any charge is always normal to the direction of the motion of
the charge, hence the magnetic field is not able to change the energy of any charge. In
other
words, the magnitude of the momentum P is constant. In the nonrelativistic case
dP q
q (v B ) ( p B ) . The direction of the momentum of the charges is changing
dt m
in a plane normal to the magnetic field. In fact the momentum of any charge is
qB
precessing around the direction of the magnetic field by a frequency = . The
m
magnetic moment of a charge in this case will be in the direction of the magnetic field.
The situation becomes more complex when the charges have originally (before applying
the magnetic field) a net magnetic moment in some other direction maybe due to another
source of magnetic field or a permanent spin as in electrons. The applied magnetic field
tends to make the magnetic moment of the charge aligned in its direction while the
angular momentum associated with the magnetic moment of the charge tends to maintain
its original direction. The net effect is that the magnetic moment precesses around the
direction of the magnetic field. The situation is analogous to the motion of a spinning
gyroscope that undergoes a precession under the force of gravity when its rotational axis
initially is not vertical. The magnetic moment of a system of moving charges is defined
1
as : M qi ( ri vi ) . Hence the magnetic moment is related to the angular momentum
2
by the formula: M L , where is the gyromagnetic ratio defined as the ratio of the
magnetic dipole moment to the mechanical angular momentum of a system. For classical
q
systems it equals . In quantum mechanics, the magnetic moment of an electron
2m
due to its spin is related to the spin by a similar relation but the gyromagnetic ratio is
multiplied by a factor slightly larger than 2. Regardless of the value of , the torque
exerted by the magnetic field on the magnetic dipole is expressed as: N M B . Since
dL
this torque is what causes the change in the angular momentum of the dipole , N ,
dt
dL
then L B . That means that the frequency of precession of the magnetic dipole
dt
B .
Figure (1)
Figure (2).
References:
1- http://www.cem.msu.edu/~reusch/VirtualText/Spectrpy/nmr/nmr1.htm
2- http://hyperphysics.phy-astr.gsu.edu
3- Classical Electrodynamics, J.Jackson
4- Fundamental of Physics, Halliday, Resnick, Walker