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Lecture 5 – Appendix A – Motion in E and B fields (Chapter 5 in K&B)
Our considerations of rotating frames of reference in Lecture 5 are particularly useful
when considering the motion of charge particles in magnetic fields. Consider first
motion in a uniform, time independent magnetic field. The equation of motion is
d q
mr qr B r B r
dt m
(A. 1 )
The second equation has a form suggestive of Eq. (5.14), where we identify the
angular velocity of the effective rotating frame has qB m
. The magnitude of
this angular velocity is called the cyclotron frequency, c
qB m
. For simplicity we
choose the z - axis along the magnetic field and write
d q
mr r B r Bz r
dt m
z
d q
Bz
dt m
(A. 2 )
Thus motion in a uniform, time independent magnetic field yields uniform motion
along the field plus rotating motion in the plane perpendicular to the field. The
general motion is along a helix. If there is initially no motion perpendicular to the
magnetic field, the only motion is parallel to the magnetic field ( i.e ., along the field
lines). If there is initially no motion along the field, the motion is only in a circle in
the plane perpendicular to the field. The radius of the circle for motion with initial
velocity v 0
is given by
0 0
c
c
v mv
qB
(A. 3 )
This description of cyclotron motion in a magnetic field is relevant to particle
accelerators until the motion becomes relativistic. In the early cyclotrons the
magnetic field was constant and split into 2 regions, the “D”s. In between the D’s an
electric field was applied to accelerate the particles. Clearly from Eq. (A. 3 ) the radius
of the orbit increases as the energy increases. For a given physical size for the “D”s
there was a maximum energy possible. In more modern synchrotrons the magnetic
field is restricted to the region of the vacuum pipe, which is typically in the shape of a
circle, and the magnetic field strength increases with the beam energy (in a
synchronous fashion) to keep the beam moving in the vacuum pipe. It is still the case
that the actual acceleration takes place (typically in the electric field in a microwave
cavity) at only one point around the circumference of the “ring”.
Consider now the case when both electric and magnetic fields are present in the same
region of space. The resulting Lorentz force takes the form
F q E r B.
(A. 4 )
If the electric and magnetic fields are parallel we will just see the circles above in the
xy - plane and uniform acceleration in the z direction. If the electric and magnetic
fields are aligned orthogonal to each other, we have, for example,
E Ey B Bz
mx qBy
my qE qBx
mz
(A. 5 )
These equations of motion can clearly be simplifying by transforming to a new
inertial frame moving with velocity E / B relative to the original one in the x direction.
We find
Now transform to a rotating frame, rotating with angular velocity
qB 2 m
.
From the results in Lecture 5 we have
2 2
inertial inertial rotating
2
rotating
2
rotating
rotating rotating
2
rotating
k k
mr r qr B r q r r t B
r r
k
r qB r r t
r
k
r m r r t
r
m r m r m r t
k m
r r
r
2
2
r t
k m q
r B B r t
r m
(A. 9 )
So in this choice for the rotating frame the velocity dependent term has cancelled and
we have only the initial 1/r potential and the effective centripetal term. Our final
simplification comes from defining what we mean by a “weak” magnetic field. If we
have
2
2 2
0
3 3
0
,
2 4
qB k qq
m mr mr
(A. 10 )
where ω 0
is the angular frequency of the orbit in the 1/ r potential. Recall from Eq.
(4.23) that
2
2
0 3
2
.
k
ma
(A. 11 )
In this weak field limit we can ignore the second term on the right hand side of Eq.
(A. 9 ) and any bound orbits, as viewed in the rotating frame, will be just the familiar
elliptical trajectory. The plane of this orbit will, in general be inclined ( i.e ., not
orthogonal to the magnetic field). In the original inertial reference frame this plane
will be observed to precess about the magnetic field at the very slow angular rate
defined above, i.e ., much more slowly than the angular motion in the orbit. This
motion is call the Larmor effect with precession frequency (Larmor frequency)
0
c
L
qB
m
(A. 12 )
Thus there will be only a small shift in the plane of the orbit during each orbit. The
corresponding QM shift in atomic Bohr energy levels in the presence of a magnetic
field leads to shifts in atomic spectral lines known as the Zeeman effect.
Another way to think about the Larmor effect is to think about the angular
momentum corresponding to the particle in the orbit. The weak magnetic field yields
a (weak) torque so that
d
J r F qr r B q r B r r r B
dt
(A. 13 )
In the last step we used our knowledge of the triple vector product. From our earlier
discussion we know that in the weak magnetic field limit the change of the angular
momentum is tiny per orbit of the particle and the only relevant variation is the
change averaged over a full period of the orbital motion. This is a typical approach in
periodic systems characterized by a fast motion and a slow motion. We are only
interested in the slow motion averaged over a period of the fast motion. The fast
motion will smoothly adapt to the perturbation due to the slow motion. (Note also
that in this case, if the orbit is circular, the second term in Eq. (A. 13 ) will vanish,
r r 0
.) To a good approximation in the current limit (Eq. (A. 12 )) we can write
av
av
L L
d
J r F qr r B
dt
q q
J B J B
m m
(A. 14 )
The discussion in K&B works out this form in detail for a specific choice of axes, but
the form should be clear by the “what else can it be” rule. If we average over the
orbital motion, the direction of the resulting vector can depend only on the (nearly)
constant vectors B
and
J
( i.e ., constant with respect to the orbital motion). The only
