z
,
y
y
z ,
E' E vB
B'
z
B vE / 2 c
z
y
z
z
y ,
,
whose more compact “semi-vector” form is
Lorentz
E' E ,
B' B ,
transform
(9.135)
of field
E ' E v B
B ' B v E c
components
,
/ 2 ,
where the indices and stand, respectively, for the field components parallel and normal to the relative velocity v of the two reference frames. In the non-relativistic limit, the Lorentz factor tends to 1, and Eqs. (135) acquire an even simpler form
1
E ' E v B,
B ' B
v E .
(9.136)
2
c
Thus we see that the electric and magnetic fields are transformed to each other even in the first order of the v/ c ratio. For example, if we fly across the field lines of a uniform, static, purely electric field E (e.g., the one in a plane capacitor) we will see not only the electric field’s renormalization (in the second order of the v/ c ratio), but also a non-zero dc magnetic field B ’ perpendicular to both the vector E and the vector v, i.e. to the direction of our motion. This is of course what might be expected from the relativity principle: from the point of view of the moving observer (which is as legitimate as that of a stationary observer), the surface charges of the capacitor’s plates, that create the field E, move back creating the dc currents (114), which induce the magnetic field B ’. Similarly, motion across a magnetic field creates, from the point of view of the moving observer, an electric field.
This fact is very important conceptually. One may say there is no such thing in Mother Nature as an electric field (or a magnetic field) all by itself. Not only can the electric field induce the magnetic field (and vice versa) in dynamics, but even in an apparently static configuration, what exactly we measure depends on our speed relative to the field sources – justifying once again the term electromagnetism for the field of physics we are studying in this course.
Another simple but very important application of Eqs. (134)-(135) is the calculation of the fields created by a charged particle moving in free space by inertia, i.e. along a straight line with constant velocity u, at the impact parameter 51 (the closest distance) b from the observer. Selecting the reference frame 0 ’ to move with the particle in its origin, and the reference frame 0 to reside in the “lab” in which the fields E and B are measured, we can use the above formulas with v = u. In this case, the fields E ’
and B ’ may be calculated from, respectively, electro- and magnetostatics: q
'
r
E '
,
B ' 0 ,
(9.137)
4
3
r'
0
because in frame 0’, the particle does not move. Selecting the coordinate axes so that at the measurement point, x = 0, y = b, z = 0 (Fig. 11a), for this point we may write x’ = – ut’, y’ = b, z’ = 0, so that r’ = ( u 2 t’ 2 + b 2)1/2, and the Cartesian components of the fields (137) are: 51 This term is very popular in the theory of particle scattering – see, e.g., CM Sec. 3.7.
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E' q
ut'
q
b
,
E'
,
E' ,
0
x
3 / 2
y
3 / 2
z
4
0 2 2
2
u t' b
4 0 2 2
2
u t' b
(9.138)
B' B' B' 0.
x
y
z
(a)
(b)
1.5
2
y
y'
E , B /
uc
1
y
z
b r '
0.5
q x
0
0
0'
v u
x'
ut'
Ex
z
z'
0.5
Fig. 9.11. The field pulses
induced by a uniformly
1 3
2
1
0
1
2
3
moving charge.
ut
/ b
Now using the last of Eqs. (19b) with x = 0, giving t’ = t, and the relations reciprocal to Eqs.
(134) for the field transform (they are similar to the direct transform but with v replaced with – v = – u), in the lab frame we get
q
u t
q
b
E E'
E E'
E
(9.139)
x
x
y
y
z
0 2 2 2
2
u
t b ,
4
3 / 2
4 0 2 2 2
2
u t b ,
,
0
3 / 2
u
u
q
b
u
B ,
0
B ,
0
B
E'
E .
(9.140)
x
y
z
2
y
c
c 2 40 u 2 2
t 2 b 2 3/2
2
y
c
These results,52 plotted in Fig. 11b in the units of q 2/40 b 2, reveal two major effects. First, the charge passage by the observer generates not only an electric field pulse but also a magnetic field pulse.
This is natural, because, as was repeatedly discussed in Chapter 5, any charge motion is essentially an electric current.53 Second, Eqs. (139)-(140) show that the pulse duration scale is
1/ 2
2
b
b
u
t
1
,
(9.141)
2
u
u
c
i.e. shrinks to virtually zero as the charge’s velocity u approaches the speed of light. This is of course a direct corollary of the relativistic length contraction. Indeed, in the frame 0 ’ moving with the charge, the longitudinal spread of its electric field at distance b from the motion line is of the order of x’ = b.
When observed from the lab frame 0, this interval, in accordance with Eq. (20), shrinks to x = x’/ =
b/, and hence so does the pulse duration scale t = x/ u = b/ u.
52 In the next chapter, we will re-derive them in a different way.
53 It is straightforward to use Eqs. (140) and the linear superposition principle to calculate, for example, the magnetic field of a string of charges moving along the same line and separated by equal distances x = a (so that the average current, as measured in frame 0, is qu/ a), and to show that the time-average of the magnetic field is given by the familiar Eq. (5.20) of magnetostatics, with b instead of .
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9.6. Relativistic particles in electric and magnetic fields
Now let us analyze the dynamics of charged particles in electric and magnetic fields. Inspired by
“our” success in forming the 4-vector (75) of energy-momentum, with the contravariant form
E
dx
p , p mc,
p m
mu ,
(9.142)
c
d
where u is the contravariant form of the 4-velocity (63) of the particle,
dx
dx
u
,
u
,
(9.143)
d
d
we may notice that the non-relativistic equation of motion, resulting from the Lorentz-force formula (5.10) for the three spatial components of p, for a charged particle’s motion in an electromagnetic field, Particle’s
p
d
equation of
qE u B,
(9.144)
motion
dt
is fully consistent with the following 4-vector equality (which is evidently form-invariant with respect to the Lorentz transform):
Particle’s
dp
dynamics:
qF u .
(9.145)
4-form
d
For example, according to Eq. (125), the = 1 component of this equation reads
1
dp
E
1
x
qF u q
c
0 ( u
) ( B )( u
) B ( u
) q E u B
(9.146)
x
z
y
y
z
,
x
d
c
and similarly for two other spatial components ( = 2 and = 3). It may look that these expressions differ from the 2nd Newton law (144) by an extra factor of . However, plugging into Eq. (146) the definition of the proper time interval, d = dt/, and canceling in both parts, we recover Eq. (144) exactly – for any velocity of the particle! The only caveat is that if u is comparable with c, the vector p in Eq. (144) has to be understood as the relativistic momentum (70), proportional to the velocity-dependent mass M = m m rather than to the rest mass m.
The only remaining general task is to examine the meaning of the 0th component of Eq. (145).
Let us spell it out:
dp 0
0
Ex
Ey
Ez
E
qF u q0 c
u
. (9.147)
x
uy uz q u
d
c
c
c
c
Recalling that p 0 = E/ c, and using the basic relation d = dt/ again, we see that Eq. (147) looks exactly like the non-relativistic relation for the kinetic energy change (what is sometimes called the work-energy principle, in our case for the Lorentz force only54):
54 See, e.g., CM Eq. (1.20) divided by dt, and with dp/ dt = F = qE. (As a reminder, the magnetic field cannot affect the particle’s energy, because the magnetic component of the Lorentz force is perpendicular to its velocity.) Chapter 9
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d
Particle’s
E E
q u ,
(9.148) energy:
dt
evolution
besides that in the relativistic case, the energy has to be taken in the general form (73).
Without question, the 4-component equation (145) of the relativistic dynamics is absolutely beautiful in its simplicity. However, for the solution of particular problems, Eqs. (144) and (148) are frequently more convenient. As an illustration of this point, let us now use these equations to explore relativistic effects at charged particle motion in uniform, time-independent electric and magnetic fields.
In doing that, we will, for the time being, neglect the contributions into the field by the particle itself.55
(i) Uniform magnetic field. Let the magnetic field be constant and uniform in the “lab” reference frame 0 that is used for measurements. Then in this frame, Eqs. (144) and (148) yield
dp
d
qu ,
E
B
.
0
(9.149)
dt
dt
From the second equation, E = const, we get u = const, u/ c = const, (1 – 2)-1/2 = const, and M
m = const, so that the first of Eqs. (149) may be rewritten as
u
d
u ω ,
(9.150)
c
dt
where c is the vector directed along the magnetic field B, with the magnitude equal to the following cyclotron frequency (sometimes called “gyrofrequency”):
qB
qB
qc 2 B
.
(9.151) Cyclotron
c
M
m
frequency
E
If the particle’s initial velocity u0 is perpendicular to the magnetic field, Eq. (150) describes its circular motion, with a constant speed u = u 0, in a plane normal to B, with the angular velocity (151). In the non-relativistic limit u << c, when 1, i.e. M m, the cyclotron frequency c equals qB/ m, i.e. is independent of the speed. However, as the kinetic energy of the particle is increased to become comparable with its rest energy mc 2, the frequency decreases, and in the ultra-relativistic limit, B
qB
qc
,
u
for c .
(9.152)
c
p
m
The cyclotron motion’s radius may be calculated as R = u/c; in the non-relativistic limit, it is proportional to the particle’s speed, i.e. to the square root of its kinetic energy. However, as Eq. (151) shows, in the general case the radius is proportional to the particle’s relativistic momentum rather than its speed:
u
Mu
m u
1 p
R
,
(9.153) Cyclotron
qB
qB
q B
radius
c
so that in the ultra-relativistic limit, when p E/ c, R is proportional to the kinetic energy.
55 As was emphasized earlier in this course, in statics this contribution is formally infinite and has to be ignored.
In dynamics, this is generally not true; these self-action effects (which are, in most cases, negligible) will be discussed in the next chapter.
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These dependencies of c and R on energy are the major factors in the design of circular accelerators of charged particles. In the simplest of these machines (the cyclotron, invented in 1929 by Ernest Orlando Lawrence), the frequency of the accelerating ac electric field is constant, so that even if it is tuned to the c of the initially injected particles, the drop of the cyclotron frequency with energy eventually violates this tuning. Due to this reason, the largest achievable particle’s speed is limited to just ~0.1 c (for protons, corresponding to the kinetic energy of just ~15 MeV). This problem may be addressed in several ways. In particular, in synchrotrons (such as Fermilab’s Tevatron and the CERN’s Large Hadron Collider, LHC56) the magnetic field is gradually increased in time to compensate for the momentum increase ( B p), so that both R (148) and c (147) stay constant, enabling proton acceleration to energies as high as ~ 7 TeV, i.e. ~2,000 mc 2.57
Returning to our initial problem, if the particle’s initial velocity has a component u along the magnetic field, then it is conserved in time, so that the trajectory is a spiral around the magnetic field lines. As Eqs. (149) show, in this case, Eq. (150) remains valid but in Eqs. (151) and (153) the full speed and momentum have to be replaced with magnitudes of their (also time-conserved) components, u and p, normal to B, while the Lorentz factor in those formulas still includes the full speed of the particle.
Finally, in the special case when the particle’s initial velocity is directed exactly along the magnetic field’s direction, it continues to move straight along the vector B. In this case, the cyclotron frequency still has the non-zero value (151) but does not correspond to any real motion, because R = 0.
(ii) Uniform electric field. This problem is (technically) more complex than the previous one because in the electric field, the particle’s energy changes. Directing the z-axis along the field E, from Eq. (144) we get
dp
dp
z qE,
0 .
(9.154)
dt
dt
If E does not change in time, the first integration of these equations is elementary, p ( t) p (0) qEt,
p ( t) const p (0) ,
(9.155)
z
z
but the further integration requires care because the effective mass M = m of the particle depends on its full speed u, with
2
2
2
u u u ,
(9.156)
z
making the two motions, along and across the field, mutually dependent.
If the initial velocity is perpendicular to the field E, i.e. if pz(0) = 0, p(0) = p(0) p 0, the easiest way to proceed is to calculate the kinetic energy first:
2
E mc 2
2
2
2
2
2
c p ( t) E c qEt ,
where E ( mc ) c p
.
(9.157)
0
2
0
2 2 2 201/2
On the other hand, we can calculate the same energy by integrating Eq. (148),
56 See https://home.cern/topics/large-hadron-collider.
57 I am sorry I have no more time/space to discuss particle accelerator physics, and have to refer the interested reader to special literature, for example, either S. Lee, Accelerator Physics, 2nd ed., World Scientific, 2004, or E. Wilson, An Introduction to Particle Accelerators, Oxford U. Press, 2001.
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d E
dz
qE u qE
,
(9.158)
dt
dt
over time, with a simple result:
E E qEz( t ,)
(9.159)
0
where (just for the notation simplicity) I took z(0) = 0. Requiring Eq. (159) to give the same E 2 as Eq.
(157), we get a quadratic equation for the function z( t),
2
2
E c qEt
E qEz t
(9.160)
0
2
( )
0
2,
whose solution (with the sign before the square root corresponding to E > 0, i.e. to z 0) is
1/ 2
2
cqEt
0
E
z( t)
1
1 .
(9.161)
qE 0
E
Now let us find the particle’s trajectory. Directing the x-axis so that the initial velocity vector (and hence the velocity vector at any further instant) is within the [ x, z] plane, i.e. that y( t) = 0
identically, we may use Eqs. (155) to calculate the trajectory’s slope, at its arbitrary point, as dz
dz / dt
Mu
p
qEt
z
z
.
(9.162)
dx
dx / dt
Mu
p
p
x
x
0
Now let us use Eq. (160) to express the numerator of this fraction, qEt, as a function of z: 1
1/
qEt
qEz
.
(9.163)
0
E
2
2
0
E 2
c
Plugging this expression into Eq. (161), we get
dz
1
1/
qEz
.
(9.164)
0
E
2
2
0
E 2
dx
cp 0
This differential equation may be readily integrated separating the variables z and x, and using the following substitution: cosh-1( qEz/ E 0 +1). Selecting the origin of axis x at the initial point, so that x(0) = 0, we finally get the trajectory:
E
qEx
z 0 cosh
1 .
(9.165)
qE
cp 0
This curve is usually called the catenary, but sometimes the “chainette”, because it (with the proper constant replacement) describes, in particular, the stationary shape of a heavy, uniform chain in a uniform gravity field directed along the z-axis. At the initial part of the trajectory, where qEx << cp 0(0), this expression may be approximated with the first non-zero term of its Taylor expansion in small x, giving the following parabola:
2
E qE x
0
z
,
(9.166)
2
cp 0
so that if the initial velocity of the particle is much lower than c (i.e. p 0 mu 0, E0 mc 2), we get the very familiar non-relativistic formula:
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qE
2
a 2
F
qE
z
x t ,
a
with
.
(9.167)
2 mu 2
2
m
m
0
The generalization of this solution to the case of an arbitrary direction of the particle’s initial velocity is left for the reader’s exercise.
(iii) Crossed uniform magnetic and electric fields (E B). In view of the somewhat bulky solution of the previous problem (i.e. the particular case of the current problem for B = 0), one might think that this problem, with B 0, should be forbiddingly complex for an analytical solution. Counterintuitively, this is not the case, due to the help from the field transform relations (135). Let us consider two possible cases.
Case 1: E/ c < B. Let us consider an inertial reference frame 0’ moving (relatively the “lab”
reference frame 0 in that the fields E and B are measured) with the following velocity: E B
v
,
(9.168)
2
B
and hence the speed v = c( E/ c)/ B < c. Selecting the coordinate axes as shown in Fig. 12, so that E ,
0
E E, E ;
0
B ,
0
B ,
0
B B ,
(9.169)
x
y
z
x
y
z
we see that the Cartesian components of this velocity are vx = v, vy = vz = 0.
y
y'
E
'
0
Fig. 9.12. Particle’s trajectory in
B
0
x
v
'
B
crossed electric and magnetic fields
x'
(at E/c < B).
z
z'
Since this choice of the coordinates complies with the one used to derive Eqs. (134), we can readily use that simple form of the Lorentz transform to calculate the field components in the moving reference frame:
E
E' ,
0
E' E vB
E
B
E'
(9.170)
x
y
,
0
,
0
B
z
2
vE
vE
v
B
B' ,
0
B' ,
0
B' B
B
1
B
1
B,
(9.171)
x
y
z
2
2
2
c
Bc
c
where the Lorentz parameter (1 – v 2/ c 2)-1/2 corresponds to the velocity (168) rather than that of the particle. These relations show that in this special reference frame, the particle only “sees” the renormalized uniform magnetic field B’ B, parallel to the initial field, i.e. normal to the velocity (168).
Using the result of the above case (i), we see that in this frame the particle moves along either a circle or a spiral winding about the direction of the magnetic field, with the angular velocity (151): Chapter 9
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qB'
'
,
(9.172)
c
2
'/c
E
and the radius (153):
p'
R'
.
(9.173)
qB'
Hence in the lab frame, the particle performs this orbital/spiral motion plus a “drift” with the constant velocity v (Fig. 12). As a result, the lab-frame trajectory of the particle (or rather its projection onto the plane normal to the magnetic field) is a trochoid-like curve58 that, depending on the initial velocity, may be either prolate (self-crossing), as in Fig. 12, or curtate (drift-stretched so much that it is not self-crossing).
Such looped motion of electrons is used, in particular, in magnetrons – very popular generators of microwave radiation. In such a device (Fig. 13), the magnetic field, usually created by specially-shaped permanent magnets, is nearly uniform (in the region of electron motion) and directed along the magnetron’s axis (in Fig. 13, normal to the plane of the drawing), while the electric field of magnitude E
<< cB, created by the dc voltage applied between the anode and the cathode, is virtually radial.
Fig. 9.13. Schematic cross-section of a typical
magnetron.
(Figure
adapted
from
https://en.wikipedia.org/wiki/Cavity_magnetron
under the Free GNU Documentation License.)
As a result, the above simple theory is only approximately valid, and the electron trajectories are close to epicycloids rather than trochoids. The applied electric field is adjusted so that these looped trajectories pass close to the anode’s surface, and hence to the gap openings of the cylindrical microwave cavities drilled in the anode’s bulk. The fundamental mode of such a cavity is quasi-lumped, with the cylindrical walls working mostly as inductances, and the gap openings as capacitances, with the microwave electric field concentrated in these openings. This is why the mode is strongly coupled to the electrons “licking” the anode’s surface, and their interaction creates large positive feedback (equivalent to negative damping), which results in intensive microwave self-oscillations at the cavities’ own frequency.59 The oscillation energy, of course, is taken from the dc-field-accelerated electrons; due to this energy loss, the looped trajectory of each electron gradually moves closer to the anode and finally 58 As a reminder, a trochoid may be described as the trajectory of a point on a rigid disk rolled along a straight line. It’s canonical parametric representation is x = + a cos , y = a sin . (For a > 1, the trochoid is prolate, if a
< 1, it is curtate, and if a = 1, it is called the cycloid.) Note, however, that for our problem, the trajectory in the lab frame is exactly trochoidal only in the non-relativistic limit v << c (i.e. E/ c << B).
59 See, e.g., CM Sec. 5.4.
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lands on its surface. The wide use of such generators (in particular, in microwave ovens, which operate in a narrow frequency band around 2.45 GHz, allocated for these devices to avoid their interference with wireless communication systems) is due to their simplicity and high (up to 65%) efficiency of the dc-to-rf energy transfer.
Case 2: E/ c > B. In this case, the speed given by Eq. (168) would be above the speed of light, so let us introduce a reference frame moving with a different velocity,
E B
v
,
(9.174)
E
2
/ c
whose direction is the same as before (Fig. 12), and magnitude v = c B/( E/ c) is again below c. A calculation absolutely similar to the one performed above for Case 1, yields
2
vB
v
E
E' ,
0
E' E vB
E
E
E
E'
(9.175)
x
y
1
1
,
,
0
2
E
c
z
vE
EB
B' ,
0
B' ,
0
B'
(9.176)
x
y
z
B
B
.
0
2
c
E
so that in the moving frame the particle “sees” only the electric field E’ E. According to the solution of our previous problem (ii), the trajectory of the particle in the moving frame is the catenary (165), so that in the lab frame it has an “open”, hyperbolic character as well.
To conclude this section, let me note that if the electric and magnetic fields are nonuniform, the particle motion may be much more complex, and in most cases, the integration of the system of equations (144) and (148) may be carried out only numerically. However, if the field’s nonuniformity is small, approximate analytical methods may be very effective. For example, if E = 0, and the magnetic field has a small transverse gradient B in a direction normal to the vector B itself, such that
B
1
,
(9.177)
B
R
where R is the cyclotron radius (153), then it is straightforward to use Eq. (150) to show60 that the cyclotron orbit drifts perpendicular to both B and B, with the drift speed
1 2
2
v
.
(9.178)
d
u u
u
2
c
The physics of this drift is rather simple: according to Eq. (153), the instant curvature of the cyclotron orbit is proportional to the local value of the field. Hence if the field is nonuniform, the trajectory bends slightly more on its parts passing through a stronger field, thus acquiring a shape close to a curate trochoid.
For experimental physics and engineering practice, the effects of longitudinal gradients of magnetic field on the charged particle motion are much more important, but it is more convenient for me to postpone their discussion until we have developed a little bit more analytical tools in the next section.
60 See, e.g., Sec. 12.4 in J. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999.
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9.7. Analytical mechanics of charged particles
The general Eq. (145) gives a full description of relativistic particle dynamics in electric and magnetic fields, just as the 2nd Newton law (1) does it in the non-relativistic limit. However, we know that in the latter case, the Lagrange formalism of analytical mechanics allows an easier solution of many problems.61 We can expect that to be true in relativistic mechanics as well, so let us expand the analysis of Sec. 3 (which was valid only for free particles) to particles in the field.
For a free particle, our main result was Eq. (68), which may be rewritten as
2
L mc ,
(9.179)
with (1 – u 2/ c 2)-1/2, showing that the product on the left-hand side is Lorentz-invariant. How can the electromagnetic field affect this relation? In non-relativistic electrostatics, we could write L T U T
q .
(9.180)
However, in relativity, the scalar potential is just one component of the potential 4-vector (116). The only way to get from this full 4-vector a Lorentz-invariant contribution to L, which would be also proportional to the first power of the particle’s velocity (to account for the magnetic component of the Lorentz force), is evidently
2
L mc const u
A ,
(9.181)
where u is the 4-velocity (63). To comply with Eq. (180) at u << c, the constant factor should be equal to (– q), so that Eq. (181) becomes
2
L mc qu
A ,
(9.182)
and with the account of Eqs. (63) and the second of Eqs. (116), we get very important equality mc 2
Particle’s
L
q qu A ,
(9.183) Lagrangian
function
whose Cartesian form is
1/ 2
2
2
2
u u u
2
x
y
z
L mc 1
q q u A u A u A .
(9.184)
2
x
x
y
y
z
z
c
Let us see whether this relation (which admittedly was derived by an educated guess rather than by a strict derivation) passes a natural sanity check. For the case of an unconstrained motion of a particle, we can select its three Cartesian coordinates rj ( j = 1, 2, 3) as the generalized coordinates, and its linear velocity components uj as the corresponding generalized velocities. In this case, the Lagrange equations of motion are
d L
L .
0
(9.185)
dt u
r
j
j
For example, for r 1 = x, Eq. (184) yields
L
mu
L
A
x
,
u
,
(9.186)
u
1 2 /
1/ 2
2
x
x
x
x
u
c
qA
p
qA
q
q
x
x
x
61 See, e.g., CM Sec. 2.2 and on.
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so that Eq. (185) takes the form
dp
A
dA
x q
qu
q
x .
(9.187)
dt
x
x
dt
In the equations of motion, the field values have to be taken at the instant position of the particle, so that the last (full) derivative has components due to both the actual field’s change (at a fixed point of space) and the particle’s motion. Such addition is described by the so-called convective derivative 62
Convective
d
derivative
u .
(9.188)
dt
t
Spelling out both scalar products, we may group the terms remaining after cancellations as follows: dp
x
Ax
Ay A
x
A
A
x
z
q
u
u
.
(9.189)
y
z
dt
x
t
x
y
z
x
But taking into account the relations (121) between the electric and magnetic fields and potentials, this expression is nothing more than
dp
x q E u B u B q E u B , (9.190)
x
y
z
z
y
x
dt
i.e. the x-component of Eq. (144). Since other Cartesian coordinates participate in Eq. (184) similarly, it is evident that the Lagrangian equations of motion along other coordinates yield other components of the same vector equation of motion.
So, Eq. (183) does indeed give the correct Lagrangian function, and we can use it for further analysis, in particular to discuss the first of Eqs. (186). This relation shows that in the electromagnetic field, the generalized momentum corresponding to the particle’s coordinate x is not px = m ux, but63
P L p qA .
(9.191)
x
x
x
u
x
Thus, as was already discussed (at that point, without proof) in Sec. 6.4, the particle’s motion in a magnetic field may be is described by two different linear momentum vectors: the kinetic momentum p defined by Eq. (70), and the canonical (or “conjugate”) momentum 64
Particle’s
canonical
P p A
q .
(9.192)
momentum
In order to facilitate discussion of this notion, let us generalize Eq. (72) for the Hamiltonian function H of a free particle to the case of a particle in the field:
mc 2
mc 2
H P u L (p qA) u
qu A q p u
q
.
(9.193)
62 Alternatively called the “Lagrangian derivative”; for its (rather simple) derivation see, e.g., CM Sec. 8.3.
63 With regrets, I have to use for the generalized momentum the same (very common) notation as was used earlier in the course for the electric polarization – which will not be discussed here and in the balance of these notes.
64 In the Gaussian units, Eq. (192) has the form P = p + qA/ c.
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Merging the first two terms of the last expression exactly as it was done in Eq. (72), we get an extremely simple result,
mc 2
H
q ,
(9.194a)
which may be spelled out as
1/ 2
2
p
H 1
2
mc q ,
(
i.e. H q)2 (
2
mc )2
2
2
c p . (9.194b)
mc
These expressions may leave the reader wondering: where is the vector potential A here – and the magnetic field effects it has to describe? The resolution of this puzzle is easy: as we know from analytical mechanics,65 for most applications, for example for an alternative derivation of the equations of motion, H has to be represented as a function of the particle’s generalized coordinates (in the case of unconstrained motion, these may be the Cartesian components of the vector r that serves as an argument for the potentials A and ), and the generalized momenta, i.e. the components of the vector P –
generally, plus time. For that, the kinematic momentum p in Eq. (194b) has to be expressed via these variables. This may be done using Eq. (192), giving us the following generalization of Eq. (78):66
Particle’s
(H q)2 (
2
mc )2
2
c (P A
q )2 .
(9.195) Hamiltonian
function
It is straightforward to verify that the Hamilton equations of motion for three Cartesian coordinates of the particle, obtained in a regular way from this H, may be merged into the same vector equation (144). In the non-relativistic limit, performing the expansion of Eqs. (194b) into the Taylor series in p 2, and limiting it to two leading terms, we get the following generalization of Eq. (74): p 2
2
2
1
H mc
q ,
i.e. H mc
(P qA)2 U ,
U
with
q .
(9.196)
2 m
2 m
These expressions for H, and Eq. (183) for L , give a clear view of the electromagnetic field effects’ description in analytical mechanics. The electric part qE of the total Lorentz force can perform mechanical work on the particle, i.e. change its kinetic energy – see Eq. (148) and its discussion. As a result, the scalar potential , whose gradient gives a contribution to E, may be directly associated with the potential energy U = q of the particle. On the contrary, the magnetic component quB of the Lorentz force is always perpendicular to the particle’s velocity u, and cannot perform a non-zero work on it, and as a result, cannot be described by a contribution to U. However, if A did not participate in the functions L and/or H at all, the analytical mechanics would be unable to describe effects of the magnetic field B = A on the particle’s motion. The relations (183) and (195)-(196) show the wonderful way in which physics (with some help from Mother Nature herself :-) solves this problem: the vector potential gives such contributions to the functions L and H that cannot be uniquely attributed to either kinetic or potential energy, but ensure both the Lagrange and Hamilton formalisms yield the correct equation of motion (144) – including the magnetic field effects.
65 See, e.g., CM Sec. 10.1.
66 Alternatively, this relation may be obtained from the expression for the Lorentz-invariant norm, p p = ( mc)2, of the 4-momentum (75), p = {E/ c, p} = {(H – q)/ c, P – qA}.
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I believe I still owe the reader some discussion of the physical sense of the canonical momentum P. For that, let us consider a charged particle moving near a region of localized magnetic field B(r, t), but not entering this region (see Fig. 14), so that on its trajectory A B = 0.
B(r, t)
Fig. 9.14. Particle’s motion around a localized
( t)
C
magnetic field with a time-dependent flux.
If there is no electrostatic fields affecting the particle (i.e. no other electric charges nearby), we may select such a local gauge that (r, t) = 0 and A = A( t), so that Eq. (144) is reduced to dp
dA
qE q
,
(9.197)
dt
dt
and Eq. (192) immediately gives
dP
dp
dA
q
0 .
(9.198)
dt
dt
dt
Hence, even if the magnetic field is changed in time, so that the induced electric field E does accelerate the particle, its canonical momentum does not change. Hence P is a variable more stable to magnetic field changes than its kinetic counterpart p. This conclusion may be criticized because it relies on a specific gauge, and generally P p + qA is not gauge–invariant, because the vector potential A is not.67
However, as was already discussed in Sec. 5.3, the integral A dr over a closed contour is gauge-invariant and is equal to the magnetic flux through the area limited by the contour – see Eq. (5.65).
So, integrating Eq. (197) over a closed trajectory of a particle (Fig. 14), and over the time of one orbit, we get
Δ p dr q
,
ΔΦ
Δ
that
so
P dr 0
,
(9.199)
C
C
where is the change of flux during that time. This gauge-invariant result confirms the above conclusion about the stability of the canonical momentum to magnetic field variations.
Generally, Eq. (199) is invalid if a particle moves inside a magnetic field and/or changes its trajectory at the field variation. However, if the field is almost uniform, i.e. its gradient is small in the sense of Eq. (177), this result is (approximately) applicable. Indeed, analytical mechanics68 tells us that for any canonical coordinate-momentum pair { qj, pj}, the corresponding action variable, 1
J
p dq ,
(9.200)
j
j j
2
remains virtually constant at slow variations of motion conditions. According to Eq. (191), for a particle in a magnetic field, the generalized momentum corresponding to the Cartesian coordinate rj is Pj rather than pj. Thus forming the net action variable J Jx + Jy + Jz , we may write 67 In contrast, the kinetic momentum p = Mu is evidently gauge- (though not Lorentz-) invariant.
68 See, e.g., CM Sec. 10.2.
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2 J
P dr p dr Φ
q const
.
(9.201)
Let us apply this relation to the motion of a non-relativistic particle in an almost uniform magnetic field, with a relatively small longitudinal velocity, u / u 0 – see Fig. 15.
u
R
B
u
Fig. 9.15. Particle in a magnetic field with
a small longitudinal gradient B B.
In this case, in Eq. (201) is the flux encircled by the particle’s cyclotron orbit, = – R 2 B, where R is its radius given by Eq. (153), and the negative sign accounts for the fact that in our case, the
“correct” direction of the normal vector n in the definition of flux, = Bn d 2 r, is antiparallel to the vector B. At u << c, the kinetic momentum is just p = mu, while Eq. (153) yields mu qBR .
(9.202)
Plugging these relations into Eq. (201), we get
2
qRB
2 J
mu 2 R
q R
B m
2 R
q R 2
B (2 )
1 q R 2
B
q .
(9.203)
m
This means that even if the circular orbit slowly moves through the magnetic field, the flux encircled by the cyclotron orbit should remain virtually constant. One manifestation of this effect is the result already mentioned at the end of Sec. 6: if a small gradient of the magnetic field is perpendicular to the field itself, then the particle orbit’s drift direction is perpendicular to B, so that stays constant.
Now let us analyze the case of a small longitudinal gradient, B B (Fig. 15). If a small initial longitudinal velocity u is directed toward the higher field region, the cyclotron orbit has to gradually shrink to keep constant. Rewriting Eq. (202) as
R
2 B
mu q
q
,
(9.204)
R
R
we see that this reduction of R (at constant ) increases the orbiting speed u. But since the magnetic field cannot perform any work on the particle, its kinetic energy,
m
E
2 2
u u ,
(9.205)
2
should stay constant, so that the longitudinal velocity u has to decrease. Hence eventually the orbit’s drift has to stop, and then it has to start moving back toward the region of lower fields, being essentially repulsed from the high-field region. This effect is very important, in particular, for plasma confinement systems. In the simplest of such systems, two coaxial magnetic coils, inducing magnetic fields of the same direction (Fig. 16), naturally form a “magnetic bottle”, which traps charged particles injected, with sufficiently low longitudinal velocities, into the region between the coils. More complex systems of this Chapter 9
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type, but working on the same basic principle, are the most essential components of the persisting large-scale efforts to achieve controllable nuclear fusion.69
B
Fig. 9.16. A simple magnetic bottle (schematically).
Returning to the constancy of the magnetic flux encircled by free particles, it reminds us of the Meissner-Ochsenfeld effect, which was discussed in Sec. 6.4, and gives a motivation for a brief revisit of the electrodynamics of superconductivity. As was emphasized in that section, superconductivity is a substantially quantum phenomenon; nevertheless, the classical notion of the conjugate momentum P
helps to understand its theoretical description. Indeed, the general rule of quantization of physical systems70 is that each canonical pair { qj, pj} of a generalized coordinate qj and the corresponding generalized momentum pj is described by quantum-mechanical operators that obey the following commutation relation:
q ˆ , p ˆ i .
(9.206)
j
j'
jj'
According to Eq. (191), for the Cartesian coordinates rj of a particle in the magnetic field, the corresponding generalized momenta are Pj, so that their operators should obey the similar commutation relations:
r ˆ P ˆ, i .
(9.207)
j
j'
jj'
In the coordinate representation of quantum mechanics, the canonical operators of the Cartesian components of the linear momentum are described by the corresponding components of the vector operator – i. As a result, ignoring the rest energy mc 2 (which gives an inconsequential phase factor exp{– imc 2 t/} in the wavefunction), we can use Eq. (196) to rewrite the usual non-relativistic Schrödinger equation,
i
H
ˆ ,
(9.208)
t
as follows:
2
p ˆ
1
i
U
i
qA2
q
.
(9.209)
t
2 m
2 m
Thus, I believe I have finally delivered on my promise to justify the replacement (6.50), which had been used in Secs. 6.4 and 6.5 to discuss the electrodynamics of superconductors, including the Meissner-Ochsenfeld effect. The Schrödinger equation (209) may be also used as the basis for the quantum-mechanical description of other magnetic field phenomena, including the so-called Aharonov-Bohm and quantum Hall effects – see, e.g., QM Secs. 3.1-3.2.
69 For further reading on this technology, the reader may be referred, for example, to the simple monograph by F.
Chen, Introduction to Plasma Physics and Controllable Fusion, vol. 1, 2nd ed., Springer, 1984, and/or the graduate-level theoretical treatment by R. Hazeltine and J. Meiss, Plasma Confinement, Dover, 2003.
70 See, e.g., CM Sec. 10.1.
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9.8. Analytical mechanics of the electromagnetic field
We have just seen that the analytical mechanics of a particle in an electromagnetic field may be used to get some important results. The same is true for the analytical mechanics of the electromagnetic field as such, and the field-particle system as a whole. For such space-distributed systems as fields, governed by local dynamics laws (in our case, the Maxwell equations), we need to apply analytical mechanics to the local densities l and h of the Lagrangian and Hamiltonian functions, defined by relations
L l d 3 r, H d 3
h r .
(9.210)
Let us start, as usual, from the Lagrange formalism. Some clues on the possible structure of the Lagrangian function density l may be obtained from that of the particle-field interaction description in this formalism, discussed in the last section. As we have seen, for the case of a single particle, the interaction is described by the last two terms of Eq. (183):
q u
q A
int
L
.
(9.211)
Obviously, if the charge q is continuously distributed over some volume, we may represent this Lint as a volume integral of the following Lagrangian function density:
Interaction
l
j A
.
(9.212)
int
j A
Lagrangian
density
Notice that this density (in contrast to Lint itself!) is Lorentz-invariant. (This is due to the contraction of the longitudinal coordinate, and hence volume, at the Lorentz transform.) Hence we may expect the density of the field’s part of the Lagrangian to be Lorentz-invariant as well. Moreover, given the local structure of the Maxwell equations (containing only the first spatial and temporal derivatives of the fields), l field should be a function of the potential’s 4-vector and its 4-derivative: l
.
(9.213)
field
lfield
A , A
Also, the density should be selected in such a way that the 4-vector analog of the Lagrangian equation of motion,
l
l
,
(9.214)
field
A
field
0
A
gave us the correct inhomogeneous Maxwell equations (127).71 The field part lfield of the total Lagrangian density l should be a scalar and a quadratic form of the field strengths, i.e. of the tensor F, so that the natural choice is
l
const
.
(9.215)
field
F F
with the implied summation over both indices. Indeed, adding to this expression the interaction Lagrangian (212),
l l
const
,
(9.216)
field
lint
F F
j A
71 Here the implicit summation over the index plays a role similar to the convective derivative (188) in replacing the full derivative over time, in a way that reflects the symmetry of time and space in special relativity. I do not want to spend more time justifying Eq. (214), because of the reasons that will be clear imminently.
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and performing the differentiations, we see that Eqs. (214)-(215) indeed yield Eqs. (127), provided that the constant factor equals (-1/40).72 So, the field’s Lagrangian density is
Field’s
2
2
1
Lagrangian
1 E
B
2
0
2
l
F F
B
E
u u ,
(9.217)
field
density
2
e
m
4
2 c
2
2
0
0
0
where u e is the electric field energy density (1.65), and u m is the magnetic field energy density (5.57).
Let me hope the reader agrees that Eq. (217) is a wonderful result because the Lagrangian function has a structure absolutely similar to the well-known expression L = T – U of classical mechanics. So, for the field alone, the “potential” and “kinetic” energies are separable again.73
Now let us explore whether we can calculate the 4-form of the field’s Hamiltonian function H.
In the generic analytical mechanics,
L
H
q .
(9.218)
j
L
j q
j
However, just as for the Lagrangian function, for a field we should find the spatial density h of the Hamiltonian, defined by the second of Eqs. (210), for which the natural 4-form of Eq. (218) is
l
h
A g l .
(9.219)
( A )
Calculated for the field alone, i.e. using Eq. (217) for l, this definition yields
h
,
(9.220)
field
D
where the tensor
Symmetric
energy-
1
1
g F F
g
F F ,
(9.221)
momentum
0
4
tensor
is gauge-invariant, while the remaining term,
1
g
,
(9.222)
D
F
A
0
is not, so that it cannot correspond to any measurable variables. Fortunately, it is straightforward to verify that the last tensor may be represented in the form
1
,
(9.223)
D
F A
0
and as a result, obeys the following relations:
,
0
(9.224)
D
0 3
d r ,
0
D
72 In the Gaussian units, this coefficient is (-1/16).
73 Since the Lagrange equations of motion are homogeneous, the simultaneous change of the signs of T and U
does not change them. Thus, it is not important which of the two energy densities, u e or u m, we count as the potential energy, and which as the kinetic energy. (Actually, such duality of the two energy components is typical for all analytical mechanics – see, e.g., the discussion of this issue in CM Sec. 2.2.)
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so it does not interfere with the conservation properties of the gauge-invariant, symmetric energy-momentum tensor (also called the symmetric stress tensor) , to be discussed below.
Let us use Eqs. (125) to express the elements of the latter tensor via the electric and magnetic fields. For = = 0, we get
00
0
2
B 2
E
u u u ,
(9.225)
e
m
2
20
i.e. the expression for the total energy density u – see Eq. (6.113). The other 3 elements of the same row/column turn out to be just the Cartesian components of the Poynting vector (6.114), divided by c: E
E
S
j 0
1
B H j ,
for j ,
1 ,
2 3 .
(9.226)
c
j c
c
0
j
The remaining 9 elements jj’ of the tensor, with j, j’ = 1, 2, 3, are usually represented as jj'
(M)
,
(9.227)
jj '
where (M) is the so-called Maxwell stress tensor:
Maxwell
(M)
jj'
1
2
jj '
E E
E
2
B B
B
,
(9.228) stress
jj '
0
j
j'
j
j'
2
2
tensor
0
so that the whole symmetric energy-momentum tensor (221) may be conveniently represented in the following symbolic way:
u
S / c
.
(9.229)
(M)
S / c
jj'
The physical meaning of this tensor may be revealed in the following way. Considering Eq.
(221) as the definition of the tensor ,74 and using the 4-vector form of Maxwell equations given by Eqs. (127) and (129), it is straightforward to verify an extremely simple result for the 4-derivative of the symmetric tensor:
F
j .
(9.230)
This expression is valid in the presence of electromagnetic field sources, e.g., for any system of charged particles and the fields they have created. Of these four equations (for four values of the index ), the temporal one (with = 0) may be simply expressed via the energy density (225) and the Poynting vector (226):
u
S j E ,
(9.231)
t
while three spatial equations (with = j = 1, 2, 3) may be represented in the form 74 In this way, we are using Eq. (219) just as a useful guess, which has led us to the definition of , and may leave its strict justification for more in-depth field theory courses.
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S
3
j
.
(9.232)
2
(M) E
j B
jj '
j
t
c
'1
j
rj'
If integrated over a volume V limited by surface S, with the account of the divergence theorem, Eq. (231) returns us to the Poynting theorem (6.111):
u
j d 3
E
r S d 2 r ,0
(9.233)
n
t
V
S
while Eq. (232) yields75
3
S
3
(M)
f d r
dA , with f E
j B,
