Classical Electrodynamics by Konstantin K. Likharev - HTML preview
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EM: Classical Electrodynamics
= g/ of the time evolution of each pair’s wavefunction = exp{– i t} is exactly the same and that the phases of the wavefunctions, defined by the relation
i
e ,
(6.47)
coincide, so the electric current is carried not by individual Cooper pairs but rather by their Bose-Einstein condensate described by a single wavefunction (47). Due to this coherence, the quantum effects (which are, in the usual Fermi-gases of single electrons, masked by the statistical spread of their energies, and hence of their phases), become very explicit – “macroscopic”.
To illustrate this, let us write the well-known quantum-mechanical formula for the probability current density of a free, non-relativistic particle,24
i
1
j
,
(6.48)
w
c.c.
i c.c.
2 m
2 m
where c.c. means the complex conjugate of the previous expression. Now let me borrow one result that will be proved later in this course (in Sec. 9.7) when we discuss the analytical mechanics of a charged particle moving in an electromagnetic field. Namely, to account for the magnetic field effects, the particle’s kinetic momentum p mv (where v dr/ dt is the particle’s velocity) has to be distinguished from its canonical momentum,25
P p A
q .
(6.49)
where A is the field’s vector potential defined by Eq. (5.27). In contrast with the Cartesian components pj = mvj of the kinetic momentum p, the canonical momentum’s components are the generalized momenta corresponding to the Cartesian components rj of the radius-vector r, considered as generalized coordinates of the particle: Pj = L / vj, where L is the particle’s Lagrangian function. According to the general rules of transfer from classical to quantum mechanics,26 it is the vector P whose operator (in the coordinate representation) equals – i, so the operator of the kinetic momentum p = P – qA is – i +
qA. Hence, to account for the magnetic field27 effects, we should make the following replacement,
i
i
A
q ,
(6.50)
in all quantum-mechanical relations. In particular, Eq. (48) has to be generalized as 1
j
.
(6.51)
w
i
qA
c.c.
2 m
This expression becomes more transparent if we take the wavefunction in form (47); then detailed, but still very readable coverage of the physics of superconductors, I can recommend the reader the monograph by M. Tinkham, Introduction to Superconductivity, 2nd ed., McGraw-Hill, 1996.
24 See, e.g., QM Sec. 1.4, in particular Eq. (1.47).
25 I am sorry to use traditional notations p and P for the momenta – the same symbols which were used for the electric dipole moment and polarization in Chapter 3. I hope there will be no confusion because the latter notions are not used in this section.
26 See, e.g., CM Sec. 10.1, in particular Eq. (10.26).
27 The account of the electric field is easier, because the related energy q of the particle may be directly included in the potential energy operator.
Chapter 6
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2
q
j
.
(6.52)
w
A
m
This relation means, in particular, that in order to keep j w gauge-invariant, the transformation (8)-(9) has to be accompanied by a simultaneous transformation of the wavefunction’s phase:
q
.
(6.53)
It is fascinating that the quantum-mechanical wavefunction (or more exactly, its phase) is not gauge-invariant, meaning that you may change it in your mind – at your free will! Again, this does not change any observable (such as j w or the probability density *), i.e. any experimental results.
Now for the electric current density of the whole superconducting condensate, Eq. (52) yields the following constitutive relation:
qn p
2
q
j j qn
,
(6.54) Supercurrent
w
p
A
m
density
The formula shows that this supercurrent may be induced by the dc magnetic field alone and does not require any electric field. Indeed, for the simple 1D geometry shown in Fig. 2a, j(r) = j( x)n z, A(r) = A( x) n z, and / z = 0, so the Coulomb gauge condition (5.48) is satisfied for any choice of the gauge function
( x). For the sake of simplicity we can choose this function to provide (r) const,28 so 2
q n
1
j
p
A
A .
(6.55)
2
m
L
where L is given by Eq. (46), and the field is assumed to be small and hence not affecting the probability 2 (here normalized to 1 in the absence of the field). This is the so-called London equation, proposed (in a different form) by F. and H. London in 1935 for the Meissner-Ochsenfeld effect’s explanation. Combining it with Eq. (5.44), generalized for a linear magnetic medium by the replacement
0 , we get
2
1
A
A ,
(6.56)
2
L
For our 1D geometry, this simple differential equation, similar to Eq. (23), has an exponential solution similar to Eq. (32):
x
x
x
(
A x) (
A 0) exp
,
B( x) B(0) exp
,
j( x) j(0) exp
,
(6.57)
L
L
L
which shows that the magnetic field and the supercurrent penetrate into a superconductor only by London’s penetration depth L, regardless of frequency.29 By the way, integrating the last result through the penetration layer, and using the vector potential’s definition, B = A (for our geometry, giving 28 This is the so-called London gauge; for our simple geometry, it is also the Coulomb gauge (5.48).
29 Since at T > 0, not all electrons in a superconductor form Cooper pairs, at any frequency 0 the unpaired electrons provide energy-dissipating Ohmic currents, which are not described by Eq. (54). These losses become very substantial when the frequency becomes so high that the skin-effect length s of the material becomes less than L. For typical metallic superconductors, this crossover takes place at frequencies of a few hundred GHz, so even for microwaves, Eq. (57) still gives a fairly accurate description of the field penetration.
Chapter 6
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B( x) = dA( x)/ dx = –L A( x)) we may readily verify that the linear density J of the surface supercurrent still satisfies the universal coarse-grain relation (38).
This universality should bring to our attention the following common feature of the skin effect (in “normal” conductors) and the Meissner-Ochsenfeld effect (in superconductors): if the linear size of a bulk sample is much larger than, respectively, s or L, than B = 0 in the dominating part of its interior.
According to Eq. (5.110), a formal description of such conductors (valid only on a coarse-grain scale much larger than either s or L), may be achieved by formally treating the sample as an ideal diamagnet, with = 0. In particular, we can use this description and Eq. (5.124) to immediately obtain the magnetic field’s distribution outside of a bulk sphere:
R 3
B H ,
with H r
cos ,
r
for R .
(6.58)
0
0
m
m
0
2 r 2
Figure 3 shows the corresponding surfaces of equal potential m. It is evident that the magnetic field lines (which are normal to the equipotential surfaces) bend to become parallel to the surface near it.
Fig. 6.3. Equipotential surfaces
H
m = const around a conducting
0
sphere of radius R >> s (or L),
placed into a uniform magnetic
field, calculated within the
coarse-grain (ideal-diamagnet)
approximation = 0.
This pattern also helps to answer the question that might arise at making the assumption (24): what happens to bulk conductors placed into a normal ac magnetic field – and to superconductors in a normal dc magnetic field as well? The answer is: the field is deformed outside of the conductor to sustain the following coarse-grain boundary condition:30
Coarse-
grain
B
0 ,
(6.59)
n surface
boundary
condition which follows from Eq. (5.118) and the coarse-grain requirement Binside = 0.
This answer should be taken with reservations. For normal conductors, it is only valid at sufficiently high frequencies where the skin depth (33) is relatively small: s << a, where a is the scale of the conductor’s linear size – for a sphere, a ~ R. In superconductors, this simple picture requires not only that s << a, but also that magnetic field is relatively low because strong fields do penetrate 30 Sometimes this boundary condition, as well as the (compatible) Eq. (38), are called “macroscopic”. However, this term may lead to confusion with the genuine macroscopic boundary conditions (5.117)-(5.118), which also ignore the atomic-scale microstructure of the “effective currents” jef = M, but (as was shown earlier in this section) still allow explicit, detailed accounts of the skin-current (34) and supercurrent (55) distributions.
Chapter 6
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superconductors, destroying superconductivity (either completely or partly), and as a result violating the Meissner-Ochsenfeld effect – see the next section.
6.5. Electrodynamics of macroscopic quantum phenomena31
Despite the superficial similarity of the skin effect and the Meissner-Ochsenfeld effect, the electrodynamics of superconductors is much richer. For example, let us use Eq. (54) to describe the fascinating effect of magnetic flux quantization. Consider a closed ring/loop (not necessarily a round one) made of a superconducting “wire” with a cross-section much larger than 2
L (Fig. 4a).
(a)
(b)
(c)
I
1
2
1
i
e
i
e
i
e
2
i
e
1
i
e
2
i
e
C
1 2
1 2
1 2
n0
Fig. 6.4. (a) A closed, flux-quantizing superconducting ring, (b) a ring with a narrow slit, and (c) a Superconducting QUantum Interference Device (SQUID).
From the last section’s discussion, we know that deep inside the wire the supercurrent is exponentially small. Integrating Eq. (54) along any closed contour C that does not approach the surface closer than a few L at any point (see the dashed line in Fig. 4), so with j = 0 at all its points, we get q
dr
A dr 0
.
(6.60)
C
C
The first integral, i.e. the difference of in the initial and final points , has to be equal to either zero or an integer number of 2 because the change + 2 n does not change the Cooper pair’s condensate’s wavefunction:
i
2 n
' e
i
e
.
(6.61)
On the other hand, according to Eq. (5.65), the second integral in Eq. (60) is just the magnetic flux
through the contour.32 As a result, we get a wonderful result:
31 The material of this section is not covered in most E&M textbooks, and will not be used in later sections of this course. Thus the “only” loss due to the reader’s skipping this section would be the lack of familiarity with one of the most fascinating fields of physics. Note also that we already have virtually all formal tools necessary for its discussion, so reading this section should not require much effort.
32 Due to the Meissner-Ochsenfeld effect, the exact path of the contour is not important, and we may discuss
just as the magnetic flux through the ring.
Chapter 6
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Magnetic
2
flux
n ,
where
, with n ,0 ,1 ,...
2
,
(6.62)
0
0
quantization
q
saying that the magnetic flux inside any superconducting loop can only take values multiple of the flux quantum 0. This effect, predicted in 1950 by the same Fritz London (who expected q to be equal to the electron charge – e), was observed experimentally in 1961,33 but with q = 2 e – so 0 2.0710-15 Wb.
Historically, this observation gave decisive support to the BCS theory of superconductivity (implying Cooper pairs with charge q = –2 e) that had been put forward just four years earlier.
Note the truly macroscopic character of this quantum effect: it has been repeatedly observed in human-scale superconducting loops, and from what is known about superconductors, there is no doubt that if we had made a giant superconducting wire loop extending, say, over the Earth’s equator, the magnetic flux through it would still be quantized – though with a very large flux quanta number n. This means that the quantum coherence of Bose-Einstein condensates may extend over, using H. Casimir’s famous expression, “miles of dirty lead wire”. (Lead is a typical superconductor, with T c 7.2 K, and indeed retains its superconductivity even being highly contaminated by impurities.) Moreover, hollow rings are not entirely necessary for flux quantization. In 1957, A. Abrikosov explained the counter-intuitive high-field behavior of superconductors with L > 2, known experimentally as their mixed (or “Shubnikov”) phase since the 1930s. He showed that a sufficiently high magnetic field may penetrate such superconductors in the form of self-formed magnetic field
“threads” (or “tubes”) surrounded by vortex-shaped supercurrents – the so-called Abrikosov vortices. In the simplest case, the core of such a vortex is a straight line, on which the superconductivity is completely suppressed ( = 0), surrounded by circular, axially-symmetric, persistent supercurrents j(), where is the distance from the vortex axis – see Fig. 5a. At the axis, the current vanishes, and with the growth of , it first rises and then falls (with j() = 0), reaching its maximum at ~ , while the magnetic field B(), directed along the vortex axis, is largest at = 0, and drops monotonically at distances of the order of L (Fig. 5b).
(a)
(b)
B
Fig. 6.5. The Abrikosov vortex:
B
j
(a) a 3D structure’s sketch, and
j
(b) the main variables as
functions of the distance
0
from the axis (schematically).
L
The total flux of the field equals exactly one flux quantum 0, given by Eq. (62).
Correspondingly, the wavefunction’s phase performs just one 2 revolution along any contour drawn around the vortex’s axis, so = n/, where n is the azimuthal unit vector.34 This topological feature of the wavefunction’s phase is sometimes called fluxoid quantization – to distinguish it from 33 Independently and virtually simultaneously by two groups: B. Deaver and W. Fairbank, and R. Doll and M.
Näbauer; their reports were published back-to-back in the same issue of the Physical Review Lett ers.
34 The last (perhaps, evident) expression for follows from MA Eq. (10.2) with f = + const.
Chapter 6
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magnetic flux quantization, which is valid only for relatively large contours, not approaching the axis by distances ~L.
A quantitative analysis of Abrikosov vortices requires, besides the equations we have discussed, one more constituent relation that would describe the suppression of the number of Cooper pairs (quantified by 2) by the magnetic field – or rather by the field-induced supercurrent. In his original work, Abrikosov used for this purpose the famous Ginzburg-Landau equation,35 which is quantitatively valid only at T T c. The equation may be conveniently represented in either of the following two forms: 2
1
q
i qA2
2
2
a
b ,
* i A
2
1 2
, (6.63)
2 m
where a and b are certain temperature-dependent coefficients, with a 0 at T T c. The first of these forms clearly shows that the Ginzburg-Landau equation (as well as the similar Gross-Pitaevskii equation describing electrically-neutral Bose-Einstein condensates) belongs to a broader class of nonlinear Schrödinger equations, differing from the usual Schrödinger equation, which is linear in , only by the additional nonlinear terms. The equivalent, second form of Eq. (63) is more convenient for applications and shows more clearly that if the superconductor’s condensate density, proportional to 2, is suppressed only locally, it self-restores to its unperturbed value (with 2 = 1) at the distances of the order of the coherence length /(2 ma)1/2.
This fact enables a simple quantitative analysis of the Abrikosov vortex in the most important limit << L. Indeed, as Fig. 5 shows, in this case. 2 = 1 at most distances ( ~ L) where the field and current are distributed, so these distributions may be readily calculated without any further involvement of Eq. (63), just from Eq. (54) with = n/ , and the Maxwell equations (21) for the magnetic field, giving B = j, and B = 0. Indeed, combining these equations just as this was done at the derivation of Eq. (23), for the only Cartesian component of the vector B(r) = B()n z (where the z-axis is directed along the vortex’ symmetry axis), we get a simple equation
22 B B
ρ
,
(6.64)
L
02 ,
at
q
which coincides with Eq. (56) at all regular points 0. Spelling out the Laplace operator for our current case of axial symmetry,36 we get an ordinary differential equation,
d dB
2 1
B ,
0
for 0 .
(6.65)
L
d
d
Comparing this equation with Eq. (2.155) with = 0, and taking into account that we need the solution decreasing at , making any contribution proportional to the function I 0 unacceptable, we get 35 This equation was derived by Vitaly Lazarevich Ginzburg and Lev Davidovich Landau from phenomenological arguments in 1950, i.e. before the advent of the “microscopic” BSC theory, and may be used for simple analyses of a broad range of nonlinear effects in superconductors. The Ginzburg-Landau and Gross-Pitaevskii equations will be further discussed in SM Sec. 4.3.
36 See, e.g., MA Eq. (10.3) with / = / z = 0.
Chapter 6
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B CK
(6.66)
0
L
– see the plot of this Bessel function on the right panel of Fig. 2.22 (black line). The constant C should be calculated by fitting the 2D delta function on the right-hand side of Eq. (64), i.e. by requiring
B 2
d 2 B
2
d 2 C K d
.
(6.67)
L
0
0
vortex
0
0
The last, dimensionless integral equals 1,37 so finally
0
B
K
,
at .
(6.68)
2 0
2
L
L
So the magnetic field of the vortex drops exponentially at distances much larger than L, and diverges at 0 – see, e.g., the second of Eqs. (2.157). However, this divergence is very slow (logarithmic), and, as was repeatedly discussed in this series, is avoided by the account of virtually any other factor. In our current case, this factor is the decrease of 2 to zero at ~ (see Fig. 5), not taken into account in Eq. (68). As a result, we may estimate the field on the axis of the vortex as
B0
0
L
ln
;
(6.69)
2
2
L
the exact (and much more involved) solution of the problem confirms this estimate with a minor correction: ln(L/) ln(L/) – 0.28, i.e. 1.3.
The current density distribution may be now calculated from the Maxwell equation B = j, giving j = j()n, with38
1 B
0
0
j
K
K
,
at ,
(6.70)
2
2
0
L
L
3 1
2
L
L
where the same identity (2.158), with Jn Kn and n = 1, was used. Now looking at Eqs. (2.157) and (2.158), with n = 1, we see that the supercurrent’s density is exponentially low at >> L (thus outlining the vortex’ periphery), and is proportional to 1/ within the broad range << << L. This rise of the current at 0 (which could be readily predicted directly from Eq. (54) with = n/, and the A-
term negligible at << L) is quenched at ~ by a rapid drop of the factor 2 in the same Eq. (54), i.e. by the suppression of the superconductivity near the axis (by the same supercurrent!) – see Fig. 5
again.
This structure of the Abrikosov vortex may be used to calculate, in a straightforward way, its energy per unit length (i.e. its linear tension)
37 This fact follows, for example, from the integration of both sides of Eq. (2.143) (which is valid for any Bessel functions, including Kn) with n = 1, from 0 to , and then using the asymptotic values given by Eqs. (2.157)-
(2.158): K 1() = 0, and K 1() 1/ at 0.
38 See, e.g., MA Eq. (10.5), with f = f = 0, and fz = B().
Chapter 6
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2
U
0
L
T
ln
,
(6.71)
l
2
4
L
and hence the so-called “first critical” value H c1 of the external magnetic field,39 at which the vortex formation becomes possible (in a long cylindrical sample parallel to the field):
0
L
T
H
ln
.
(6.72)
1
c
4
0
2
L
Let me leave the proof of these two formulas for the reader’s exercise.
The flux quantization and the Abrikosov vortices discussed above are just two of several macroscopic quantum effects in superconductivity. Let me discuss just one more, but perhaps the most interesting of such effects. Let us consider a superconducting ring/loop interrupted with a very narrow slit (Fig. 4b). Integrating Eq. (54) along any current-free path from point 1 to point 2 (see, e.g., dashed line in Fig. 4b), we get
2
q
q
0 A dr
.
Φ
(6.73)
2
1
1
Using the flux quantum definition (62), this result may be rewritten as
2
Josephson
,
(6.74) phase
1
2
difference
0
where is called the Josephson phase difference. Note that in contrast to each of the phases 1,2, their difference is gauge-invariant: Eq. (74) directly relates it to the gauge-invariant magnetic flux .
Can this be measured? Yes, for example, using the Josephson effect.40 Let us consider two (for the argument simplicity, similar) superconductors, connected with some sort of weak link, for example, a small tunnel junction, or a point contact, or a narrow thin-film bridge, through which a weak Cooper-pair supercurrent can flow. (Such a system of two weakly coupled superconductors is called a Josephson junction.) Let us think about what this supercurrent I may be a function of. For that, reverse thinking is helpful: let us imagine that we change the current; what parameter of the superconducting condensate can it affect? If the current is very weak, it cannot perturb the superconducting condensate’s density, proportional to 2; hence it may only change the Cooper condensate phases 1,2. However, according to Eq. (53), the phases are not gauge-invariant, while the current should be. Hence the current may affect (or, if you like, may be affected by) only the phase difference defined by Eq. (74). Moreover, just has already been argued during the flux quantization discussion, a change of any of 1,2 (and hence of ) by 2 or any of its multiples should not change the current. Also, if the wavefunction is the same in both superconductors ( = 0), the supercurrent should vanish due to the system’s symmetry. Hence the function I() should satisfy the following conditions:
39 This term is used to distinguish H c1 from the higher “second critical field” H c2, at which the Abrikosov vortices are pressed to each other so tightly (to distances d ~ ) that they merge, and the remains of superconductivity vanish: 0. Unfortunately, I do not have time/space to discuss these effects; the interested reader may be referred, for example, to Chapter 5 of M. Tinkham’s monograph cited above.
40 It was predicted in 1961 by Brian David Josephson (then a PhD student!) and observed experimentally by several groups soon after that.
Chapter 6
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I(0) =0,
I( + 2) = I().
(6.75)
With these conditions on hand, we should not be terribly surprised by the following Josephson’s result that for the weak link provided by tunneling,41
Josephson
(super)current
I
( ) I sin ,
(6.76)
c
where constant I c, which depends on the weak link’s strength and temperature, is called the critical current. Actually, Eqs. (54) and (63) enable not only a straightforward calculation of this relation but even obtaining a simple expression of the critical current I c via the link’s normal-sate resistance – the task left for the (creative :-) reader’s exercise.
Now let us see what happens if a Josephson junction is placed into the gap in a superconductor loop – see Fig. 4c. In this case, we may combine Eqs. (74) and (76), getting
Macroscopic
quantum
interference
I I sin 2
.
(6.77)
c
0
This effect of a periodic dependence of the current on the magnetic flux is called macroscopic quantum interference,42 while the system shown in Fig. 4c, the superconducting quantum interference device –
SQUID (with all letters capitalized, please :-). The low value of the magnetic flux quantum 0, and hence the high sensitivity of to external magnetic fields, allows using such SQUIDs as ultrasensitive magnetometers. Indeed, for a superconducting ring of area ~1 cm2, one period of the change of the supercurrent (77) is produced by a magnetic field change of the order of 10-11 T (10-7 Gs), while sensitive electronics allows measuring a tiny fraction of this period – limited by thermal noise at a level of the order of a few fT. Such sensitivity allows measurements, for example, of the miniscule magnetic fields induced outside of the body by the beating human heart, and even by brain activity.43
An important aspect of quantum interference is the so-called Aharonov-Bohm (AB) effect –
which actually takes place for single quantum particles as well.44 Let the magnetic field lines be limited to the central, hollow part of the SQUID loop so that no appreciable magnetic field ever touches the ring itself. (This may be done experimentally with very good accuracy, for example using high- magnetic cores – see their discussion in Sec. 5.6.) As predicted by Eq. (77), and confirmed by several careful experiments carried out in the mid-1960s,45 this restriction does not matter – the interference is observed 41 For some other types of weak links, the function I() may deviate from the sinusoidal form Eq. (76) rather considerably, while still satisfying the general conditions (75).
42 The name is due to a deep analogy between this phenomenon and the interference between two coherent waves, to be discussed in detail in Sec. 8.4.
43 Other practical uses of SQUIDs include MRI signal detectors, high-sensitive measurements of magnetic properties of materials, and weak field detection in a broad variety of physical experiments – see, e.g., J. Clarke and A. Braginski (eds.), The SQUID Handbook, vol. II, Wiley, 2006. For a comparison of these devices with other sensitive magnetometers see, e.g., the review collection by A. Grosz et al. (eds.), High Sensitivity Magnetometers, Springer, 2017.
44 For a more detailed discussion of the AB effect see, e.g., QM Sec. 3.2.
45 Similar experiments have been carried out with single (unpaired) electrons – moving either ballistically, in vacuum, or in “normal” (non-superconducting) conducting rings. In the last case, the effect is much harder to observe than in SQUIDs: the ring size has to be very small, and temperature very low, to avoid the so-called Chapter 6
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anyway. This means that not only the magnetic field B but also the vector potential A represents physical reality, albeit in a quite peculiar way – remember the gauge transformation (5.46), which you may carry out in your head, without changing any physical reality? (Fortunately, this transformation does not change the contour integral participating in Eq. (5.65), and hence the magnetic flux , and hence the interference pattern.)
Actually, the magnetic flux quantization (62) and the macroscopic quantum interference (77) are not completely different effects, but just two manifestations of the interrelated macroscopic quantum phenomena. To show that, one should note that if the critical current I c (or rather its product by the loop’s self-inductance L) is high enough, the flux in the SQUID loop is due not only to the external magnetic field flux ext but also has a self-field component – cf. Eq. (5.68):46
Φ Φ
LI,
Φ
where
(
)
.
(6.78)
ext
ext
B d 2 r
ext n
S
Now the relation between and ext may be readily found by solving this equation together with Eq.
(77). Figure 6 shows this relation for several values of the dimensionless parameter 2 LI c/0.
2
2
c
LI
3
.
0
3
10
0
1
0
0
Fig. 6.6. The function (ext) for SQUIDs
with various values of the normalized LI c
product. Dashed arrows show the flux
1
leaps as the external field is changed. (The
1
0
1
2
3
branches with d/ dext < 0 are unstable.)
Φ / Φ
ext
0
These plots show that if the critical current (and/or the inductance) is low, << 1, the self-field effects are negligible, and the total flux follows the external field (i.e., ext) faithfully. However, at > 1, the function (ext) becomes hysteretic, and at >> 1, its stable (positive-slope) branches are nearly flat, with the total flux values corresponding to Eq. (62). Thus, a superconducting ring closed with a high- I c Josephson junction exhibits a nearly-perfect flux quantization.
The self-field effects described by Eq. (78) create certain technical problems for SQUID
magnetometry, but they are the basis for one more useful application of these devices: ultrafast dephasing effects due to unavoidable interactions of the electrons with their environment – see, e.g., QM Chapter 7.
46 The sign before LI would be positive, as in Eq. (5.70), if I was the current flowing into the inductance.
However, in order to keep the sign in Eq. (76) intact, I should mean the current flowing into the Josephson junction, i.e. from the inductance, thus changing the sign of the LI term in Eq. (78).
Chapter 6
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computing. Indeed, Fig. 6 shows that at the values of modestly above 1 (e.g., 3), and within a certain range of applied field, the SQUID has two stable flux states, which differ by 0 and may be used for coding binary 0 and 1. For practical superconductors (like Nb), the time of switching between these states (see dashed arrows in Fig. 4) is of the order of a picosecond, while the energy dissipated at such event may be as low as ~10-19 J. (This bound is determined not by device’s physics, by the fundamental requirement for the energy barrier between the two states to be much higher than the thermal fluctuation energy scale k B T, ensuring a sufficiently long information retention time.) While the picosecond switching speed may be also achieved with some semiconductor devices, the power consumption of the SQUID-based digital devices may be 5 to 6 orders of magnitude lower, enabling large-scale digital integrated circuits with 100-GHz-scale clock frequencies. Unfortunately, the range of practical applications of these Rapid Single-Flux-Quantum (RSFQ) digital circuits is still very narrow, due to the inconvenience of their deep refrigeration to temperatures below T c.47
Since we have already got the basic relations (74) and (76) describing the macroscopic quantum phenomena in superconductivity, let me mention in brief two other prominent members of this group, called the dc and ac Josephson effects. Differentiating Eq. (74) over time, and using the Faraday induction law (2), we get48
Josephson
d
e
2
phase-to-
V .
(6.79)
voltage
dt
relation
This famous Josephson phase-to-voltage relation should be valid regardless of the way how the voltage V has been created,49 so let us apply Eqs. (76) and (79) to the simplest circuit with a non-superconducting source of dc voltage – see Fig. 7.
1
2
I ( t)
Fig. 6.7. DC-voltage-biased
Josephson junction.
V
If the current’s magnitude is below the critical value, Eq. (76) allows phase to have the time-independent value
I
1
sin
,
if -I I I ,
(6.80)
c
c
I c
and hence, according to Eq. (79), a vanishing voltage drop across the junction: V = 0. This dc Josephson effect is not quite surprising – indeed, we have postulated from the very beginning that the Josephson junction may pass a certain supercurrent. Much more fascinating is the so-called ac Josephson effect that occurs if the voltage across the junction has a non-zero average (dc) component V 0. For simplicity, let us 47 For more on that technology, see, e.g., the review paper by P. Bunyk et al., Int. J. High Speed Electron. Syst.
11, 257 (2001), and references therein.
48 Since the induced e.m.f. Vind cannot drop on the superconducting path between the Josephson junction electrodes 1 and 2 (see Fig. 4c), it should be equal to (- V), where V is the voltage across the junction.
49 Indeed, it may be also obtained from simple Schrödinger-equation-based arguments – see, e.g., QM Sec. 1.6.
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assume that this is the only voltage component: V( t) = V 0 = const;50 then Eq. (79) may be easily integrated to give = J t + 0, where
2 e
Josephson
V .
(6.81)
J
0
oscillation
frequency
This result, plugged into Eq. (76), shows that the supercurrent oscillates,
I I sin t ,
(6.82)
c
J
0
with the so-called Josephson frequency J (81) proportional to the applied dc voltage. For practicable voltages (above the typical noise level), the frequency f J = J/2 corresponds to the GHz or even THz ranges, because the proportionality coefficient in Eq. (81) is very high: f J/ V 0 = e/ 483 MHz/V.51
An important experimental fact is the universality of this coefficient. For example, in the mid-1980s, a Stony Brook group led by J. Lukens proved that this factor is material-independent with a relative accuracy of at least 10-15. Very few experiments, especially in solid-state physics, have ever reached such precision. This fundamental nature of the Josephson voltage-to-frequency relation (81) allows an important application of the ac Josephson effect in metrology. Namely, phase-locking52 the Josephson oscillations with an external microwave signal from an atomic frequency standard, one can get a more precise dc voltage than from any other source. In NIST and other metrological institutions around the globe, this effect is used for the calibration of simpler “secondary” voltage standards that can operate at room temperature.
6.6. Inductors, transformers, and ac Kirchhoff laws
Let a wire coil (meaning either a single loop illustrated in Fig. 5.4b or a series of such loops, such as one of the solenoids shown in Fig. 5.6) have a self-inductance L much larger than that of the wires connecting it to other components of our system: ac voltage sources, voltmeters, etc. (Since, according to Eq. (5.75), L scales as the square of the number N of wire turns, this condition is easier to satisfy at N >> 1.) Then in a quasistatic system consisting of such lumped induction coils, external wires, and other lumped circuit elements such as resistors, capacitances, etc., we may neglect the electromagnetic induction effects everywhere outside the coil, so the electric field in those external regions is potential. Then the voltage V between the coil’s terminals may be defined, just as in electrostatics, as the difference of values of between the terminals, i.e. as the integral V E r
d
(6.83)
between the coil terminals along any path outside the coil. This voltage has to be balanced by the induction e.m.f. (2) in the coil, so if the Ohmic resistance of the coil is negligible, we may write 50 In experiment, this condition is hard to implement, due to the relatively high inductances of the current leads providing the dc voltage supply. However, this technical complication does not affect the main conclusion of the simple analysis described here.
51 This 1962 prediction (by the same B. Josephson) was confirmed experimentally – in 1963 indirectly, by phase-locking of the oscillations (82) with an external microwave signal, and in 1967 explicitly, by the direct detection of the emitted microwave radiation.
52 For a discussion of this very important (and general) effect, see, e.g., CM Sec. 5.4.
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d
V
,
(6.84)
dt
where is the magnetic flux in the coil.53 If the flux is due to the current I in the same coil only (i.e. if it is magnetically uncoupled from other coils), we may use Eq. (5.70) to get the well-known relation Voltage
drop on
dI
inductance
V L
,
(6.85)
coil
dt
where compliance with the Lenz sign rule is achieved by selecting the relations between the assumed voltage polarity and the current direction as shown in Fig. 8a.
(a)
(b)
(c)
I
M
( t)
Fig. 6.8. Some lumped ac circuit
elements: (a) an induction coil,
V
L
L
L
N
N
1
2
1
2
(b) two inductively coupled
coils, and (c) an ac transformer.
If similar conditions are satisfied for two magnetically coupled coils (Fig. 8b), then, in Eq. (84), we need to use Eqs. (5.69) instead, getting
dI
dI
dI
dI
V L
1 M
2 ,
V L
2 M
1 .
(6.86)
1
1 dt
dt
2
2 dt
dt
Such systems of inductively coupled coils have numerous applications in electrical engineering and physical experiment. Perhaps the most important of them is the ac transformer, in which the coils share a common soft-ferromagnetic core of the toroidal (“doughnut”) topology – see Fig. 8c.54 As we already know from the discussion in Sec. 5.6, such cores, with >> 0, “try” to absorb all magnetic field lines, so the magnetic flux ( t) in the core is nearly the same in each of its cross-sections. With this, Eq. (84) yields
d
d
V N
, V N
,
(6.87)
1
1
2
2
dt
dt
so the voltage ratio is completely determined by the ratio N 1/ N 2 of the number of wire turns.
Now we may generalize, to the ac current case, the Kirchhoff laws already discussed in Sec. 4.1
– see Fig. 4.3 reproduced in Fig. 9a below. Let not only inductances but also capacitances and resistances of the wires be negligible in comparison with those of the lumped (compact) circuit elements, whose list now would include not only resistors and current sources (as in the dc case), but also the induction coils (including magnetically coupled ones) and capacitors – see Fig. 9b. In the quasistatic approximation, the current flowing in each wire is conserved, so the “node rule”, i.e. the 1st Kirchhoff law (4.7a),
53 If the resistance is substantial, it may be represented by a separate lumped circuit element (resistor) connected in series with the coil.
54 The first practically acceptable form of this device, called the Stanley transformer, was invented in 1886. In it, multi-turn windings could be easily mounted onto a toroidal ferromagnetic (at that time, silicon-steel-plate) core.
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I 0 .
(6.88a)
j
j
remains valid. Also, if the electromagnetic induction effect is restricted to the interior of lumped induction coils as discussed above, the voltage drops Vk across each circuit element may be still represented, just as in dc circuits, with differences between the adjacent node potentials. As a result, the
“loop rule”, i.e. 2nd Kirchhoff law (4.7b),
V 0,
(6.88b)
k
k
is also valid. Now, in contrast to the dc case, Eqs. (88) may be the (ordinary) differential equations.
However, if all circuit elements are linear (as in the examples presented in Fig. 9b), these equations may be readily reduced to linear algebraic equations, using the Fourier expansion. (In the common case of sinusoidal ac sources, the final stage of the Fourier series summation is unnecessary.) (a)
(b)
“circuit
“wire”
element”
“node”
~
dI
1
V L
V RI
V
Idt V V ( t)
dt
C
Fig. 6.9. (a) A typical quasistatic ac circuit obeying the
Kirchhoff laws, and (b) the simplest lumped circuit
“loop”
elements.
My teaching experience shows that the potential readers of these notes are well familiar with the application of Eqs. (88) to such problems from their undergraduate studies, so I will save time/space by skipping discussions of even the simplest examples of such circuits, such as LC, LR, RC, and LRC loops and periodic structures.55 However, since such problems are very important for practice, my sincere advice to the reader is to carry out a self-test by solving a few problems of this type, provided in Sec. 9
below, and if they cause any difficulty, pursue some remedial reading.
6.7. Displacement currents
Electromagnetic induction is not the only new effect arising in non-stationary electrodynamics.
Indeed, though Eqs. (21) are adequate for the description of quasistatic phenomena, a deeper analysis shows that one of these equations, namely H = j, cannot be exact. To see that, let us take the divergence of both sides:
H j .
(6.89)
But, as the divergence of any curl,56 the left-hand side should equal zero. Hence we get 55 Curiously enough, these effects include wave propagation in periodic LC circuits, even within the quasistatic approximation! However, the speed 1/( LC)1/2 of these waves in lumped circuits is much lower than the speed 1/()1/2 of electromagnetic waves in the surrounding medium – see Sec. 8 below.
56 Again, see MA Eq. (11.2) – if you need it.
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j 0 .
(6.90)
This is fine in statics, but in dynamics, this equation forbids any charge accumulation, because according to the continuity relation (4.5),
j
.
(6.91)
t
This discrepancy had been recognized by James Clerk Maxwell who suggested, in the 1860s, a way out of this contradiction. If we generalize the equation for H by adding to the term j (that describes the density of real electric currents) the so-called displacement current density term, Displacement
current
D
j
,
(6.92)
d
density
t
(which of course vanishes in statics), then the equation takes the form
D
H j j j
.
(6.93)
d
t
In this case, due to the equation (3.22), D = , the divergence of the right-hand side equals zero due to the continuity equation (92), and the discrepancy is removed. This incredible theoretical feat,57
confirmed by the 1886 experiments carried out by Heinrich Hertz (see below) was perhaps the main triumph of theoretical physics of the 19th century.
Maxwell’s displacement current concept, expressed by Eq. (93), is so important that it is worthwhile to have one more look at its derivation using a particular model shown in Fig. 10.58
C
Q
Q
I
I
S
Fig. 6.10. The Ampère law applied
1
S
to capacitor recharging.
2
D
Neglecting the fringe field effects, we may use Eq. (4.1) to describe the relationship between the current I flowing through the wires and the electric charge Q of the capacitor:59
dQ
I .
(6.94)
dt
57 It looks deceivingly simple now – after the fact, and with the current mathematical tools (especially the del operator), which are much superior to those that were available to J. Maxwell.
58 No physicist should be ashamed of doing this. For example, J. Maxwell’s main book, A Treatise of Electricity and Magnetism, is full of drawings of plane capacitors, inductance coils, and voltmeters. More generally, the whole history of science teaches us that snobbery regarding particular examples and practical systems is a virtually certain path toward producing nothing of either practical value or fundamental importance.
59 This is of course just the integral form of the continuity equation (91).
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Now let us consider a closed contour C drawn around the wire. (Solid points in Fig. 10 show the places where the contour intercepts the plane of the drawing.) This contour may be seen as the line limiting either surface S 1 (crossed by the wire) or surface S 2 (avoiding such crossing by passing through the capacitor’s gap). Applying the macroscopic Ampère law (5.116) to the former surface, we get
H d j d 2
r
r I ,
(6.95)
n
C
S 1
while for the latter surface the same law gives a different result,
H d j d 2
r
r 0 ,
[WRONG!]
(6.96)
n
C
S 2
for the same integral. This is just an integral-form manifestation of the discrepancy outlined above, but it shows clearly how serious the problem is (or rather it was – before Maxwell).
Now let us see how the introduction of the displacement currents saves the day, considering for the sake of simplicity a plane capacitor of area A, with a small and constant electrode spacing. In this case, as we already know, the field inside it is uniform, with D = , so the total capacitor’s charge Q =
A = AD, and the current (94) may be represented as
dQ
dD
I
A
.
(6.97)
dt
dt
So, instead of the wrong Eq. (96), the Ampère law modified following Eq. (93), gives D
n
dD
H d ( j )
2
2
d r
d r
A I,
r
(6.98)
d n
t
dt
C
S
S
2
2
i.e. the Ampère integral becomes independent of the choice of the surface limited by the contour C – as it has to be.
6.8. Finally, the full Maxwell equation system
.
This is a very special moment in this course: with the displacement currents in, i.e. with the replacement of Eq. (5.107) with Eq. (93), we have finally arrived at the full set of macroscopic Maxwell equations for time-dependent fields,60
B
D
E
,
0
H
, j
(6.99a)
Macroscopic
t
t
Maxwell
equations
D ,
B ,
0
(6.99b)
whose validity has been confirmed by an enormous body of experimental data. Indeed, despite numerous efforts, no other corrections (e.g., additional terms) to the Maxwell equations have been ever found, and these equations are still considered exact within the range of their validity, i.e. while the electric and magnetic fields may be considered classically. Moreover, even in quantum theory, these 60 This vector form of the Maxwell equations, magnificent in its symmetry and simplicity, was developed in 1884-85 by Oliver Heaviside, with substantial contributions by H. Lorentz. (The original Maxwell’s result circa 1864 looked like a system of 20 equations for Cartesian components of the vector and scalar potentials.) Chapter 6
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equations are believed to be strictly valid as relations between the Heisenberg operators of the electric and magnetic fields.61 (Note that the microscopic Maxwell equations for the genuine fields E and B may be formally obtained from Eqs. (99) by the substitutions D = 0E and H = B/0, and the simultaneous replacement of the stand-alone charge and current densities on their right-hand sides with the full ones.) Perhaps the most striking feature of these equations is that, even in the absence of stand-alone charges and currents inside the region of our interest, when the equations become fully homogeneous,
B
D
E
,
H
,
(6.100a)
t
t
D ,
0
B ,
0
(6.100b)
they still describe something very non-trivial: electromagnetic waves, including light. The physics of the waves may be clearly seen from Eqs. (100a): according to the first of them, the change of the magnetic field in time creates a vortex-like (divergence-free) electric field. On the other hand, the second of Eqs.
(100a) describes how the changing electric field, in turn, creates a vortex-like magnetic field. So-coupled electric and magnetic fields may propagate as waves – even very far from their sources.
We will carry out a detailed quantitative analysis of the waves in the next chapter, and here I will only use this notion to make good on the promise given in Sec. 3, namely to establish the condition of validity of the quasistatic approximation (21). For simplicity, let us consider an electromagnetic wave with a time period T, velocity v, and hence the wavelength62 = vT in a linear medium with D = E, B =
H. Then the magnitude of the left-hand side of the first of Eqs. (100a) is of the order of E/ = E/ vT, while that of its right-hand side may be estimated as B/ T ~ H/ T. Using similar estimates for the second of Eqs. (100a), we arrive at the following two requirements:63
E
1
~ v
~
.
(6.101)
H
v
To ensure the compatibility of these two relations, the wave’s speed should satisfy the estimate 1
v ~
,
(6.102)
1/2
reduced to v ~ 1/(00)1/2 c in free space, while the ratio of the electric and magnetic field amplitudes should be of the following order:
1/ 2
E
1
~ v ~
.
(6.103)
H
1/ 2
(In the next chapter we will see that for plane electromagnetic waves, these results are exact.) Now, let a system of a linear size ~ a carry currents producing a certain magnetic field H. Then, according to Eqs. (100a), their magnetic field Faraday-induces the electric field of magnitude E ~
Ha/ T, whose displacement currents, in turn, produce an additional magnetic field with magnitude 61 See, e.g., QM Chapter 9.
62 Let me hope the reader knows that the relation = vT is universal and valid for waves of any nature – see, e.g., CM Chapter 6. (In the case of substantial dispersion, v means the phase velocity.) 63 The fact that T has canceled, shows that these estimates are valid for waves of any frequency.
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a
a a
a
2
a 2
H' ~
E ~
H
H H .
(6.104)
T
T T
vT
Hence, the displacement current effects are negligible for a system of size a << .64
In particular, the quasistatic picture of the skin effect, discussed in Sec. 3, is valid while the skin depth (33) remains much smaller than the corresponding wavelength,
1/ 2
2
2 v
4
vT
.
(6.105)
2
The wavelength decreases with the frequency as 1/, i.e. faster than s 1/1/2, so they become comparable at the crossover frequency
,
(6.106)
r
0
which is nothing else than the reciprocal charge relaxation time (4.10). As was discussed in Sec. 4.2, for good metals this frequency is extremely high (about 1018 s-1), so the validity of Eq. (33) is typically limited by the anomalous skin effect (which was briefly discussed in Sec. 3), rather than the wave effects.
Before going after the analysis of the full Maxwell equations for particular situations (that will be the main goal of the next chapters of this course), let us have a look at the energy balance they yield for a certain volume V, which may include both some charged particles and the electromagnetic field.
Since, according to Eq. (5.10), the magnetic field performs no work on charged particles even if they move, the total power P being transferred from the field to the particles inside the volume is due to the electric field alone – see Eq. (4.38):
P p 3
d r,
with j E
p
,
(6.107)
V
Expressing j from the corresponding Maxwell equation of the system (99), we get
D
P
E ( H)
3
E
d r.
(6.108)
t
V
Let us pause here for a second, and transform the divergence of EH, using the well-known vector algebra identity:65
E H H E E H.
(6.109)
The last term on the right-hand side of this equality is exactly the first term in the square brackets of Eq.
(108), so we may rewrite that formula as
D
P
E H H E
3
E
d r.
(6.110)
t
V
64 Let me emphasize that if this condition is not fulfilled, the lumped-circuit representation of the system (see Fig.
9 and its discussion) is typically inadequate – besides some special cases, to be discussed in the next chapter.
65 See, e.g., MA Eq. (11.7) with f = E and g = H.
Chapter 6
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However, according to the Maxwell equation for E, this curl is equal to –B/ t, so the second term in the square brackets of Eq. (110) equals –HB/ t and, according to Eq. (14), is just the (minus) time derivative of the magnetic energy per unit volume. Similarly, according to Eq. (3.76), the third term under the integral is the (minus) time derivative of the electric energy per unit volume. Finally, we can use the divergence theorem to transform the integral of the first term in the square brackets to a 2D
integral over the surface S limiting the volume V. As a result, we get the so-called Poynting theorem 66
for the power balance in the system:
Poynting
u
3
2
theorem
p
d r S d r 0
.
(6.111)
t
n
V
S
Here u is the density of the total (electric plus magnetic) energy of the electromagnetic field, with Field’s
energy
u E D
H B
(6.112)
variation
– just the sum of the expressions given by Eqs. (3.76) and (14). For the particular case of an isotropic, linear, and dispersion-free medium, with D( t) = E( t), B( t) = H( t), Eq. (112) yields Field’s
E D
H B
2
2
E
B
energy
u
.
(6.113)
2
2
2
2
Another key notion participating in Eq. (111) is the Poynting vector, defined as67
Poynting
vector
S E H .
(6.114)
The first integral in Eq. (111) is evidently the net change of the energy of the system (particles + field) per unit time, so the second (surface) integral has to be the power flowing out from the system through the surface. As a result, it is tempting to interpret the Poynting vector S locally, as the power flow density at the given point. In many cases, such a local interpretation of vector S is legitimate; however, in other cases, it may lead to wrong conclusions. Indeed, let us consider the simple system shown in Fig.
11: a charged plane capacitor placed into a static and uniform external magnetic field, so that the electric and magnetic fields are mutually perpendicular.
S
E
B S
Fig. 6.11. The Poynting vector paradox.
In this static situation, with no charges moving, both p and / t are equal to zero, and there should be no power flow in the system. However, Eq. (114) shows that the Poynting vector is not equal 66 It is named after John Henry Poynting for his work published in 1884, though this fact was independently discovered by O. Heaviside in 1885 in a simpler form, while a similar result for the intensity of mechanical elastic waves had been obtained earlier (in 1874) by Nikolay Alekseevich Umov – see, e.g., CM Sec. 7.7.
67 Actually, an addition to S of the curl of an arbitrary vector function f(r, t) does not change Eq. (111). Indeed, we may use the divergence theorem to transform the corresponding change of the surface integral in Eq. (111) to a volume integral of scalar function (f) that equals zero at any point – see, e.g., MA Eq. (11.2).
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to zero inside the capacitor, being directed as the red arrows in Fig. 11 show. From the point of view of the only unambiguous corollary of the Maxwell equations, Eq. (111), there is no contradiction here, because the fluxes of the vector S through the side boundaries of the volume shaded in Fig. 11 are equal and opposite (and they are zero for other faces of this rectilinear volume), so the total flux of the Poynting vector through the volume boundary equals zero, as it should. It is, however, useful to recall this example each time before giving a local interpretation of the vector S.
The paradox illustrated in Fig. 11 is closely related to the radiation recoil effects, due to the electromagnetic field’s momentum – more exactly, it linear momentum. Indeed, acting as at the Poynting theorem derivation, it is straightforward to use the microscopic Maxwell equations68 to prove that, neglecting the boundary effects, the vector sum of the mechanical linear momentum of the particles in an arbitrary volume, and the integral of the following vector,
Electro-
S
g
,
(6.115) magnetic
2
c
field’s
momentum
over the same volume, is conserved, enabling an interpretation of g as the density of the linear momentum of the electromagnetic field. (It will be more convenient for me to prove this relation, and discuss the related issues, in Sec. 9.8, using the 4-vector formalism of special relativity.) Due to this conservation, if some static fields coupled to mechanical bodies are suddenly decoupled from them and are allowed to propagate in space, i.e. to change their local integral of g, they give the bodies an equal and opposite impulse of force.
Finally, to complete our initial discussion of the Maxwell equations,69 let us rewrite them in terms of potentials A and , because this is more convenient for the solution of some (though not all!) problems.
Even when dealing with the system (99) of the more general Maxwell equations than discussed before, Eqs. (7) are still used for the definition of the potentials. It is straightforward to verify that with these definitions, the two homogeneous Maxwell equations (99b) are satisfied automatically. Plugging Eqs.
(7) into the inhomogeneous equations (99a), and considering, for simplicity, a linear, uniform medium with frequency-independent and , we get
2
A
2
A ,
2
A
A
. j
(6.116)
2
t
t
t
This is a more complex result than what we would like to get. However, let us select a special gauge, which is frequently called (especially for the free space case, when v = c) the Lorenz gauge condition 70
Lorenz
A
,
0
(6.117) gauge
t
condition
68 The situation with the macroscopic Maxwell equations is more complex, and is still a subject of some lingering discussions (usually called the Abraham-Minkowski controversy, despite contributions by many other scientists including A. Einstein), because of the ambiguity of the momentum’s division between its field and particle components – see, e.g., the review paper by R. Pfeiffer et al., Rev. Mod. Phys. 79, 1197 (2007).
69 We will return to their general discussion (in particular, to the analytical mechanics of the electromagnetic field, and its stress tensor) in Sec. 9.8, after we get equipped with the special relativity theory.
70 This condition, named after Ludwig Lorenz, should not be confused with the so-called Lorentz invariance condition of relativity, due to Hendrik Lorentz, to be discussed in Sec. 9.4. (Note the last names’ spelling.) Chapter 6
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which is a natural generalization of the Coulomb gauge (5.48) to time-dependent phenomena. With this condition, Eqs. (107) are reduced to a simpler, beautifully symmetric form:
Potentials’
2
1
2
2
1
2 A
dynamics
,
A
j
,
(6.118)
2
2
2
2
v t
v t
where v 2 1/. Note that these equations are essentially a set of 4 similar equations for 4 scalar functions (namely, and three Cartesian components of A) and thus clearly invite the 4-component vector formalism of the relativity theory; it will be discussed in Chapter 9.71
If and A depend on just one spatial coordinate, say z, then in a region without field sources:
= 0, j = 0, Eqs. (118) are reduced to the following 1D wave equations
2
1
2
2
A
1
2
A
,
0
0.
(6.119)
2
2
2
2
2
2
z v t
z
v t
It is well known72 that these equations describe waves, with arbitrary waveforms (including sinusoidal waves of any frequency), propagating with the same speed v in either of the z-axis directions.
According to the definitions of the constants 0 and 0, in free space, v is just the speed of light: 1
v
.
(6.120)
0
0
c
1/ 2
Historically, the experimental observation of relatively low-frequency (GHz-scale) electromagnetic waves, with their speed equal to that of light, was the decisive proof (actually, a real triumph!) of the Maxwell theory and his prediction of such waves.73 This was first accomplished in 1886 by Heinrich Rudolf Hertz, using the electronic circuits and antennas he had invented for this purpose.
Before proceeding to the detailed analysis of these waves in the following chapters, let me mention that the invariance of Eqs. (119) with respect to the wave propagation direction is not occasional; it is just a manifestation of one more general property of the Maxwell equations (99), called the Lorentz reciprocity. We have already met its simplest example, for time-independent electrostatic fields, in one of the problems of Chapter 1. In a much more general case when two monochromatic electromagnetic fields of the same frequency, with complex amplitudes, say, {E1(r), H1(r)} and {E2(r), 71 Here I have to mention in passing the so-called Hertz vector potentials e and m (whose introduction may be traced back at least to the 1904 work by E. Whittaker). They may be defined by the following relations: Π
1
e
A
Π ,
Π ,
m
e
t
which make the Lorentz gauge condition (117) automatically satisfied. These potentials are especially convenient for the solution of problems in which the electromagnetic field is induced by sources characterized by field-independent electric and magnetic polarizations P and M – rather than by field-independent charge and current densities and j. Indeed, it is straightforward to check that both e and m satisfy the equations similar to Eqs.
(118), but with their right-hand sides equal to, respectively, –P and –M. Unfortunately, I would not have time/space to discuss such problems and have to refer interested readers elsewhere – for example, to a classical text by J. Stratton, Electromagnetic Theory, Adams Press, 2008.
72 See, e.g., CM Secs. 6.3-6.4 and 7.7-7.8.
73 By that time, the speed of light (estimated very reasonably by Ole Rømer as early as 1676) has been experimentally measured, by Hippolyte Fizeau and then Léon Foucault, with an accuracy better than 1%.
Chapter 6
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H2(r)} are induced, separately, by stand-alone currents with complex amplitudes j1(r) and j2(r) of their densities. Then it may be proved74 that if the medium is linear and either isotropic or even anisotropic but with symmetric tensors jj’ and jj’, then for any volume V limited by a closed surface S,
j E j E
E H E H
.
(6.121)
1
2
2
1 d 3 r
1 2 2 1 d 2 r
n
V
S
This property implies, in particular, that the waves propagate similarly in two reciprocal directions even in situations much more general than the 1D case described by Eqs. (119). For some important practical applications (e.g., for low-noise amplifiers and detectors) such reciprocity is rather inconvenient. Fortunately, Eq. (121) may be violated in anisotropic media with asymmetric tensors jj’
and/or jj’. The simplest case of such an anisotropy, the Faraday rotation of the wave polarization in plasma, will be discussed in the next chapter.
6.9. Exercise problems
6.1. Prove that the electromagnetic induction e.m.f. Vind in a conducting loop may be measured as shown on two panels of Fig. 1:
(i) by measuring the current I = Vind/ R induced in the loop closed with an Ohmic resistor R, or (ii) using a voltmeter inserted into the loop.
6.2. The flux of the magnetic field that pierces a resistive ring
V ?
is being changed in time, while the field outside of the ring is negligibly
low. A voltmeter is connected to a part of the ring, as shown in the Φ( t)
figure on the right. What would the voltmeter show?
6.3. A weak constant magnetic field B is applied to an axially-symmetric permanent magnet with the dipole magnetic moment m directed along its axis, rapidly rotating about the same axis, with an angular momentum L. Calculate the electric field resulting from the magnetic field’s application, and formulate the conditions of your result’s validity.
6.4. The similarity of Eq. (5.53) obtained in Sec. 5.3 without any use of the Faraday induction law, and Eq. (5.54) proved in Sec. 2 of this chapter using it, implies that the law may be derived from magnetostatics. Prove that this is indeed true for a particular case of a current loop being slowly deformed in a fixed magnetic field B(r).
6.5. Could Problem 5.2 (i.e. the semi-quantitative analysis of the
I
mechanical stability of the system shown in the figure on the right) be solved
1
using potential energy arguments?
I 2
74 It will be more convenient for me to give this proof (or rather offer it for the reader’s exercise :-) in the next chapter, after we have discussed the Fourier expansion of the fields in linear media.
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6.6. Use energy arguments to calculate the pressure exerted by the magnetic field B inside a long uniform solenoid of length l, and a cross-section of area A << l 2, with N >> l/ A 1/2 >> 1 turns, on its
“walls” (windings), and the forces exerted by the field on the solenoid’s ends, for two cases: (i) the current through the solenoid is fixed by an external source, and
(ii) after the initial current setting, the ends of the solenoid’s wire, with negligible resistance, are connected, so that it continues to carry a non-zero current.
Compare the results, and give a physical interpretation of the direction of these forces.
6.7. The electromagnetic railgun is a
V
I
l
projectile launch system consisting of two
F, v
long parallel conducting rails and a sliding
(a)
conducting projectile shorting the current I
I
fed into the system by a powerful source –
v
t
(b)
see panel (a) in the figure on the right.
Calculate the force exerted on the projectile,
w
using two approaches:
(i) by a direct calculation, assuming that the cross-section of the system has the simple shape shown on panel (b) of the figure above, with t << w, l, and (ii) by using the energy balance (for simplicity, neglecting the Ohmic resistances in the system), and compare the results.
6.8. A uniform, static magnetic field B is applied along the axis of a
,
long thin pipe of a radius R and wall thickness << R, made of a material M ,
with Ohmic conductivity . A sphere of mass M and radius R’ << R, made R
R'
B
of a linear magnetic material with permeability >> 0, is launched, with v0
an initial velocity v 0, to fly ballistically along the pipe’s axis – see the figure on the right. Use the quasistatic approximation to calculate the
distance the sphere would pass before it stops. Formulate the conditions of validity of your result.
6.9. A planar thin-wire loop with inductance L, resistance R, and area A is launched to fly ballistically from field-free space into a region where the magnetic field B is constant. Calculate the final change of the kinetic energy of the loop, assuming that the time of its entry into the field region is much shorter than the relaxation time constant L/ R and that the loop cannot rotate.
6.10. AC current of frequency is being passed through a long uniform wire with a round cross-section of a radius R comparable with the skin depth s. In the quasistatic approximation, find the current’s distribution across the cross-section, and analyze it in the limits R << s and s << R. Calculate the effective ac resistance of the wire (per unit length) in these two limits.
6.11. A long round cylinder of radius R, made of a uniform conductor with an Ohmic conductivity and magnetic permeability , is placed into a uniform ac magnetic field Hext( t) =
H0cos t directed along its symmetry axis. Calculate the spatial distribution of the magnetic field’s amplitude and, in particular, its value on the cylinder’s axis. Spell out the last result in the limits of relatively small and large R.
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6.12.* Define and calculate an appropriate spatial-temporal Green’s function for Eq. (25), and then use this function to analyze the dynamics of propagation of the external magnetic field that is suddenly turned on at t = 0 and then kept constant:
H
,
0
at
t
,
0
x ,
0 t
H ,
at t ,
0
0
into an Ohmic conductor occupying the semi-space x > 0 – see Fig. 2.
Hint: Try to use a function proportional to exp{–( x– x’)2/2( x)2}, with a suitable time dependence of the parameter x and a properly selected pre-exponential factor.
6.13. Solve the previous problem using the variable separation method, and compare the results.
6.14. Calculate the average force exerted by ac current I( t) of z R'
amplitude I 0, flowing in a planar round coil of radius R, on a conducting sphere with a much smaller radius R’ (which is still much larger than the skin depth s at the ac current’s frequency), located on the loop’s axis, at distance z from its center – see the figure on the right.
0 R
I t
6.15. A small planar wire loop carrying current I is located relatively
far from a planar surface of a superconductor. Within the coarse-grain (ideal-diamagnetic) description of the Meissner-Ochsenfeld effect, calculate:
(i) the energy of the loop-superconductor interaction,
(ii) the force and torque acting on the loop, and
(iii) the distribution of supercurrents on the superconductor surface.
6.16. A straight uniform magnet of length 2 l, cross-section area A
l
2
<< l 2, and mass m, with a permanent longitudinal magnetization M 0, is placed over a horizontal surface of a superconductor – see the figure on the
M0
right. Within the ideal-diamagnet description of superconductivity, find the
g
stable equilibrium position of the magnet.
6.17. A plane superconducting wire loop of area A and
inductance L may turn, without static friction, about a horizontal axis 0
0
(in the figure on the right, normal to the plane of the drawing) passing
through its center of mass. Initially, the loop had been horizontal (with
B
= 0) and carried supercurrent I 0 in such a direction that its magnetic
dipole vector had been directed down. Then a uniform magnetic field B,
directed vertically up, was applied. Using the ideal-diamagnet description of the Meissner-Ochsenfeld effect, find all possible equilibrium positions of the loop, analyze their stability, and give a physical interpretation of the results.
6.18. Use the London equation to analyze the penetration of a uniform external magnetic field into a thin ( t ~ L) planar superconducting film whose plane is parallel to the field.
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6.19. Use the London equation to calculate the distribution of supercurrent density j inside a long straight superconducting wire with a circular cross-section of radius R ~ L, carrying current I.
6.20. Use the London equation to calculate the
w t, L
inductance (per unit length) of a long uniform superconducting
I
d
strip placed close to the surface of a similar superconductor –
L
t ~ L
see the figure on the right, which shows the structure’s cross-
section.
b c
6.21. Calculate the inductance (per unit length) of a superconducting
a
cable with the round cross-section shown in the figure on the right, in the following limits:
I
(i) L << a, b, c – b, and
(ii) a << L << b, c – b.
I
6.22. Use the London equation to analyze the magnetic field shielding by a superconducting thin film of thickness t << L, by calculating the penetration of the field induced by current I in a thin wire that runs parallel to a wide planar thin film, at a distance d >> t from it, into the space behind the film.
6.23. Assuming that the magnetic monopole does exist and has a magnetic charge q m, calculate the change I of current in a superconducting loop due to a passage of a single monopole through its area. Evaluate I for a monopole with the charge conjectured by P. Dirac, q m = nq 0 n(2/ e) with an integer n, and compare the result with the magnetic flux quantum 0 (62). Review your result for a similar passage of a single quasi-monopole magnetic charge formed at one of the ends of a permanent-magnet needle – see, e.g., Fig. 19 and the accompanying discussion.
Hint: To simplify calculations, you may consider the monopole’s passage along the symmetry axis of a round ring of radius R, made of a superconducting wire with a cross-section’s area A satisfying the conditions 2
L << A << R 2.
6.24. Use the Ginzburg-Landau equations (54) and (63) to calculate the largest (“critical”) value of supercurrent in a uniform superconducting wire with a cross-section area much smaller than 2
L .
6.25. Use the discussion of a long straight Abrikosov vortex, in the limit << L, in Sec. 5 to prove Eqs. (71)-(72) for its energy per unit length and the first critical field.
6.26.* Use the Ginzburg-Landau equations (54) and (63) to prove the Josephson relation (76) for a small superconducting weak link, and express its critical current I c via the Ohmic resistance R n of the same weak link in its normal state.
6.27. Use Eqs. (76) and (79) to calculate the coupling energy of a Josephson junction and the full potential energy of the SQUID shown in Fig. 4c.
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6.28. Analyze the possibility of wave propagation in a long
L
L
L
uniform chain of lumped inductances and capacitances – see the figure
on the right.
C
C
Hint: Readers without prior experience in electromagnetic
wave analysis may like to use a substantial analogy between this effect and mechanical waves in a 1D
chain of elastically coupled particles.75
6.29. A sinusoidal e.m.f. of amplitude V
R
R
R
0 and frequency
is applied to an end of a long chain of similar lumped resistors
V t
and capacitors, shown in the figure on the right. Calculate the
C
C
law of decay of the ac voltage amplitude along the chain.
6.30. As was discussed in Sec. 7, the displacement current concept allows one to extend the Ampère law to time-dependent processes as
H d I
D d 2
r
r .
S
n
t
C
S
We also have seen that this generalization makes the integral H dr over an external contour, such as the one shown in Fig. 10, independent of the choice of the surface S limited by the C
contour. However, it may look like the situation is different for a contour drawn
I
I
inside a capacitor – see the figure on the right. Indeed, if the contour’s size is much larger than the capacitor’s thickness, the magnetic field H created by the linear current I on the contour’s line is virtually the same as that of a continuous S
wire, and hence the integral H dr along the contour apparently does not depend on its area, while the magnetic flux Dnd 2 r does, so the equation displayed above seems invalid. (The current IS piercing this contour evidently equals zero.) Resolve this paradox, for simplicity considering an axially-symmetric system.
6.31. A straight, uniform, long wire with a circular cross-section of radius R, made of an Ohmic conductor with conductivity , carries dc current I. Calculate the flux of the Poynting vector through its surface, and compare it with the Joule rate of energy dissipation.
75 See, e.g., CM Sec. 6.3.
Chapter 6
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Chapter 7. Electromagnetic Wave Propagation
This (rather extensive) chapter focuses on the most important effect that follows from the time-dependent Maxwell equations, namely the electromagnetic waves, at this stage avoiding the issue of their origin, i.e. of the wave radiation process – which will be the subject of Chapters 8 and 10. We will start from the simplest, plane waves in uniform and isotropic media, and then proceed to a discussion of nonuniform systems, bringing up such effects as reflection and refraction. Then we will discuss the so-called guided waves, propagating along various transmission lines – such as cables, waveguides, and optical fibers. Finally, the end of the chapter is devoted to final-length fragments of such lines, serving as resonant cavities, and to the effects of energy dissipation in transmission lines and cavities.
7.1. Plane waves
Let us start by considering a spatial region that does not contain field sources ( = 0, j = 0), and is filled with a uniform, isotropic, linear medium, which therefore obeys Eqs. (3.46) and (5.110): D E,
B H
.
(7.1)
Moreover, let us assume for a while that these constitutive equations hold for all frequencies of interest.
(Of course, these relations are exactly valid for the very important particular case of free space, where we may formally use the macroscopic Maxwell equations (6.100), but with = 0 and = 0.) As was already shown in Sec. 6.8, in this case, the Lorenz gauge condition (6.117) allows the Maxwell equations to be recast into the wave equations (6.118) for the scalar and vector potentials. However, for most purposes, it is more convenient to use the homogeneous Maxwell equations (6.100) for the electric and magnetic fields – which are independent of the gauge choice. After an elementary elimination of D
and B using Eqs. (1),1 these equations take a simple, very symmetric form:
Maxwell
H
E
equations
E
,
0 H
,
0
(7.2a)
for uniform
t
t
linear
media
E ,
0
H 0.
(7.2b)
Now, let us act by the operator on each of Eqs. (2a), i.e. take their curl, and then use the vector algebra identity (5.31). The appearing terms E and H vanish due to Eqs. (2b), so the first terms of Eqs. (2a) turn into the Laplace operators of these vectors (with the minus sign). Now swapping, in the second terms, the operators / t and , and using Eqs. (2a) again, we get fully similar wave equations for the electric and magnetic fields:2
EM wave
2
1
2
2
1
2
equations
E ,
0
H ,
0
(7.3)
2
2
2
2
v t
v t
1 Though in a medium, B rather than H is the actual macroscopic magnetic field, mathematically it is a bit more convenient (just as it was in Sec. 6.2) to use the vector pair {E, H} in the following discussion, because at sharp media boundaries, it is H that obeys the boundary condition (5.117) similar to that for E – cf. Eq. (3.37).
2 The two vector equations (3) are of course just a shorthand for six similar equations for three Cartesian components of E and H.
© K. Likharev
EM: Classical Electrodynamics
where the parameter v is defined as
2
1
v
.
(7.4)
Wave
velocity
with v 2 = 1/00 c 2 in free space – see Eq. (6.120) again.
These equations allow, in particular, solutions of the following type;
E H f ( z vt),
(7.5)
Plane
wave
where z is the Cartesian coordinate along a certain ( arbitrary) direction n, and f is an arbitrary function of one argument. Note that this solution, first of all, describes a traveling wave – meaning a certain field pattern moving, without deformation, along the z-axis, with the constant velocity v. Second, according to Eq. (5), both E and H have the same values at all points of each plane perpendicular to the direction n
n z of the wave propagation; hence the second name – plane wave.
According to Eqs. (2), the independence of the wave equations (3) for vectors E and H does not mean that their plane-wave solutions are independent. Indeed, plugging any solution of the type (5) into Eqs. (2a), we get
n E
H
,
E
i.e.
Z H n ,
(7.6)
Field vector
Z
relation
where
1/ 2
E
Z
.
(7.7)
Wave
H
impedance
The vector relationship (6) means, first of all, that at any point of space and at any time instant, the vectors E and H are perpendicular not only to the propagation vector n (such waves are called transverse) but also to each other –– see Fig. 1.
k
n
0
Fig. 7.1. Field vectors in a plane electromagnetic
H
wave propagating along direction n.
E
Second, this equality does not depend on the function f, meaning that the electric and magnetic fields increase and decrease simultaneously. Finally, the field magnitudes are related by the constant Z
called the wave impedance of the medium. Very soon we will see that this impedance plays a pivotal role in many problems, in particular at the wave reflection from the interface between two media. Since the dimensionality of E, in SI units, is V/m, and that of H is A/m, Eq. (7) shows that Z has the dimensionality of V/A, i.e. ohms ().3 In particular, in free space,
3 In the Gaussian units, E and H have a similar dimensionality (in particular, in a free-space wave, E = H), making the (very useful) notion of the wave impedance less manifestly exposed – so in some older physics textbooks it is not mentioned at all!
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Wave
1/ 2
impedance
of free
0
Z Z
4 107
c 377 Ω .
(7.8)
0
space
0
Next, plugging Eq. (6) into Eqs. (6.113) and (6.114), we get:
Wave’s
energy
2
2
u E
H
,
(7.9a)
2
Wave’s
E
power
2
S E H n
n ZH ,
(7.9b)
Z
so, according to Eqs. (4) and (7), the wave’s energy and power densities are universally related as S n uv .
(7.9c)
In view of the Poynting vector paradox discussed in Sec. 6.8 (see Fig. 6.11), one may wonder whether the last equality may be interpreted as the actual density of power flow. In contrast to the static situation shown in Fig. 6.11, which limits the electric and magnetic fields to the vicinity of their sources, waves may travel far from them. As a result, they can form wave packets of a finite length in free space
– see Fig. 2.
n
S 0
packet
wave
Fig. 7.2. Interpreting the Poynting
vector in a plane electromagnetic
wave. (Horizontal lines show
V
equal-field planes.)
S 0
Let us apply the Poynting theorem (6.111) to the cylinder shown with dashed lines in Fig. 2, with one lid inside the wave packet, and another lid in the region already passed by the wave. Then, according to Eq. (6.111), the rate of change of the full field energy E inside the volume is d E / dt = – SA (where A is the lid area), so S may be indeed interpreted as the power flow (per unit area) from the volume. Making a reasonable assumption that the finite length of a sufficiently long wave packet does not affect the physics inside it, we may indeed interpret the S given by Eqs. (9b-c) as the power flow density inside a plane electromagnetic wave.
As we will see later in this chapter, the free-space value Z 0 of the wave impedance, given by Eq.
(8), establishes the scale of Z of virtually all wave transmission lines, so we may use it, together with Eq. (9), to get a better feeling of how much different are the electric and magnetic field amplitudes in the waves – on the scale of typical electrostatics and magnetostatics experiments. For example, according to Eqs. (9), a wave of a modest intensity S = 1 W/m2 (this is what we get from a usual electric bulb a few meters away from it) has E ~ ( SZ 0)1/2 ~ 20 V/m, quite comparable with the dc field created by a standard AA battery right outside it. On the other hand, the wave’s magnetic field H = ( S/ Z 0)1/2 0.05 A/m. For this particular case, the relation following from Eqs. (1), (4), and (7),
Chapter 7
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E
E
B H
1/2
E
,
(7.10)
Z
/
E
1/ 2
v
gives B = 0 H = E/ c ~ 710-8T, i.e. a magnetic field a thousand times lower than the Earth’s field, and about 7 orders of magnitude lower than the field of a typical permanent magnet. This huge difference may be interpreted as follows: the scale B ~ E/ c of magnetic fields in the waves is “normal” for electromagnetism, while the permanent magnet fields are abnormally high because they are due to the ferromagnetic alignment of electron spins, essentially relativistic objects – see the discussion in Sec. 5.5.
The fact that Eq. (5) is valid for an arbitrary function f means, in the standard terminology, that a medium with frequency-independent and supports the propagation of plane waves without either decay ( attenuation) or waveform deformation ( dispersion). However, for any real medium but pure vacuum, this approximation is valid only within limited frequency intervals. We will discuss the effects of attenuation and dispersion in the next section and will see that all our prior formulas remain valid even for an arbitrary linear media, provided that we limit them to single-frequency (i.e. sinusoidal, frequently called monochromatic) waves. Such waves may be most conveniently represented as4
Mono-
i kz t
f Re
f e
,
chromatic
(7.11)
wave
where f is the complex amplitude of the wave, and k is its wave number (the magnitude of the wave vector k n k), sometimes called the spatial frequency. The last term is justified by the fact, evident from Eq. (11), that k is related to the wavelength exactly as the usual (“temporal”) frequency is related to the time period T:
2
2
Spatial and
k
,
.
(7.12)
temporal
T
frequencies
In the dispersion-free case (5), the compatibility of that relation with Eq. (11) requires the argument ( kz
– t) k[ z – (/ k) t] to be proportional to ( z – vt), so / k = v, i.e.
k
1/2 ,
(7.13) Dispersion
v
relation
so in that particular case, the dispersion relation ( k) is linear.
Now note that Eq. (6) does not mean that the vectors E and H retain their direction in space.
(The wave in which they do, is called linearly polarized.5) Indeed, nothing in the Maxwell equations prevents, for example, a joint rotation of this vector pair around the fixed vector n, while still keeping all these three vectors perpendicular to each other at any instant – see Fig. 1. However, an arbitrary rotation law or even an arbitrary constant frequency of such rotation would violate the single-frequency (monochromatic) character of the elementary sinusoidal wave (11). To understand what is the most general type of polarization the wave may have without violating that condition, let us represent two 4 As we have already seen in the previous chapter (see also CM Sec. 5.1), such complex-exponential representation of sinusoidally changing variables is more convenient for mathematical manipulation than by using sine and cosine functions, especially because in all linear relations, the operator Re may be omitted (implied) until the very end of the calculation. Note, however, that this is not valid for the quadratic forms such as Eqs. (9).
5 The possibility of different polarizations of electromagnetic waves was discovered (for light) in 1699 by Rasmus Bartholin, a.k.a. Erasmus Bartholinus.
Chapter 7
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Cartesian components of one of these vectors (say, E) along any two fixed axes x and y, perpendicular to each other and the z-axis (i.e. to the vector n), in the same form as used in Eq. (11):
i kz t
i kz t
E Re E e
,
E
.
(7.14)
x
E e
Re
x
y
y
To keep the wave monochromatic, the complex amplitudes E x and E y have to be constant in time; however, they may have different magnitudes and an arbitrary phase shift between them.
In the simplest case when the arguments of these complex amplitudes are equal,
i
E
E
.
(7.15)
x, y
e
x, y
the real field components have the same phase:
E E
cos( kz t ),
E E
cos( kz t ),
(7.16)
x
x
y
y
so their ratio is constant in time – see Fig. 3a. This means that the wave is linearly polarized, with the polarization plane defined by the relation
tan E / E .
(7.17)
y
x
E
(a)
E
(b)
E
(c)
y
y
y
E y
E
E y
( t)
( t)
0
0
0
E
E
x
E
E
E
x
E
x
x
x