of the order of the Bohr magneton B, similar to the one sketched for electric dipoles in Fig. 3.7a.
5.19.* Use the classical picture of the orbital (“Larmor”) diamagnetism, discussed in Sec. 5, to calculate its (small) contribution B(0) to the magnetic field B felt by an atomic nucleus, treating the electrons of the atom as a spherically-symmetric cloud with an electric charge density ( r). Express the result via the value (0) of the electrostatic potential of the electron cloud, and use this expression for a crude numerical estimate of the ratio B(0)/ B for the hydrogen atom.
5.20. Calculate the self-inductance of a toroidal solenoid
z
(see Fig. 6b) with the round cross-section of radius r ~ R (see the
I
I
r
figure on the right), with many ( N >> 1, R/ r) wire turns uniformly
distributed along the perimeter, and filled with a linear magnetic
0
R
material of permeability . Check your results by analyzing the
limit r << R.
Chapter 5
Page 41 of 42
Essential Graduate Physics
EM: Classical Electrodynamics
5.21. A long straight, thin wire carrying current I, passes parallel
I
to the plane boundary between two uniform, linear magnetic media – see
1
d
the figure on the right. Calculate the magnetic field everywhere in the
system, and the force (per unit length) exerted on the wire.
2
5.22. Solve the magnetic shielding problem similar to that discussed in Sec. 5.6 of the lecture notes, but for a spherical rather than cylindrical shell, with the same central cross-section as shown in Fig. 16. Compare the efficiency of those two shields, for the same shell’s permeability , and the same b/a ratio.
5.23. Calculate the magnetic field distribution around a spherical permanent magnet with a uniform magnetization M0 = const.
5.24. A limited volume V is filled with a magnetic material with a fixed (field-independent) magnetization M(r). Write explicit expressions for the magnetic field induced by the magnetization, and its potential, and recast these expressions into the forms more convenient when M(r) = M0 = const inside the volume V.
5.25. Use the results of the previous problem to calculate the
distribution of the magnetic field H along the axis of a straight
M
R
0
l
permanent magnet of length 2 l, with a round cross-section of radius
l
0
z
R, and a uniform magnetization M
l
0 parallel to the axis – see the
figure on the right.
5.26. A flat end of a long straight permanent magnet, similar to that considered in the previous problem but of an arbitrary cross-section of area A, is stuck to a flat surface of a large sample of a linear magnetic material with a very high permeability >> 0. Calculate the normally-directed force needed to detach them.
5.27. A permanent magnet with a uniform magnetization M0 has the form of a spherical shell with an internal radius R 1 and an external radius R 2 > R 1. Calculate the magnetic field inside the shell.
5.28. A very broad film of thickness 2 t is permanently magnetized normally to its plane, with a periodic checkerboard pattern, with the square of area a a:
M
n M , ,
with
,
sgn cos
cos
.
z
x y
M x y
x
y
M
0
z t
a
a
Calculate the magnetic field’s distribution in space.
5.29.* Based on the discussion of the quadrupole electrostatic lens in Sec. 2.4, suggest the permanent-magnet systems that may similarly focus particles moving close to the system’s axis, for the cases when each particle carries:
(i) an electric charge,
(ii) no net electric charge, but a spontaneous magnetic dipole moment m of a certain orientation.
Chapter 5
Page 42 of 42
EM: Classical Electrodynamics
Chapter 6. Electromagnetism
This chapter discusses two major effects that arise when electric and magnetic fields change over time: the “electromagnetic induction” of an additional electric field by changing magnetic field, and the reciprocal effect of the “displacement currents”– actually, the induction of an additional magnetic field by changing electric field. These two phenomena, which make time-dependent electric and magnetic fields inseparable (hence the term “electromagnetism” 1 ), are reflected in the full system of Maxwell equations, valid for an arbitrary electromagnetic process. On the way toward this system, I will make a brief detour to review the electrodynamics of superconductivity, which (besides its own significance), provides a perfect platform for discussion of the important general issue of gauge invariance.
6.1. Electromagnetic induction
As Eqs. (5.36) show, in static situations (/ t = 0) the Maxwell equations describing the electric and magnetic fields are independent – more exactly, coupled only implicitly, via the continuity equation (4.5) relating their right-hand sides and j. In dynamics, when the fields change in time, the situation is different.
Historically, the first discovered explicit coupling between the electric and magnetic fields was the effect of electromagnetic induction. Although this effect was discovered independently by Joseph Henry, it was a brilliant series of experiments by Michael Faraday, carried out mostly in 1831, that resulted in the first general formulation of the induction law. The summary of Faraday’s numerous experiments has turned out to be very simple: if the magnetic flux defined by Eq. (5.65),
B d 2
Φ
r ,
(6.1)
n
S
through a surface S limited by a closed contour C, changes in time by whatever reason (e.g., either due to a change of the magnetic field B (as in Fig.1), or the contour’s motion, or its deformation, or any combination of the above), it induces an additional, vortex-like electric field Eind directed along the contour – see Fig. 1.
B t
B t
C
C
E
Eind
I
/ R
ind
ind
V
S
S
ind
V
Fig. 6.1. Two simplest ways to observe the Faraday electromagnetic induction.
The exact distribution of Eind in space depends on the system’s details, but its integral along the contour C, called the inductive electromotive force (e.m.f.), obeys a very simple Faraday induction law: 1 It was coined by H. Ørsted in 1820 in the context of his experiments – see the previous chapter.
© K. Likharev
EM: Classical Electrodynamics
d Φ
Faraday
V E d
.
(6.2)
ind
r
ind
induction
dt
C
law
(In the Gaussian units, the right-hand side of this formula has an additional coefficient of 1/ c.) It is straightforward (and hence left for the reader’s exercise) to show that this e.m.f. may be measured, for example, either by inserting a voltmeter into a conducting loop following the contour C or by measuring the small current I = Vind/ R it induces in a thin wire with a sufficiently large Ohmic resistance R,2 whose shape follows that contour – see Fig. 1. (Actually, these methods are not entirely different, because a typical voltmeter measures voltage by the small Ohmic current it drives through the pre-calibrated high internal resistance of the device.) In the context of the latter approach, the minus sign in Eq. (2) may be described by the following Lenz rule: the magnetic field of the induced current I provides a partial compensation of the change of the original flux ( t) with time.3
In order to recast Eq. (2) in a differential form, more convenient in many cases, let us apply to the contour integral in it the Stokes theorem, which was repeatedly used in Chapter 5. The result is V E
.
(6.3)
ind
ind
d 2 r
n
S
Now combining Eqs. (1)-(3), for a contour C whose shape does not change in time (so that the integration along it is interchangeable with the time derivative), we get
B
2
E
.
(6.4)
ind
d r 0
t
S
n
Since the induced electric field is an addition to the gradient field (1.33) created by electric charges, for the net field we may write E = Eind – . However, since the curl of any gradient field is zero,4 () = 0, Eq. (4) remains valid even for the net field E. Since this equation should be correct for any closed area S, we may conclude that
B
Faraday law:
E
0
(6.5)
differential
t
form
at any point. This is the final (time-dependent) form of this Maxwell equation. Superficially, it may look that Eq. (5) is less general than Eq. (2); for example, it does not describe any electric field, and hence any e.m.f. in a moving loop, if the field B is constant in time, even if the magnetic flux (1) through the loop does change in time. However, this is not true; in Chapter 9 we will see that in the reference frame moving with the loop such e.m.f. does appear.5
2 Such induced current is sometimes called the eddy current, though most often this term is reserved for the distributed currents induced by changing magnetic fields in bulk conductors – see Sec. 3 below.
3 Let me also hope that the reader is familiar with the paradox arising at attempts to measure Vind with a voltmeter without its insertion into the wire loop; if not, I would highly recommend them to solve the offered Problem 2.
4 See, e.g., MA Eq. (11.1).
5 I have to admit that from the beginning of the course, I was carefully sweeping under the rug a very important question: in what exactly reference frame(s) all the equations of electrodynamics are valid? I promise to discuss this issue in detail later in the course (in Chapter 9), and for now would like to get away with a very short answer: all the formulas discussed so far are valid in any inertial reference frame, as defined in classical mechanics – see, e.g., CM Sec. 1.3; however, the fields E and B have to be measured in the same reference frame.
Chapter 6
Page 2 of 38
EM: Classical Electrodynamics
Now let us reformulate Eq. (5) in terms of the vector potential A. Since the induction effect does not alter the fundamental relation B = 0, we still may represent the magnetic field as prescribed by Eq. (5.27), i.e. as B = × A. Plugging this expression into Eq. (5), and changing the order of the temporal and spatial differentiation, we get
A
E
0 .
(6.6)
t
Hence we can use the same argumentation as in Sec. 1.3 (there applied to the vector E alone) to represent the expression in the parentheses as –, so that we get
Fields via
A
E
,
B A .
(6.7)
potentials
t
It is very tempting to interpret the first term of the right-hand side of the expression for E as the one describing the electromagnetic induction alone, and the second term as representing a purely electrostatic field induced by electric charges. However, the separation of these two terms is, to a certain extent, conditional. Indeed, let us consider the gauge transformation already mentioned in Sec. 5.2, A A ,
(6.8)
that, as we already know, does not change the magnetic field. According to Eq. (7), to keep the full electric field intact ( gauge-invariant) as well, the scalar electric potential has to be transformed simultaneously, as
,
(6.9)
t
leaving the choice of an addition to restricted only by the Laplace equation – since the full should satisfy the Poisson equation (1.41) with a gauge-invariant right-hand side. We will return to the discussion of the gauge invariance in Sec. 4.
6.2. Magnetic energy revisited
Now we are sufficiently equipped to revisit the issue of magnetic energy, in particular, to finally prove Eqs. (5.57) and (5.140), and discuss the dichotomy of the signs in Eqs. (5.53) and (5.54). For that, let us consider a sufficiently slow and small magnetic field variation B. If we want to neglect the kinetic energy of the system of electric currents under consideration, as well as the wave radiation effects, we need to prevent its significant acceleration by the arising induction field Eind. Let us suppose that we do this by virtual balancing of this field by an external electric field Eext = –Eind. According to Eq. (4.38), the work of that field6 on the stand-alone currents of the system during a small time interval
t, and hence the change of the potential energy of the system, is
U
δt jE d 3 r,
that
so
j E
,
(6.10)
ext
U δt d 3 r
ind
V
V
6 As a reminder, the magnetic component of the Lorentz force (5.10), vB, is always perpendicular to the particle velocity v, so the magnetic field B itself cannot perform any work on moving charges, i.e. on currents.
Chapter 6
Page 3 of 38
EM: Classical Electrodynamics
where the integral is over the volume of the system. Now expressing the current density j from the macroscopic Maxwell equation (5.107), j = H, and then applying the vector algebra identity7
HE H
,
(6.11)
ind
Eind E H
ind
we get
U
δt H E d 3 r δt E H
d 3 r .
(6.12)
V
V
According to the divergence theorem, the second integral in the right-hand of this equality is equal to the flux of the so-called Poynting vector S E H through the surface limiting the considered volume V. Later in the course we will see that this flux represents, in particular, the power of electromagnetic radiation through the surface. If such radiation is negligible (as it always is if the field variation is sufficiently slow), the surface may be selected sufficiently far, so that the flux of S vanishes.
In this case, we may express E from the Faraday induction law (5) to get
B
U
δt H d 3 r H B
d 3 r .
(6.13)
t
V
V
Just as in the electrostatics (see Eqs. (1.65) and (3.73), and their discussion), this relation may be interpreted as the variation of the magnetic field energy U of the system, and represented in the form Magnetic
U
u
r d 3 r,
with u H δB
.
(6.14) energy’s
variation
V
This is a keystone result; let us discuss it in some detail.
First of all, for a system filled with a linear and isotropic magnetic material, we may use Eq. (14) together with Eq. (5.110): B = H. Integrating the result over the variation of the field from 0 to a certain final value B, we get Eq. (5.140) – so important one that it deserves rewriting it again: 2
B
U u 3
d r, with u
r
.
(6.15)
2
V
In the simplest case of free space (no magnetics at all, so that j above is the complete current density), we may take = 0, and reduce Eq. (15) to Eq. (5.57). Now performing backward the transformations that took us, in Sec. 5.3, to derive that relation from Eq. (5.54), we finally have the latter formula proved
– as was promised in the last chapter.
It is very important, however, to understand the limitations of Eq. (15). For example, let us try to apply it to a very simple problem, which was already analyzed in Sec. 5.6 (see Fig. 5.15): a very long cylindrical sample of a linear magnetic material placed into a fixed external field Hext parallel to the sample’s length. It is evident that in this simple geometry, the field H and hence the field B = H have to be uniform inside the sample, besides negligible regions near its ends, so that Eq. (15) is reduced to B 2
U
V ,
(6.16)
2
7 See, e.g., MA Eq. (11.7) with f = Eind and g = H.
Chapter 6
Page 4 of 38
EM: Classical Electrodynamics
where V = Al is the cylinder’s volume. Now if we try to calculate the static (equilibrium) value of the field from the minimum of this potential energy, we get evident nonsense: B = 0 (WRONG!).8
The situation may be readily rectified by using the notion of the Gibbs potential energy, just as it was done for the electric field in Sec. 3.5 (and implicitly in the end of Sec. 1.3). According to Eq. (14), in magnetostatics, the Cartesian components of the field H(r) play the role of the generalized forces, while those of the field B(r), of the generalized coordinates (per unit volume).9 As the result, the Gibbs potential energy, whose minimum corresponds to the stable equilibrium of the system under the effect of a fixed generalized force (in our current case, of the fixed external field Hext), is
Gibbs
potential
U u
d r,
with u
u
,
(6.17)
G
r 3
G
G r
r H rBr
ext
energy
V
– the expression parallel to Eq. (3.78). For a system with linear magnetics, we may use for u our result (15), getting the following Gibbs energy’s density:
1
1
u (r)
B B H
B
B H
,
(6.18)
G
ext
ext 2
const
2
2
where “const” means a term independent of the field B inside the sample. For our simple cylindrical system, with its uniform fields, Eqs. (17)-(18) gives the following full Gibbs energy of the sample:
B H
int
ext 2
U
V const ,
(6.19)
G
2
whose minimum immediately gives the correct stationary value Bint = Hext, i.e. Hint Bint/ = Hext, which was already obtained in Sec. 5.6 in a different way, from the boundary condition (5.117).
Now notice that with this result on hand, Eq. (18) may be rewritten in a different form: 1
2
B
B
u (r)
B B
B
,
(6.20)
G
2
2
similar to Eq. (15) for u(r), but with an opposite sign. This sign dichotomy explains that of Eqs. (5.53) and Eq. (5.54); indeed, as was already noted in Sec. 5.3, the former of these expressions gives the potential energy whose minimum corresponds to the equilibrium of a system with fixed currents. (In our current example, these are the external stand-alone currents inducing the field Hext.) So, the energy Uj given by Eq. (5.53) is essentially the Gibbs energy U G defined by Eqs. (17) and (for the equilibrium state of linear magnetic media) by Eq. (20), while Eq. (5.54) is just another form of Eq. (15) – as was explicitly shown in Sec. 5.3.10
8 This erroneous result cannot be corrected by just adding the energy of the field outside the cylinder because in the limit A 0, this field is not affected by the internal field B.
9 Note an aspect in that the analogy with electrostatics is not quite complete. Indeed, according to Eq. (3.76), in electrostatics, the role of a generalized coordinate is played by the “would-be” field D, and that of the generalized force, by the actual (if macroscopic) electric field E. This difference may be traced back to the fact that the electric field E may perform work on a moving charged particle, while the magnetic field cannot. However, this difference does not affect the full analogy of the expressions (3.73) and (15) for the field energy density in linear media.
10 As was already noted in Sec. 5.4, one more example of the energy Uj (i.e. U G) is given by Eq. (5.100).
Chapter 6
Page 5 of 38
EM: Classical Electrodynamics
Let me complete this section by stating that the difference between the energies U and U G is not properly emphasized (or even left obscure) in some textbooks, so that the reader is advised to get additional clarity by solving a few additional simple problems – for example, by spelling out these energies for a long straight solenoid (Fig. 5.6a), and then using the results to calculate the pressure exerted by the magnetic field on the solenoid’s walls (windings) and the longitudinal forces exerted on its ends.
6.3. Quasistatic approximation and skin effect
Perhaps the most surprising experimental fact concerning the time-dependent electromagnetic phenomena is that unless they are so fast that one more new effect of the displacement currents (to be discussed in Sec. 7 below) becomes noticeable, all formulas of electrostatics and magnetostatics remain valid, with the only exception: the generalization of Eq. (3.36) to Eq. (5), describing the Faraday induction. As a result, the system of macroscopic Maxwell equations (5.109) is generalized to
B
E
0,
H j,
Quasistatic
t
(6.21) approximation
D ,
B 0.
(As it follows from the discussions in chapters 3 and 5, the corresponding system of microscopic Maxwell equations for the genuine, “microscopic” fields E and B may be obtained from Eq. (21) by the formal substitutions D = 0E and H = B/0, and the replacement of the stand-alone charge and current densities and j with their full densities.11) These equations, whose range of validity will be quantified in Sec. 7, define the so-called quasistatic approximation of electromagnetism and are sufficient for an adequate description of a broad range of physical effects.
In order to form a complete system of equations, Eqs. (21) should be augmented by constituent equations describing the medium under consideration. For a linear isotropic material, they may be taken in the simplest (and simultaneously, most common) linear and isotropic forms already discussed in Chapters 4 and 5:
j E
,
B H
.
(6.22)
If the conductor is uniform, i.e. the coefficients and are constant inside it, the whole system of Eqs.
(21)-(22) may be reduced to just one simple equation. Indeed, a sequential substitution of these equations into each other, using a well-known vector-algebra identity12 in the middle, yields: B
1
1
1
1
E j ( H)
( B)
(B) 2
B
t
(6.23)
1
2
B.
Thus we have arrived, without any further assumptions, at a rather simple partial differential equation. Let us use it for an analysis of the so-called skin effect, the phenomenon of an Ohmic 11 Obviously, in free space, the last replacement is unnecessary, because all charges and currents may be treated as
“stand-alone” ones.
12 See, e.g., MA Eq. (11.3).
Chapter 6
Page 6 of 38
Essential Graduate Physics
EM: Classical Electrodynamics
conductor’s self-shielding from the alternating ( ac) magnetic field. In its simplest geometry (Fig. 2a), an external source (which, at this point, does not need to be specified) produces, near a plane surface of a bulk conductor, a spatially-uniform ac magnetic field H(0)( t) parallel to the surface.13
y
(a)
(b)
0
,
C
C
1
2
0
Fig. 6.2. (a) The skin effect in
0
the simplest, planar geometry,
H 0
H
H 0
(0)
H
and (b) two Ampère contours,
H x
C 1 and C 2, for deriving the
n
J
“macroscopic” ( C 1) and the
0
x
“coarse-grain” ( C 2) boundary
s
s
conditions for H.
Selecting the coordinate system as shown in Fig. 2a, we may express this condition as
0
H
H
t
( n
)
x
.
(6.24)
0
y
The translational symmetry of our simple problem within the surface plane [ y, z] implies that inside the conductor, / y = / z = 0 as well, and H = H( x, t)n y even at x 0, so that Eq. (23) for the conductor’s interior is reduced to a differential equation for just one scalar function H( x, t) = B( x, t)/:
H
1
2
H
,
for x 0 .
(6.25)
2
t
x
This equation may be further simplified by noticing that due to its linearity, we may use the linear superposition principle for the time dependence of the field,14 via expanding it, as well as the external field (24), into the Fourier series:
H ( x, t) H ( x)
e i t
, for . x ,0
(6.26)
0
0
H ( t)
H e i t ,
for
x ,
0
and arguing that if we know the solution for each frequency component of the series, the whole field may be found through the straightforward summation (26) of these solutions.
For each single-frequency component, Eq. (25) is immediately reduced to an ordinary differential equation for the complex amplitude H( x):15
13 Due to the simple linear relation B = H between the fields B and H, it does not matter too much which of them is used for the solution of this problem, with a slight preference for H, due to the simplicity of Eq. (5.117) –
the only boundary condition relevant for this simple geometry.
14 Another way to exploit the linearity of Eq. (6.25) is to use the spatial-temporal Green’s function approach to explore the dependence of its solutions on various initial conditions. Unfortunately, because of a lack of time, I have to leave an analysis of this opportunity for the reader’s exercise.
Chapter 6
Page 7 of 38
EM: Classical Electrodynamics
d 2
1
i H
H .
(6.27)
2
dx
From the theory of linear ordinary differential equations, we know that Eq. (27) has the following general solution:
x
x
H ( x
H e H e
)
,
(6.28)
where the constants are the roots of the characteristic equation that may be obtained by the substitution of any of these two exponents into the initial differential equation. For our particular case, the characteristic equation following from Eq. (27) is
2
i
(6.29)
and its roots are
1 i
.
(6.30)
i 1/2
1/2
2
For our problem, the field cannot grow exponentially at x +, so only one of the coefficients, namely the H– corresponding to the decaying exponent, with Re – < 0, may be different from zero, i.e.
H( x) = H(0)exp{ –x}. To find the constant factor H(0), we can integrate the macroscopic Maxwell equation H = j along a pre-surface contour – say, the contour C 1 shown in Fig. 2b. The right-hand side’s integral is negligible because the stand-alone current density j does not include the “genuinely-surface” currents responsible for the magnetic permeability – see Fig. 5.12. As a result, we get the boundary condition similar to Eq. (5.117) for the stationary magnetic field: H = const at x = 0, giving us
H ,
0 t
0
H t,
i.e. H
0
0
H ,
(6.31)
so that the final solution of our boundary problem may be represented as
0
0
x
x
H ( x) H
exp x
H
exp
exp
i t
,
(6.32)
s
s
where the constant s, with the dimension of length, is called the skin depth:
1/ 2
1
2
.
(6.33)
Skin
s
Re
depth
This solution describes the skin effect: the penetration of the ac magnetic field, and the eddy currents j, into a conductor only to a finite depth of the order of s. Let me give a few numerical examples of this depth: for copper at room temperature, s 1 cm at the usual ac power distribution frequency of 60 Hz, and is of the order of just 1 m at a few GHz, i.e. at typical frequencies of cell 15 Let me hope that the reader is not intimidated by the (very convenient) use of such complex variables for describing real fields; their imaginary parts always disappear at the final summation (26). For example, if the external field is purely sinusoidal, with the actual (positive) frequency , each sum in Eq. (26) has just two terms, with complex amplitudes H and H- = H*, so that their sum is always real. (For a more detailed discussion of this issue, see, e.g., CM Sec. 5.1.)
Chapter 6
Page 8 of 38
EM: Classical Electrodynamics
phone signals and kitchen microwave magnetrons. On the other hand, for lightly salted water, s is close to 250 m at just 1 Hz (with significant implications for radio communications with submarines), and of the order of 1 cm at a few GHz (explaining, in particular, the nonuniform heating of a soup bowl in a microwave oven).16
Let me hope that the equality chain (23) makes the physics of this effect very clear: the external electric field E, which is Faraday-induced by an external ac magnetic field, drives the eddy currents j, which in turn induce their own magnetic field that eventually (at x ~ s) compensates the external one.
Let us quantify these E and j. Since we have used, in particular, relations j = H = × B/, and E =
j/, and spatial differentiation of an exponent yields a similar exponent, the electric field and current density have the same spatial dependence as the magnetic field, i.e. penetrate the conductor only by distances of the order of s(). Their vectors are directed normally to B, while still being parallel to the conductor’s surface:17
j
x
H ( x) n ,
E x
H ( x)n
.
(6.34)
z
z
We may use these expressions, in particular, to calculate the time-averaged power density (4.39) of the energy dissipation, for the important case of a sinusoidal (“monochromatic”) field H( x, t) = H( x)
cos( t + ), and hence sinusoidal eddy currents: j( x, t) = j( x) cos( t + ’): 2
2
j x, t
j x cos t
2
2
'
j x
2
2
2
H x
H x
p x
.
(6.35)
2
2
2
s
Now the (elementary) integration of this expression along the x-axis (through all the skin depth), using the exponential law (6.32), gives us the following average power of the energy loss per unit area: Energy
loss
d
P
1
2
at skin
p x
0
s
2
0
dx
H
H
.
(6.36)
effect
dA
2
4
0
s
We will extensively use this expression in the next chapter to calculate the energy losses in microwave waveguides and resonators with conducting (practically, metallic) walls, and for now let me note only that according to Eqs. (33) and (36), for a fixed magnetic field amplitude, the losses grow with frequency as 1/2.
One more important remark concerning Eqs. (34): integrating the first of them over x, with the help of Eq. (32), we may see that the linear density J of the surface currents (measured in A/m), is simply and fundamentally related to the applied magnetic field:
J j
x
0
dx
H n
.
(6.37)
z
0
Since this relation does not have any frequency-dependent factors, we may sum it up for all frequency components, and get a universal relation
16 Let me hope that the reader’s physical intuition makes it evident that the skin effect remains conceptually the same for samples of any shape, besides possibly some quantitative details of the field distribution.
17 The loop (vortex) character of the induced current lines, responsible for the term “eddy”, is not very apparent in the 1D geometry explored above, with the near-surface currents (Fig. 2b) looping only implicitly, at z .
Chapter 6
Page 9 of 38
EM: Classical Electrodynamics
J t
0
H t
0
n H
0
0
,
(6.38a)
z
t n n
y
x
H t n x
H t n
(where n = –n x is the outer normal to the surface – see Fig. 2b) or, in a different form, Coarse-grain
H t n J( t),
(6.38b) boundary
relation
where H is the full change of the field through the skin layer. This simple coarse-grain relation (independent of the choice of coordinate axes), is also independent of the used constituent relations (22), and is by no means occasional. Indeed, it may be readily obtained from the macroscopic Ampère law (5.116), by applying it to a contour drawn around a fragment of the surface, extending under it substantially deeper than the skin depth – see the contour C 2 in Fig. 2b. Hence, Eq. (38) is valid regardless of the exact law of the field penetration.
For the skin effect, this fundamental relationship between the linear current density and the external magnetic field implies that the skin effect’s implementation does not necessarily require a dedicated ac magnetic field source. For example, the effect takes place in any wire that carries an ac current, leading to a current’s concentration in a surface sheet of thickness ~s. (Of course, the quantitative analysis of this problem in a wire with an arbitrary cross-section may be technically complicated, because it requires solving Eq. (23) for the corresponding 2D geometry; even for the round cross-section, the solution involves the Bessel functions.) In this case, the ac magnetic field outside the conductor, which still obeys Eq. (38), may be better interpreted as the effect, rather than the cause, of the ac current flow.
Finally, please mind the limited validity of all the above results. First, for the quasistatic approximation to be valid, the field frequency should not be too high, so that the displacement current effects are negligible. (Again, this condition will be quantified in Sec. 7 below; it will show that for metals, the condition is violated only at extremely high frequencies above ~1018 s-1.) A more practical upper limit on is that the skin depth s should stay much larger than the mean free path l of charge carriers , 18 because beyond this point, the constituent relation between the vectors j(r) and E(r) becomes essentially non-local. Both theory and experiment show that at s below l, the skin effect persists, but acquires a frequency dependence slightly different from Eq. (33): s –1/3 rather than –1/2.
Historically, this anomalous skin effect has been very useful for the measurements of the Fermi surfaces of metals.19
6.4. Electrodynamics of superconductivity, and the gauge invariance
The effect of superconductivity20 takes place (in certain materials only, mostly metals) when temperature T is reduced below a certain critical temperature T c specific for each material. For most metallic superconductors, T c is of the order of typically a few kelvins, though several compounds (the so-called high-temperature superconductors) with T c above 100 K have been found since 1987. The most notable property of superconductors is the absence, at T < T c, of measurable resistance to (not very high) dc currents. However, the electromagnetic properties of superconductors cannot be described by 18 A discussion of the mean free path may be found, for example, in SM Chapter 6. In very clean metals at very low temperatures, s may approach l at frequencies as low as ~1 GHz, but at room temperature, the crossover between the normal to the anomalous skin effect takes place only at ~ 100 GHz.
19 See, e.g., A. Abrikosov, Introduction to the Theory of Normal Metals, Academic Press, 1972.
20 Discovered experimentally in 1911 by Heike Kamerlingh Onnes.
Chapter 6
Page 10 of 38
EM: Classical Electrodynamics
just taking = in our previous results. Indeed, for this case, Eq. (33) would give s = 0, i.e., no ac magnetic field penetration at all. Experiment shows something substantially different: weak magnetic fields do penetrate into superconductors by a material-specific distance L ~ 10-7-10-6 m, the so-called London’s penetration depth,21 which is virtually frequency-independent until the skin depth s, of the same material in its “normal” state, i.e. the absence of superconductivity, becomes less than L. (This crossover happens typically at frequencies ~ 1013-1014 s-1.) The smallness of L on the human scale means that the magnetic field is pushed out from macroscopic samples at their transition into the superconducting state.
This Meissner-Ochsenfeld effect, discovered experimentally in 1933,22 may be partly understood using the following classical reasoning. Our discussion of the Ohm law in Sec. 4.2 implied that the current’s (and hence the electric field’s) frequency is either zero or sufficiently low. In the classical Drude reasoning, this is acceptable while << 1, where is the effective carrier scattering time participating in Eqs. (4.12)-(4.13). If this condition is not satisfied, we should take into account the charge carrier inertia; moreover, in the opposite limit >> 1, we may neglect the scattering at all.
Classically, we can describe the charge carriers in such a “perfect conductor” as particles with a nonzero mass m, which are accelerated by the electric field following the 2nd Newton law (4.11), v
m F E
q ,
(6.39)
so that the current density j = qnv that they create, changes in time as q 2 n
j
v
qn
E .
(6.40)
m
In terms of the Fourier amplitudes of the functions j( t) and E( t), this means q 2 n
i
j
E .
(6.41)
m
Comparing this formula with the relation j = E implied in the last section, we see that we can use all its results with the following replacement:
q 2 n
i
.
(6.42)
m
This change replaces the characteristic equation (29) with
2
m
q 2
2
n
i
,
i.e.
,
(6.43)
iq 2 n
m
i.e. replaces the skin effect with the field penetration by the following frequency-independent depth: 1/ 2
1
m
.
(6.44)
2
q
n
Superficially, this means that the field decay into the superconductor does not depend on frequency: 21 Named so to acknowledge the pioneering theoretical work of brothers Fritz and Heinz London – see below.
22 It is hardly fair to shorten this name to just the “Meissner effect” as it is frequently done, because of the reportedly crucial contribution by Robert Ochsenfeld, then a Walther Meissner’s student, to the discovery.
Chapter 6
Page 11 of 38
EM: Classical Electrodynamics
x /
H ( x, t) H ( ,
0 t) e
,
(6.45)
thus explaining the Meissner-Ochsenfeld effect.
However, there are two problems with this result. First, for the parameters typical for good metals ( q = – e, n ~ 1029 m-3, m ~ m e, 0), Eq. (44) gives ~ 10-8 m, one or two orders of magnitude lower than the experimental values of L. Experiment also shows that the penetration depth diverges at T
T c, which is not predicted by Eq. (44).
The second, much more fundamental problem with Eq. (44) is that it has been derived for
>> 1. Even if we assume that somehow there is no scattering at all, i.e. = , at 0 both parts of the characteristic equation (43) vanish, and we cannot make any conclusion about . This is not just a mathematical artifact we could ignore. For example, let us place a non-magnetic metal into a static external magnetic field at T > T c. The field would completely penetrate the sample. Now let us cool it.
As soon as temperature is decreased below T c, the above calculations would become valid, forbidding the penetration into the superconductor of any change of the field, so that the initial field would be
“frozen” inside the sample. The Meissner-Ochsenfeld experiments have shown something completely different: as T is lowered below T c, the initial field is being expelled out of the sample.
The resolution of these contradictions is provided by quantum mechanics. As was explained in 1957 in a seminal work by J. Bardeen, L. Cooper, and J. Schrieffer (commonly referred to as the BCS
theory), superconductivity is due to the correlated motion of electron pairs, with opposite spins and nearly opposite momenta. Such Cooper pairs, each with the electric charge q = –2 e and zero spin, may form only in a narrow energy layer near the Fermi surface, of a certain thickness ( T). This parameter
( T), which may be also interpreted as the binding energy of the pair, tends to zero at T T c, while at T
<< T c it has a virtually constant value (0) 3.5 k B T c, of the order of a few meV for most superconductors. This fact readily explains the relatively low spatial density of the Cooper pairs: n p ~
n( T)/F ~ 1026 m-3. With the correction n n p, Eq. (44) for the penetration depth becomes 1/ 2
m
London’s
.
(6.46) penetration
L
2
q
n
depth
p
This result diverges at T T c, and generally fits the experimental data reasonably well, at least for the so-called “clean” superconductors with the mean free path l = v F (where v F ~ (2 mF)1/2 is the r.m.s.
velocity of electrons on the Fermi surface) much longer than the Cooper pair size – see below.
The smallness of the coupling energy ( T) is also a key factor in the explanation of the Meissner-Ochsenfeld effect. Because of Heisenberg’s quantum uncertainty relation r p ~ , the spatial extension of the Cooper-pair’s wavefunction (the so-called coherence length of the superconductor) is relatively large: ~ r ~ / p ~ v F/( T) ~ 10-6 m. As a result, n p3 >> 1, meaning that the wavefunctions of the pairs are strongly overlapped in space. Due to their integer spin, Cooper pairs behave like bosons, which means in particular that at low temperatures they exhibit the so-called Bose-Einstein condensation onto the same ground energy level g.23 This means that the quantum frequency
23 A quantitative discussion of the Bose-Einstein condensation of bosons may be found in SM Sec. 3.4, though the full theory of superconductivity is more complicated because it has to describe the condensation taking place simultaneously with the formation of effective bosons (Cooper pairs) from fermions (single electrons). For a Chapter 6
Page 12 of 38
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 that the electric current is carried not by individual Cooper pairs but rather 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 that 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
Page 13 of 38
EM: Classical Electrodynamics
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 that 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 that 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
This simple differential equation, similar to Eq. (23), for our 1D geometry 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 that even for microwaves, Eq. (57) still gives a fairly accurate description of the field penetration.
Chapter 6
Page 14 of 38
EM: Classical Electrodynamics
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 sufficiently 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
Page 15 of 38
EM: Classical Electrodynamics
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 that 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
Page 16 of 38
EM: Classical Electrodynamics
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 that 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 that = n/, where n is the azimuthal unit vector.34 This topological feature of the wavefunction’s phase is sometimes called the fluxoid quantization – to 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
Page 17 of 38
EM: Classical Electrodynamics
distinguish it from the 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 only by the additional nonlinear term from the usual Schrödinger equation, which is linear in . 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 that 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
Page 18 of 38
EM: Classical Electrodynamics
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 from 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 that 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
Page 19 of 38
EM: Classical Electrodynamics
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 that 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 that 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
Page 20 of 38
EM: Classical Electrodynamics
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 capital, 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
Page 21 of 38
EM: Classical Electrodynamics
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
Page 22 of 38
EM: Classical Electrodynamics
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.
Chapter 6
Page 23 of 38
EM: Classical Electrodynamics
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 that 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 that if the Ohmic resistance of the coil is negligible, we may write 50 In experiment, this condition is hard to implement, due to 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.
Chapter 6
Page 24 of 38
EM: Classical Electrodynamics
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
