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 that 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 that 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 that 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.
Chapter 6
Page 25 of 38
EM: Classical Electrodynamics
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 its 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.
Chapter 6
Page 26 of 38
Essential Graduate Physics
EM: Classical Electrodynamics
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).
Chapter 6
Page 27 of 38
EM: Classical Electrodynamics
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 that 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
Page 28 of 38
EM: Classical Electrodynamics
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 insure 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, 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.
Chapter 6
Page 29 of 38
EM: Classical Electrodynamics
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 that 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
Page 30 of 38
Essential Graduate Physics
EM: Classical Electrodynamics
However, according to the Maxwell equation for E, this curl is equal to –B/ t, so that 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 that 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).
Chapter 6
Page 31 of 38
EM: Classical Electrodynamics
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 that 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 the 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 have got 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
Page 32 of 38
EM: Classical Electrodynamics
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
Page 33 of 38
EM: Classical Electrodynamics
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, uniform, time-independent 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 the law, 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.
6.5. Could Problem 5.2 (i.e. the analysis of the mechanical stability of
I
the system shown in the figure on the right) be solved using potential energy
1
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.
Chapter 6
Page 34 of 38
Essential Graduate Physics
EM: Classical Electrodynamics
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) 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 round pipe of a radius R and a very small thickness , made of a M ,
material with Ohmic conductivity . A sphere of mass M and radius R’ << R
R'
B
R, made of a linear magnetic material with permeability >> 0, is v0
launched, with 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 plane thin-wire loop with inductance L, resistance R, and area A is launched to fly ballistically 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 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 very long, round cylinder of radius R, made of a uniform conductor with an Ohmic conductivity and magnetic permeability , has been 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.
Chapter 6
Page 35 of 38
Essential Graduate Physics
EM: Classical Electrodynamics
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 far from
a plane 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 the Meissner-Ochsenfeld
g
effect, find the stable equilibrium position of the magnet.
6.17. A plane superconducting wire loop of area A and
inductance L may rotate, without friction, about a horizontal axis 0 (in
0
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.
Chapter 6
Page 36 of 38
Essential Graduate Physics
EM: Classical Electrodynamics
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 dc 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. Use the Ginzburg-Landau equations (54) and (63) to calculate the largest (“critical”) value of supercurrent in a uniform, long superconducting wire of a small cross-section A
2
w << L .
6.24. 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.
6.25.* 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.
6.26. 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.
6.27. 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 with 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
6.28. 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.
75 See, e.g., CM Sec. 6.3.
Chapter 6
Page 37 of 38
Essential Graduate Physics
EM: Classical Electrodynamics
6.29. As was discussed in Sec. 7, the displacement current concept allows one to generalize 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 such 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
Q Q
contour. However, it may look like the situation is different for a contour drawn
inside the capacitor – see the figure on the right. Indeed, if the contour’s size is
C
much larger than the capacitor’s thickness, the magnetic field H created by the I
I
linear current I on the contour’s line is virtually the same as that of a continuous wire, and hence the integral H dr along the contour apparently does not depend S
on its area, while the magnetic flux Dnd 2 r does, so that the equation displayed above seems invalid. (The current IS piercing this contour evidently equals zero.) Resolve the paradox, for simplicity considering an axially-symmetric system.
d
6.30. A straight, uniform, long wire with a circular cross-section of radius R is made of an Ohmic conductor with conductivity , and carries dc current I. Calculate the flux of the Poynting vector through its surface, and compare it with the Joule rate of energy dissipation.
Chapter 6
Page 38 of 38
EM: Classical Electrodynamics
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 linear, uniform, isotropic medium, which 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, acting by operator on each of Eqs. (2a), i.e. taking their curl, and then using the vector algebra identity (5.31), whose first term, for both E and H, vanishes due to Eqs. (2b), 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
where the parameter v is defined as
1 Though B rather than H is the actual 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, and hence for their magnitudes E and H.
© K. Likharev
EM: Classical Electrodynamics
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
relationship
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 that in some older physics textbooks it is not mentioned at all!
Chapter 7
Page 2 of 68
EM: Classical Electrodynamics
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 that, 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 that 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
Page 3 of 68
EM: Classical Electrodynamics
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 that / k = v, i.e.
k
1/2 ,
(7.13) Dispersion
v
relation
so that 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 that 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. 1), such complex-exponential representation of sinusoidally-changing variables is more convenient for mathematical manipulation with than 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
Page 4 of 68
EM: Classical Electrodynamics
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 that 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
E
0
0
E
x
E
E
E
x
E
x
x
x
Fig. 7.3. Time evolution of the instantaneous electric field vector in monochromatic waves with: (a) a linear polarization, (b) a circular polarization, and (c) an elliptical polarization.
Another simple case is when the moduli of the complex amplitudes E x and E y are equal, but their phases are shifted by +/2 or –/2:
i
/2
E
i
E e , E
E e
.
(7.18)
x
y
In this case
E E cos kz t , E E cos
sin
.
(7.19)
x
y
kz t E
kz t
2
This means that on the wave’s plane (normal to n), the end of the vector E moves, with the wave’s frequency , either clockwise or counterclockwise around a circle – see Fig. 3b:
( t) ( t ) .
(7.20)
Chapter 7
Page 5 of 68
EM: Classical Electrodynamics
Such waves are called circularly polarized. In the dominant convention, the wave is called right-polarized (RP) if it is described by the lower sign in Eqs. (18)-(20), i.e. if the vector of the angular frequency of the field vector’s rotation coincides with the wave propagation’s direction n, and left-polarized (LP) in the opposite case. These particular solutions of the Maxwell equations are very convenient for quantum electrodynamics, because single electromagnetic field quanta with a certain (positive or negative) spin direction may be considered as elementary excitations of the corresponding circularly-polarized wave.6 (This fact does not exclude, from the quantization scheme, waves of other polarizations, because any monochromatic wave may be presented as a linear combination of two opposite circularly-polarized waves – just as Eqs. (14) represent it as a linear combination of two linearly-polarized waves.)
Finally, in the general case of arbitrary complex amplitudes E x and E y, the field vector’s end moves along an ellipse (Fig. 3c); such wave is called elliptically polarized. The elongation (“eccentricity”) and orientation of the ellipse are completely described by one complex number, the ratio E x/ E y, i.e. by two real numbers, for example, E x/ E y and = arg( E x/ E y).7
7.2. Attenuation and dispersion
Let me start the discussion of the dispersion and attenuation effects by considering a particular case of the time evolution of the electric polarization P( t) of a dilute, non-polar medium, with negligible interaction between its elementary dipoles p( t). As was discussed in Sec. 3.3, in this case, the local electric field acting on each elementary dipole, may be taken equal to the macroscopic field E( t). Then, the dipole moment p( t) may be caused not only by the values of the field E at the same moment of time ( t), but also by those at the earlier moments t’ < t. Due to the linear superposition principle, the macroscopic polarization P( t) = np( t) should be a sum (practically, an integral) of the values of E( t’) at all moments t’ t, weighed by some function of t and t’: 8
t
Temporal
P t
( ) E t'
( G
) t
( , t' ) dt' .
(7.21) Green’s
function
6 This issue is closely related to that of the wave’s angular momentum; it will be more convenient for me to discuss it later in this chapter (in Sec. 7).
7 Note that the same information may be expressed via four so-called Stokes parameters s 0, s 1, s 2, and s 3, which are popular in practical optics, because they may be used for the description of not only completely coherent waves that are discussed here, but also of party coherent or even fully incoherent waves – including the natural light emitted by thermal sources such as our Sun. (In contrast to the coherent waves (14), whose complex amplitudes are deterministic numbers, the amplitudes of incoherent waves should be treated as random variables.) For more on the Stokes parameters, as well as many other optics topics I will not have time to cover, I can recommend the classical text by M. Born et al., Principles of Optics, 7th ed., Cambridge U. Press, 1999.
8 In an isotropic media, the vectors E, P, and hence D = 0E + P, are all parallel, and for notation simplicity, I will drop the vector sign in the following formulas of this section. I am also assuming that P at any point r is only dependent on the electric field at the same point, and hence drop the factor exp{ ikz}, the same for all variables.
This last assumption is valid if the wavelength is much larger than the elementary dipole’s size a. In most systems of interest, the scale of a is atomic (~10-10m), so that this approximation is valid up to extremely high frequencies, ~ c/ a ~ 1018 s-1, corresponding to hard X-rays.
Chapter 7
Page 6 of 68
EM: Classical Electrodynamics
The condition t’ t, which is implied by this relation, expresses a keystone principle of all science, the causal relation between a cause (in our case, the electric field E( t’) applied to each dipole) and its effect (the polarization P( t) it creates). The function G( t, t’) is called the temporal Green’s function for the electric polarization.9 To reveal its physical sense, let us consider the case when the applied field E( t) is a very short pulse at the moment t 0 < t, which may be well approximated with Dirac’s delta function:
E( t) ( t t" ) .
(7.22)
Then Eq. (21) yields just P( t) = G( t, t” ), so that the Green’s function G( t, t’) is just the polarization at moment t, created by a unit -functional pulse of the applied field at moment t’ (Fig. 4).
E( t)
E( t)
( t t' )
P( t)
P( t) G( t, t' )
Fig. 7.4. An example of the temporal
0
Green’s function for the electric
t'
t
polarization (schematically).
What are the general properties of the temporal Green’s function? First, for systems without infinite internal “memory”, G should tend to zero at t – t’ , although the type of this approach (e.g., whether the function G oscillates approaching zero, as in Fig. 4, or not) depends on the medium’s properties. Second, if parameters of the medium do not change in time, the polarization response to an electric field pulse should be dependent not on its absolute timing, but only on the time difference t
– t’ between the pulse and observation instants, when Eq. (21) is reduced to
t
P( t) E( t' ) G( t t' ) dt' E( t ) G() d .
(7.23)
0
For a sinusoidal waveform, E( t) = Re [ E e- i t], this equation yields
P t
( )
i( t
Re E e
) G () d
Re
i
E
i t
G
( ) e
d e
.
(7.24)
0
0
The expression in the last parentheses is of course nothing else than the complex amplitude P of the polarization. This means that though even if the static linear relation (3.43), P = e0 E, is invalid for an arbitrary time-dependent process, we may still keep its Fourier analog,
1
P
i
E ,
with
G( )
e
d ,
(7.25)
e
0
e
0 0
for each sinusoidal component of the process, using it as the definition of the frequency-dependent electric susceptibility e(). Similarly, the frequency-dependent electric permittivity may be defined using the Fourier analog of Eq. (3.46):
9 The idea of these functions is very similar to that of the spatial Green’s functions (see Sec. 2.10), but with a new twist, due to the causality principle. A discussion of the temporal Green’s functions in application to classical mechanics (which to some extent overlaps with our current discussion) may be found in CM Sec. 5.1.
Chapter 7
Page 7 of 68
EM: Classical Electrodynamics
Complex
D
electric
E .
(7.26a) permittivity
Then, according to Eq. (3.47), the complex permittivity is related to the temporal Green’s function by the usual Fourier transform:
P
() G( ) i
e
d .
(7.26b)
0
0
E
0
This relation shows that the function () may be complex,
() ' () i " (), with ' G( )cos d , " () G( )sin d , (7.27)
0
0
0
and that its real part ’() is always an even function of frequency, while the imaginary part ” () is an odd function of . Note that though the particular causal relationship (21) between P( t) and E( t) is conditioned by the elementary dipole independence, the frequency-dependent complex electric permittivity () may be introduced, in a similar way, if any two linear combinations of these variables are related by a similar formula.
Absolutely similar arguments show that magnetic properties of a linear, isotropic medium may be characterized by a frequency-dependent, complex permeability (). Now rewriting Eqs. (1) for the complex amplitudes of the fields at a particular frequency, we may readily repeat all calculations of Sec.
1, and verify that all its results are valid for monochromatic waves even for a dispersive (but necessarily linear!) medium. In particular, Eqs. (7) and (13) now become
1/ 2
()
Complex
Z ()
,
k() ()()1/2
,
(7.28) Z and k
()
so that the wave impedance and the wave number may be both complex functions of frequency.10
This fact has important consequences for electromagnetic wave propagation. First, plugging the representation of the complex wave number as the sum of its real and imaginary parts, k() k’() +
ik” (), into Eq. (11):
f
i[ k( ) z t]
Re f e
k" ( ) z
e
i[ k' ( ) z t]
Re f e
,
(7.29)
we see that k” () describes the rate of wave attenuation in the medium at frequency .11 Second, if the waveform is not sinusoidal (and hence should be represented as a sum of several/many sinusoidal components), the frequency dependence of k’() provides for wave dispersion, i.e. the waveform deformation at the propagation, because the propagation velocity (4) of component waves is now different.12
10 The first unambiguous observations of dispersion (for the case of light refraction) were described by Sir Isaac Newton in his Optics (1704) – even though this genius has never recognized the wave nature of light!
11 It may be tempting to attribute this effect to wave absorption, i.e. the dissipation of the wave’s energy, but we will see very soon that wave attenuation may be due to different effects as well.
12 The reader is probably familiar with the most noticeable effect of the dispersion: the difference between the group velocity v gr d/ dk’ giving the speed of the envelope of a wave packet with a narrow frequency spectrum, and the phase velocity v ph / k’ of the component waves. The second-order dispersion effect, proportional to d 2/ d 2 k’, leads to the deformation (gradual broadening) of the envelope itself. Following tradition, these effects Chapter 7
Page 8 of 68
EM: Classical Electrodynamics
As an example of such a dispersive medium, let us consider a simple but very representative Lorentz oscillator model.13 In dilute atomic or molecular systems (e.g., gases), electrons respond to the external electric field especially strongly when its frequency is close to certain frequencies j corresponding to the spectrum of quantum interstate transitions of a single atom/molecule. A phenomenological description of this behavior may be obtained from a classical model of several externally-driven harmonic oscillators, generally with non-zero damping. For a single oscillator, driven by the electric field’s force F( t) = qE( t), we can write the 2nd Newton law as m x 2
2
x x qE t ,
(7.30)
0
0
( )
where 0 is the own frequency of the oscillator, and 0 is its damping coefficient. For the electric field of a monochromatic wave, E( t) = Re [ Eexp{– i t}], we may look for a particular, forced-oscillation solution of this equation in a similar form x( t) = Re [ xexp{ –i t}].14 Plugging this solution into Eq. (30), we readily find the complex amplitude of these oscillations:
q
E
x
.
(7.31)
2
2
m ( ) 2 i
0
0
Using this result to calculate the complex amplitude of the dipole moment as p = qx, and then the electric polarization P = np of a dilute medium with n independent oscillators for unit volume, for its frequency-dependent permittivity (26) we get
2
Lorentz
q
1
oscillator
() n
.
(7.32)
0
2
2
model
m ( ) 2 i 0
0
This result may be readily generalized to the case when the system has several types of oscillators with different masses and frequencies:
f
( ) nq 2
,
(7.33)
0
j
( 2 2 ) 2
j m
i
j
j
j
where fj nj/ n is the fraction of oscillators with frequency j, so that the sum of all fj equals 1. Figure 5
shows a typical behavior of the real and imaginary parts of the complex dielectric constant, described by Eq. (33), as functions of frequency. The oscillator resonances’ effect is clearly visible, and dominates the media response at j, especially in the case of low damping, j << j. Note that in the low-damping limit, the imaginary part of the dielectric constant ” , and hence the wave attenuation k” , are negligibly small at all frequencies besides small vicinities of j, where the derivative d ’()/ d is are discussed in more detail in the quantum-mechanics part of this series (QM Sec. 2.2), because they are a crucial factor of Schrödinger’s wave mechanics. (See also a brief discussion in CM Sec. 6.3.)
13 This example is focused on the frequency dependence of rather than , because electromagnetic waves interact with “usual” media via their electric field much more than via the magnetic field. Indeed, according to Eq.
(7), the magnetic field of the wave is of the order of E/ c, so that the magnetic component of the Lorentz force (5.10), acting on a non-relativistic particle, F m ~ quB ~ ( u/ c) qE, is much smaller than that of its electric component, F e = qE, and may be neglected. However, as will be discussed in Sec. 6, forgetting about the possible dispersion of () may result in missing some remarkable opportunities for manipulating the waves.
14 If this point and Eq. (30) are not absolutely clear, please see CM Sec. 5.1 for a more detailed discussion.
Chapter 7
Page 9 of 68
EM: Classical Electrodynamics
negative.15 Thus, for a system of weakly-damped oscillators, Eq. (33) may be well approximated by a sum of singularities (“poles”):
2
q
f
()
j
n
,
for .
(7.34)
0
2
j m j
j
j
j
j
j
j'
()
'
)
0
(
"
0
0
Fig.
7.5.
Typical
frequency
dependence of the real and imaginary
parts of the complex electric
permittivity,
according
to
the
1
2
3
generalized Lorentz oscillator model.
This result is especially important because according to quantum mechanics,16 Eq. (34) (with all mj equal) is also valid for a set of non-interacting, similar quantum systems (whose dynamics may be completely different from that of a harmonic oscillator!), provided that j are replaced with frequencies of possible quantum interstate transitions, and coefficients fj are replaced with the so-called oscillator strengths of the transitions – which obey the same sum rule, j fj = 1.
At 0, the imaginary part of the complex permittivity (33) also vanishes (for any j), while its real part approaches its electrostatic (“dc”) value
n
(0) q 2
.
(7.35)
0
j 2
j m j
j
Note that according to Eq. (30), the denominator of the fraction in Eq. (35) is just the effective spring constant
2
j = mj j of the j th oscillator, so that the oscillator masses mj as such are actually (and quite naturally) not involved in the static dielectric response.
In the opposite limit of very high frequencies, >> j, j, the permittivity also becomes real and may be represented as
2
p
2
q 2
n
() 1
,
where
j .
(7.36) () in
0
2
p
plasma
0 j mj
15 In optics, such behavior is called anomalous dispersion.
16 See, e.g., QM Chapters 5-6.
Chapter 7
Page 10 of 68
EM: Classical Electrodynamics
This result is very important because it is also valid at all frequencies if all j and j vanish, for example for gases of free charged particles, in particular for plasmas – ionized atomic gases, provided that the ion collision effects are negligible. (This is why the parameter p defined by Eq. (36) is called the plasma frequency.) Typically, the plasma as a whole is neutral, i.e. the density n of positive atomic ions is equal to that of the free electrons. Since the ratio nj/ mj for electrons is much higher than that for ions, the general formula (36) for the plasma frequency is usually well approximated by the following simple expression:
2
ne
2
.
(7.37)
p
m
0
e
This expression has a simple physical sense: the effective spring constant
2
ef m ep = ne 2/0
describes the Coulomb force that appears when the electron subsystem of the plasma is shifted, as a whole, from its positive-ion subsystem, thus violating the electroneutrality.17 Hence, there is no surprise that the function () given by Eq. (36) vanishes at = p: at this resonance frequency, the polarization electric field E may oscillate, i.e. have a non-zero amplitude E = D/(), even in the absence of external forces induced by external (stand-alone) charges, i.e. in the absence of the field D these charges induce – see Eq. (3.32).
The behavior of electromagnetic waves in a medium that obeys Eq. (36), is very remarkable. If the wave frequency is above p , the dielectric constant () and hence the wave number (28) are positive and real, and waves propagate without attenuation, following the dispersion relation, Plasma
1
dispersion
k() ()
,
(7.38)
0 1/ 2
2 2p1/2
relation
c
whose plot is shown in Fig. 6.
3
2
1
p
1
ck
Fig. 7.6. The plasma dispersion law (solid
line) in comparison with the linear dispersion
00
1
2
3
in the free space (dashed line).
k /( / c)
p
At p the wave number k tends to zero. Beyond that point (i.e. at < p) , we still can use Eq. (38), but it is instrumental to rewrite it in the mathematically equivalent form
17 Indeed, let us consider such a small shift x, perpendicular to the plane surface of a broad, plane slab filled with plasma. The uncompensated ion charges, with equal and opposite surface densities = en x, that appear at the slab surfaces, create inside it, according to Eq. (2.3), a uniform electric field with Ex = en x/0. This field exerts the force – eEx = –( ne 2/0) x –ef x on each electron, pulling it back to its equilibrium position.
Chapter 7
Page 11 of 68
EM: Classical Electrodynamics
i
i
c
k() 2
2
,
where
.
