2 1/2
C ( )
cos( )
d ,
S ( )
sin( 2
)
d ,
(8.112) Fresnel
integrals
0
0
whose plots are shown in Fig. 13a. As was mentioned above, at large values of their argument (), both functions tend to ½.
(a)
(b)
0.8
1.5
C ( )
0.6
1
S ( ) ½
0.5
0.4
0.2
0
S ( )
0
0.5
0
2
4
6
8
10
0.5
0
0.5
1
1.5
C
( ) ½
Fig. 8. 13. (a) The Fresnel integrals and (b) their parametric representation.
42 Note also that since in this limit ka 2 << z, Eq. (97) shows that even the maximum value S(0, z) of the diffracted wave’s intensity is much lower than that ( S 0) of the incident wave. This is natural because the incident power S 0 a per unit length of the slit is now distributed over a much larger width x >> a, so that S(0, z) ~ S 0 ( a/ x) << S 0.
43 Slightly different definitions of these functions, affecting the constant factors, may also be met in literature.
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Plugging this expression into Eqs. (105) and (111), for the diffracted wave intensity, in the Fresnel limit (i.e. at x + a/2 << a), we get
2
2
1/ 2
1/ 2
Fresnel
S( x, z)
1 k
a
1
k
a
1
diffraction
C
x
S
x
.
(8.113)
pattern
S
2
2 z
2
2
2 z
2
2
0
A plot of this function (Fig. 14) shows that the diffraction pattern is very peculiar: while in the “dark”
region x < – a/2 the wave intensity fades monotonically, the transition to the “bright” region within the gap ( x > – a/2) is accompanied by intensity oscillations, just as at the Fraunhofer diffraction – cf. Fig. 8.
1.5
1
S
S 0 0.5
Fig. 8.14. The Fresnel
0 5
0
5
10
diffraction pattern.
k / 2 z1/2 x a / 2
This behavior, which is described by the following asymptotes,
1 sin( 2
/ 4)
k 1/2
a
1
, for
x ,
S
2 z
2
(8.114)
S 0
1 ,
for ,
4 2
is essentially an artifact of “observing” just the wave intensity (i.e. its real amplitude) rather than its phase as well. Indeed, as may be seen even more clearly from the parametric representation of the Fresnel integrals, shown in Fig. 13b, these functions oscillate similarly at large positive and negative values of their argument. (This famous pattern is called either the Euler spiral or the Cornu spiral.) Physically, this means that the wave diffraction at the slit edge leads to similar oscillations of its phase at x < – a/2 and x > – a/2; however, in the latter region (i.e. inside the slit) the diffracted wave overlaps the incident wave passing through the slit directly, and their interference reveals the phase oscillations, making them visible in the observed intensity as well.
Note that according to Eq. (113), the linear scale x of the Fresnel diffraction pattern is of the order of (2 z/ k)1/2, i.e. is complies with the estimate given by Eq. (90). If the slit is gradually narrowed so that its width a becomes comparable to x,44 the Fresnel diffraction patterns from both edges start to
“collide” (interfere). The resulting wave, fully described by Eq. (107), is just a sum of two contributions of the type (111) from both edges of the slit. The resulting interference pattern is somewhat complicated, and only when a becomes substantially less than x, it is reduced to the simple Fraunhofer pattern (110).
44 Note that this condition may be also rewritten as a ~ x, i.e. z/ a ~ a/.
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Of course, this crossover from the Fresnel to Fraunhofer diffraction may be also observed, at fixed wavelength and slit width a, by increasing z, i.e. by measuring the diffraction pattern farther and farther away from the slit.
Note also that the Fraunhofer limit is always valid if the diffraction is measured as a function of the diffraction angle alone. This may be done, for example, by collecting the diffracted wave with a
“positive” (converging) lens and observing the diffraction pattern in its focal plane.
8.7. Geometrical optics placeholder
I would not like the reader to miss, behind all these details, the main feature of the Fresnel diffraction, which has an overwhelming practical significance. Namely, besides narrow diffraction
“cones” (actually, parabolic-shaped regions) with the lateral scale x ~ ( z)1/2, the wave far behind a slit of width a >> , x, reproduces the field just behind the slit, i.e. reproduces the unperturbed incident wave inside it, and has a negligible intensity in the shade regions outside it. An evident generalization of this fact is that when a plane wave (in particular an electromagnetic wave) passes any opaque object of a large size a >> , it propagates around it, by distances z up to ~ a 2/, along straight lines, with virtually negligible diffraction effects. This fact gives the strict foundation for the notion of the wave ray (or beam), as the line perpendicular to the local front of a quasi-plane wave. In a uniform media such a ray follows a straight line,45 but it refracts in accordance with the Snell law at the interface of two media with different values of the wave speed v, i.e. different values of the refraction index. The concept of rays enables the whole field of geometric optics, devoted mostly to ray tracing in various (sometimes very sophisticated) optical systems.
This is why, at this point, an E&M course that followed the scientific logic more faithfully than this one, would give an extended discussion of the geometric and quasi-geometric optics, including (as a minimum46) such vital topics as
- the so-called lensmaker’s equation expressing the focus length f of a lens via the curvature radii of its spherical surfaces and the refraction index of the lens material,
- the thin lens formula relating the image distance from the lens via f and the source distance,
- the concepts of basic optical instruments such as glasses, telescopes, and microscopes,
- the concepts of the spherical, angular, and chromati c aberrations (image distortions).
However, since I have made a (possibly, wrong) decision to follow the common tradition in selecting the main topics for this course, I do not have time/space left for such discussion. Still, I am using this “placeholder” pseudo-section to relay my deep conviction that any educated physicist has to know the geometric optics basics. If the reader has not been exposed to this subject during their undergraduate studies, I highly recommend at least browsing one of the available textbooks.47
45 In application to optical waves, this notion may be traced back to at least the work by Hero (a.k.a. Heron) of Alexandria (circa 170 AD). Curiously, he correctly described light reflection from one or several plane mirrors, starting from the completely wrong idea of light propagation from the eye of the observer to the observed object.
46 Admittedly, even this list leaves aside several spectacular effects, including such a beauty as conical refraction in biaxial crystals – see, e.g., Chapter 15 of the textbook by M. Born and E. Wolf, cited in the end of Sec. 7.1.
47 My top recommendation for that purpose would be Chapters 3-6 and Sec. 8.6 in Born and Wolf. A simpler alternative is Chapter 10 in G. Fowles, Introduction to Modern Optics, 2nd ed., Dover, 1989. Note also that the venerable field of optical microscopy is currently revitalized by holographic/tomographic methods, using the Chapter 8
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8.8. Fraunhofer diffraction from more complex scatterers
So far, our quantitative analysis of diffraction has been limited to a very simple geometry – a single slit in an otherwise opaque screen (Fig. 12). However, in the most important Fraunhofer limit, z
>> ka 2, it is easy to get a very simple expression for the plane wave diffraction/interference by a plane orifice (with a linear size scale a) of arbitrary shape. Indeed, the evident 2D generalization of the approximation (106)-(107) is
ik x x' 2 y y' 2
I I
exp
x y
dx'dy'
2 z
orifice
(8.115)
ik( 2
2
x y )
kxx'
kyy'
exp
exp i
i
dx'dy' ,
2
z
z
z
orifice
so that besides the inconsequential total phase factor, Eq. (105) is reduced to
General
Fraunhofer
f (ρ) f
exp iκ '
ρ d
'
f
T ( '
ρ ) exp iκ '
ρ d
'
.
(8.116)
0
2
diffraction
2
0
orifice
screen
all
pattern
Here the 2D vector (not to be confused with wave vector k, which is virtually perpendicular to !) is defined as
ρ
κ k
q k - k ,
(8.117)
0
z
and = { x, y} and ’ = { x’, y’} are 2D radius-vectors in the, respectively, observation and orifice planes
– both nearly normal to the vectors k and k0.48 In the last form of Eq. (116), the function T( ’) describes the screen’s transparency at point ’, and the integral is over the whole screen plane z’ = 0. (Though the two forms of Eq. (116) are strictly equivalent only if T( ’ ) is equal to either 1 or 0, its last form may be readily obtained from Eq. (91) with f(r ’) = T( ’ ) f 0 for any transparency profile, provided that T( ’ ) is any function that changes substantially only at distances much larger than 2/ k.) From the mathematical point of view, the last form of Eq. (116) is just the 2D spatial Fourier transform of the function T( ’), with the variable defined by the observation point’s position:
( z/ k) ( z/2). This interpretation is useful because of the experience we all have with the Fourier transform, if only in the context of its time/frequency applications. For example, if the orifice is a single small hole, T( ’) may be approximated by a delta function, so that Eq. (116) yields f() const. This result corresponds (at least for the small diffraction angles / z, for which the Huygens approximation is valid) to a spherical wave spreading from the point-like orifice. Next, for two small holes, Eq. (116) immediately gives the interference pattern (103). Let me now use Eq. (116) to analyze other simplest (and most important) 1D transparency profiles, leaving a few 2D cases for the reader’s exercise.
(i) A single slit of width a (Fig. 12) may be described by transparency
scattered wave’s phase information. These methods are especially productive in biology and medicine – see, e.g., M. Brezinski, Optical Coherence Tomography, Academic Press, 2006, and G. Popescu, Quantitative Phase Imaging of Cells and Tissues, McGraw-Hill (2011).
48 Note that for a thin uniform plate of the same shape as the orifice we are discussing now, the Born phase integral (63) with q << k gives a result functionally similar to Eq. (116).
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,
1
for x' a / ,
2
T ( '
ρ )
(8.118)
,
0
otherwise.
Its substitution into Eq. (116) yields
a / 2
exp i a / 2 exp i a / 2
x
x
a
x
kxa
f (ρ) f
i x' dx' f
(8.119)
x
0
exp
sinc
sinc
,
0
i
z
a
x
2
2
/ 2
naturally returning us to Eqs. (64) and (110), and hence to the red lines in Fig. 8 for the wave intensity.
(Please note again that Eq. (116) describes only the Fraunhofer, but not the Fresnel diffraction!) (ii) Two infinitely narrow, similar, parallel slits with a larger distance a between them (i.e. the simplest model of Young’s two-slit experiment) may be described by taking
a
a
T ( '
ρ ) x' x' ,
(8.120)
2
2
so that Eq. (116) yields the generic 1D interference pattern,
i a
i a
a
x
x
x
kxa
f (ρ) f exp
exp
cos
cos
,
(8.121)
0
2
2
2
2 z
whose intensity is shown with the blue line in Fig. 8.
(iii) In a more realistic model of Young’s experiment, each slit has a width (say, w) that is much larger than the light wavelength , but still much smaller than the slit spacing a. This situation may be described by the following transparency function
,
1
for x' a / 2 w / ,
2
T ( '
ρ )
(8.122)
,
0
otherwise,
for which Eq. (116) yields a natural combination of the results (119) (with a replaced with w) and (121):
kxw
kxa
f (r) sinc
cos
.
(8.123)
2 z
2 z
This is the usual interference pattern, but modulated with a Fraunhofer-diffraction envelope – shown in Fig. 15 with the dashed blue line. Since the function sinc2 decreases very fast beyond its first zeros at
= , the practical number of observable interference fringes is close to 2 a/ w.
a 10 w
0.8
2
f (ρ) 0.6
f (0) 0.4
0.2
0
5
0
5
x /(2 z /
)
kw
Fig. 8.15. Young’s double-slit interference pattern for a finite-width slit.
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(iv) A structure very useful for experimental and engineering practice is a set of many parallel, similar slits, called the diffraction grating.49 If the slit’s width is much smaller than period d of the grating, its transparency function may be approximated as
T ( '
ρ )
( x' nd) ,
(8.124)
n
and Eq. (116) yields
n
n
nkxd
f (ρ)
exp in d
exp
i
.
(8.125)
x
n
n
z
This sum vanishes for all values of xd that are not multiples of 2, so that the result describes sharp intensity peaks at the following diffraction angles:
x
x
2
m
m .
(8.126)
m
z
k
kd
d
m
m
Taking into account that this result is only valid for small angles m << 1, it may be interpreted exactly as Eq. (59) – see Fig. 6a. However, in contrast with the interference (121) from two slits, the destructive interference from many slits kills the net wave as soon as the angle is even slightly different from each value (60). This is very convenient for spectroscopic purposes because the diffraction lines produced by multi-frequency waves do not overlap even if the frequencies of their adjacent components are very close.
Two unavoidable features of practical diffraction gratings make their properties different from this simple, ideal picture. First, the finite number N of slits, which may be described by limiting the sum (125) to the interval n = [- N/2, + N/2], results in a non-zero spread, / ~ 1/ N, of each diffraction peak, and hence in the reduction of the grating’s spectral resolution. (Unintentional variations of the inter-slit distance d have a similar effect, so that before the advent of high-resolution photolithography, special high-precision mechanical tools had been used for grating fabrication.)
Second, a finite slit width w leads to the diffraction peak pattern modulation by the sinc2( kw/2) envelope, similar to the pattern shown in Fig. 15. Actually, for spectroscopic purposes, such modulation is sometimes a plus, because only one diffraction peak (say, with m = 1) is practically used, and if the frequency spectrum of the analyzed wave is very broad (covers more than one octave), the higher peaks produce undesirable hindrance. Because of this reason, w is frequently selected to be equal exactly to d/2, thus suppressing each other diffraction maximum. Moreover, sometimes semi-transparent films are used to make the transparency function T(r ’) continuous and close to a sinusoidal one: 2 x'
T
1
2 x'
2 x'
T ( '
ρ ) T T cos
T
.
(8.127)
0
1
0
exp i
exp i
d
2
d
d
Plugging the last expression into Eq. (116) and integrating, we see that the output wave consists of just 3
components: the direct-passing wave (proportional to T 0) and two diffracted waves (proportional to T 1) propagating in the directions of the two lowest Bragg angles, 1 = / d.50
49 The rudimentary diffraction grating effect, produced by the parallel fibers of a bird’s feather, was discovered as early as 1673 by James Gregory (who also invented the “Gregorian” telescope – one of the basic designs for reflecting telescopes).
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The same Eq. (116) may be also used to obtain one more general (and rather curious) result, called the Babinet principle.51 Consider two experiments with the diffraction of similar plane waves on two “complementary” screens – such that together they would cover the whole plane, without a hole or an overlap. (Think, for example, about an opaque disk of radius R and a large opaque screen with a round orifice of the same radius.) Then, according to the Babinet principle, the diffracted wave patterns produced by these two screens in all directions with 0 are identical.
The proof of this principle is straightforward: since the transparency functions produced by the screens are complementary:
T ( '
ρ ) T (ρ ' ) T ( '
ρ ) ,
1
(8.128)
1
2
and the diffracted wave is (in the Fraunhofer approximation only!) a Fourier transform of T( ’), which is a linear operation, we get
f (ρ) f (ρ) f (ρ),
(8.129)
1
2
0
where f 0 is the wave “scattered” by the composite screen with T 0( ’) 1, i.e. the unperturbed initial wave propagating in the initial direction ( = 0). In all other directions, f 1 = – f 2, i.e. the diffracted waves are indeed similar besides the difference in sign – which is equivalent to a phase shift by . However, it is important to remember that the Babinet principle notwithstanding, in real experiments, with screens at finite distances, the diffracted waves may interfere with the unperturbed plane wave f 0(), leading to different diffraction patterns in cases 1 and 2 – see, e.g., Fig. 14 and its discussion.
8.9. Magnetic dipole and electric quadrupole radiation
Throughout this chapter, we have seen how many important results may be obtained from Eq.
(26) for the electric dipole radiation by a small-size source (Fig. 1). Only in rare cases when this radiation is absent, for example, if the dipole moment p of the source equals zero (or does not change in time – either at all or at the frequency of our interest), higher-order effects may be important. I will now discuss the main two of them, quadrupole electric radiation and dipole magnetic radiation.
In Sec. 2 above, the electric dipole radiation was calculated by plugging the expansion (19) into the exact formula (17b) for the retarded vector potential A(r, t). Let us make a more exact calculation, by keeping the second term of that expansion as well:
R
r
'
r n
'
r n
r
j '
r , t
j '
r , t
j '
r , t'
,
t'
where
t .
(8.130)
v
v
v
v
v
Since the expansion is only valid if the last term in the time argument of j is relatively small, in the Taylor expansion of j with respect to that argument we may keep just two leading terms:
'
r n
j '
r t'
'
r n
j '
r , t'
,
j '
r , t'
,
(8.131)
v
t'
v
50 Similar tricks are used in the so-called phased-array antennas, broadly used in radar systems and radioastronomy, in which electronically controlled mutual phase shifts of microwave signals feeding many similar component antennas are used to steer the direction of the resulting narrow beam. For more on this important technology, see, e.g. T. Milligan, Modern Antenna Design, 2nd ed., Wiley (2005).
51 Named after Jacques Babinet (1784-1874) who made several important contributions to optics.
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so that Eq. (17b) yields A = Ad + A ’, where Ad is the electric dipole contribution as given by Eq. (23), and A ’ is the new term of the next order in the small parameter r’ << r:
'
A r, t
j ' r t' ' r n
d 3
,
r' .
(8.132)
4 rv t'
Just as it was done in Sec. 2, let us evaluate this term for a system of non-relativistic particles with electric charges qk and radius vectors r k( t):
d
'
A r, t
q
r r n
.
(8.133)
k k k
4 rv dt
k
t t'
Using the “bac minus cab” identity of the vector algebra again,52 the vector operand of Eq. (133) may be rewritten as
r r n r r n r n r
r r n r n r r n r
k k
1 k k 1 k k 1
1
(
)
k
k
k
k
1
k
k
2
2
2
2
2
(8.134)
1
1 d
(r r ) n
r n r
k
k
k k ,
2
2 dt
so that the right-hand side of Eq. (133) may be represented as a sum of two terms, A ’ = Am + Aq, where
r
1
A r, t
m
t' n
m
t n
m t r t q r t (8.135)
m
, with
k
k k ;
4 rv
4 rv
v
2 k
d 2
A
,
.
(8.136)
q r t
2 q
r n r
k k
k
8 rv dt k
t t'
Comparing the second of Eqs. (135) with Eq. (5.91), we see that m is just the total magnetic moment of the source. On the other hand, the first of Eqs. (135) is absolutely similar in structure to Eq.
(23), with p replaced with (mn)/ v, so that for the corresponding component of the magnetic field it gives (in the same approximation r >> ) a result similar to Eq. (24):
Magnetic
dipole
r
r
radiation:
B r
( , t)
m
.
(8.137)
m
t n
n
2
m t n
field
4 rv
v
4 rv
v
According to this expression, just as at the electric dipole radiation, the vector B is perpendicular to the vector n, and its magnitude is also proportional to sin, where is now the angle between the direction toward the observation point and the second time derivative of the vector m – rather than p:
r
B
m t
.
(8.138)
m
sin
4
2
rv
v
As the result, the intensity of this magnetic dipole radiation has a similar angular distribution: Magnetic
dipole
2
Z
r 2
radiation:
S ZH
(8.139)
r
m t
2
sin
2
2
power
(4 v r)
v
52 If you still need it, see MA Eq. (7.5).
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- cf. Eq. (26), besides the (generally) different meaning of the angle .
Note, however, that this radiation is usually much weaker than its electric-dipole counterpart.
For example, for a non-relativistic particle with electric charge q, moving on a trajectory of linear size
~ a, the electric dipole moment is of the order of qa, while its magnetic moment scales as qa 2, where
is the motion frequency. As a result, the ratio of the magnetic and electric dipole radiation intensities is of the order of ( a/ v)2, i.e. the squared ratio of the particle’s speed to the speed of the emitted waves –
that has to be much smaller than 1 for our non-relativistic calculation to be valid.
The angular distribution of the electric quadrupole radiation, described by Eq. (136), is more complicated. To show this, we may add to Aq a vector parallel to n (i.e. along the wave’s propagation), getting
r
A r, t
Q
t ,
where Q q 3r n r
n r 2 ,
(8.140)
q
k
k
k k
24 rv
v
k
because this addition does not give any contribution to the transverse component of the electric and magnetic fields, i.e. to the radiated wave. According to the above definition of the vector Q, its Cartesian components may be represented as
3
Q
Q n ,
(8.141)
j
jj' j'
j'1
where Q jj’ are the elements of the electric quadrupole tensor of the system – see the last of Eqs. (3.4):53
Q
q
r
3 r
r 2
.
(8.142)
jj '
k
j j'
jj ' k
k
Now taking the curl of the first of Eqs. (140) at r >> , we get
r
Electric
B
,
.
(8.143) quadrupole
q r t
n Q
t
24 rv 2
v
radiation:
field
This expression is similar to Eqs. (24) and (137), but according to Eqs. (140) and (142), components of the vector Q do depend on the direction of the vector n, leading to a different angular dependence of Sr.
As the simplest example, let us consider the system of two equal point electric charges moving symmetrically, at equal distances d( t) << from a stationary center – see Fig. 16.
n
x
q
q
z
Fig. 8.16. The simplest system emitting
d ( t)
d ( t)
electric quadrupole radiation.
Due to the symmetry of the system, its dipole moments p and m (and hence its electric and magnetic dipole radiation) vanish, but the quadrupole tensor (142) still has non-zero elements. With the coordinate choice shown in Fig. 16, these elements are diagonal:
53 Let me hope that the reader has already acquired some experience in the calculation of this tensor’s elements –
e.g., for the simple systems specified in Problems 3.2-3.4.
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Q Q 2 2
qd ,
Q 4
2
qd .
(8.144)
xx
yy
zz
With the x-axis selected within the common plane of the z-axis and the direction n toward the observation point (Fig. 16), so that nx = sin, ny = 0, and nz = cos, Eq. (141) yields Q 2 2
qd sin ,
Q ,
0
Q 4
2
qd cos ,
(8.145)
x
y
z
and the vector product in Eq. (143) has only one non-vanishing Cartesian component:
3
d
n
Q n Q
n Q
6
q sin cos
d t .
(8.146)
y
z
x
x z
3
2 ( )
dt
As a result, the quadrupole radiation intensity, S B 2
q , is proportional to sin2cos2, i.e. vanishes not
only along the symmetry axis of the system (as the electric-dipole and the magnetic-dipole radiations would), but also in all directions perpendicular to this axis, reaching its maxima at = /4.
For more complex systems, the angular distribution of the electric quadrupole radiation may be different, but it may be proved that its total (instant) power always obeys the following simple formula: Electric
3
quadrupole
Z
2
radiation:
P
.
(8.147)
q
4 jj'
Q
720 v
power
j, j' 1
Let me finish this section by giving, also without proof, one more fact important for some applications: due to their different spatial structure, the magnetic-dipole and electric-quadrupole radiation fields do not interfere, i.e. the total power of radiation (neglecting the electric-dipole and higher multipole terms) may be found as the sum of these components, calculated independently. On the contrary, the electric-dipole and magnetic-dipole radiations of the same system they typically interfere coherently, so that their radiation fields (rather than powers) should be summed up.
8.10. Exercise problems
8.1. Equation (8.8) of the lecture notes obviously has standing-wave solutions ( r, t) = Re[ C sin kr exp{– i t}], turning the scalar potential = / r into a finite constant at r = 0 and into zero at kr = n, with n = 0, 1, 2,… This fact seems to imply that a metallic cavity of radius R has resonant modes with a purely radial electric field E(r) = nr E( r), and the lowest nonvanishing of them, with k = / R, gives the lowest (fundamental) frequency vk = ( v/ R) of the cavity. Is this conclusion correct?
8.2. Simplify the Lorentz reciprocity theorem (6.121) for space-localized field sources. Then find out what it says about the fields of two compact, well-separated sources of the electric-dipole radiation.
8.3. In the electric-dipole approximation, calculate the angular distribution and the total power of electromagnetic radiation by the hydrogen atom within the following classical model: an electron rotates, at a constant distance R, about a much heavier proton. Use this result to calculate the law of a gradual reduction of R in time. Finally, evaluate the classical lifetime of the atom, borrowing the initial value of R from quantum mechanics: R(0) = r B 0.5310-10 m.
8.4. A non-relativistic particle of mass m, with electric charge q, is placed into a uniform, time-independent magnetic field B. Derive the law of decrease of the particle’s kinetic energy due to its Chapter 8
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electromagnetic radiation at the cyclotron frequency c = qB/ m. Evaluate the rate of such radiation cooling of electrons in a magnetic field of 1 T, and estimate the energy interval in which this result is quantitatively correct.
Hint: The cyclotron motion will be discussed in detail (for arbitrary particle velocities v ~ c) in Sec. 9.6 below, but I hope that the reader knows that in the non-relativistic case ( v << c) the above formula for
2
c may be readily obtained by combining the 2nd Newton law mv / R = qv B for the circular motion of the particle under the effect of the magnetic component of the Lorentz force (5.10), and the geometric relation v = Rc. (Here v is the particle’s velocity in the plane normal to the vector B.)
8.5. A particle with mass m and electric charge q and kinetic energy T collides head-on with a much more massive particle of charge Z q, in free space. Calculate the total energy of electromagnetic radiation during this collision, assuming it to much lower than T.
8.6. Solve the dipole antenna radiation problem discussed in Sec. 2 (see Fig. 3) for the optimal length l = /2, assuming that the current distribution in each of its arms is sinusoidal: I( z, t) = I 0cos( z/ l) cos t. 54
8.7. A plane wave is scattered by a localized object in free space. Relate the differential cross-section of the wave’s scattering to the average force it exerts on the object. Use this general relation to calculate the force exerted by a plane monochromatic wave on a free non-relativistic particle, compare the result with those obtained in Problems 7.4 and 7.5, and discuss the comparison.
8.8. Use the Lorentz oscillator model of a bound charge, given by Eq. (7.30), to explore the transition between the two scattering limits discussed in Sec. 3, and in particular, the resonant scattering taking place at 0. In this context, discuss the contribution of scattering to the oscillator’s damping.
8.9.* A sphere of radius R, made of a material with a uniform permanent electric polarization P0
and a constant mass density , is free to rotate about its center. Calculate the average total cross-section of scattering, by the sphere, of a linearly polarized electromagnetic wave of frequency << R/ c, propagating in free space, in the limit of small wave amplitude, assuming that the initial orientation of the polarization vector P0 is random.
8.10. Use Eq. (56) to analyze the interference/diffraction pattern produced by a
plane wave’s scattering on a set of N similar, equidistant points on a straight line k0 d
N
normal to the direction of the incident wave’s propagation – see the figure on the right.
d
Discuss the trend(s) of the pattern in the limit N .
8.11. Use the Born approximation to calculate the differential cross-section of the plane wave scattering by a uniform dielectric sphere of an arbitrary radius R. In the limits kR << 1 and 1 << kR
(where k is the wave number), analyze the angular dependence of the differential cross-section and calculate the total cross-section of scattering.
54 As was emphasized in Sec. 2, this is a reasonable guess rather than a controllable approximation. The exact (rather involved!) theory shows that this assumption gives errors ~5%, depending on the wire’s diameter.
Chapter 8
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8.12. A sphere of radius R is made of a uniform dielectric material, with an arbitrary dielectric constant. Derive an exact expression for its total cross-section of scattering of a linearly-polarized low-frequency ( k << 1/ R) wave and compare the result with the solution of the previous problem.
8.13. Use the Born approximation to calculate the differential cross-section of the plane wave scattering on a right, circular cylinder of length l and radius R, for an arbitrary angle of incidence.
8.14. Formulate the quantitative condition of the Born approximation’s validity for a uniform dielectric scatterer, with all linear dimensions of the order of the same scale a.
8.15. If a scatterer absorbs some part of the incident wave’s power, it may be characterized by an absorption cross-section a defined similarly to Eq. (39) for the scattering cross-section: a
P
,
a
2
E
/ 2 Z
0
where the numerator is the time-averaged power absorbed is the scatterer. Use two different approaches to calculate a of a very small sphere of radius R << k-1,s, made of a nonmagnetic material with an Ohmic conductivity and the high-frequency permittivity opt = 0. Can a of such a sphere be larger than its geometric cross-section R 2?
8.16. Use the Huygens principle to calculate the wave’s intensity on the symmetry plane of the slit diffraction experiment (i.e. at x = 0 in Fig. 12), for arbitrary ratio z/ ka 2.
8.17. A plane wave with wavelength is normally
S( z) ?
incident on an opaque, plane screen, with a round orifice of
orifice
radius R >> . Use the Huygens principle to calculate the
2 R
passing wave’s intensity distribution along the system’s
0
z
symmetry axis, at distances z >> R from the screen (see the
S 0
figure on the right), and analyze the result.
8.18. A plane monochromatic wave is now
S( z) ?
disk
normally incident on an opaque circular disk of radius R
2 R
>> . Use the Huygens principle to calculate the wave’s
z
intensity at a distance z >> R behind the disk’s center
S
0
0
(see the figure on the right). Discuss the result.
8.19. Use the Huygens principle to analyze the Fraunhofer diffraction of a plane wave normally incident on a square-shaped hole, of size a a, in an opaque screen. Sketch the diffraction pattern you would observe at a sufficiently large distance, and quantify the expression “sufficiently large” for this case.
8.20. Use the Huygens principle to analyze the propagation of a monochromatic Gaussian beam described by Eq. (7.181), with the initial characteristic width a 0 >> , in a uniform, isotropic medium.
Use the result for a semi-quantitative derivation of the so-called Abbe limit for the spatial resolution of Chapter 8
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EM: Classical Electrodynamics
an optical system: w min = /2sin, where is the half-angle of the wave cone propagating from the object, and captured by the system.
T
8.21. Within the Fraunhofer approximation,
1
analyze the pattern produced by a 1D diffraction
grating with the periodic transparency profile shown
w
in the figure on the right, for the normal incidence of
a plane, monochromatic wave.
x
d
0
d
q
8.22. N equal point charges are attached, at equal intervals, to a circle rotating with a constant angular velocity about its center – see the figure on the R
right. For what values of N does the system emit:
(i) the electric dipole radiation?
2 / N
(ii) the magnetic dipole radiation?
(iii) the electric quadrupole radiation?
8.23. What general statements can you make about:
(i) the electric dipole radiation, and
(ii) the magnetic dipole radiation,
due to a collision of an arbitrary number of similar classical, non-relativistic particles?
8.24. Calculate the angular distribution and the total power radiated by a small round loop antenna with radius R, fed with ac current I( t) with frequency and amplitude I 0, into the free space.
8.25. The orientation of a magnetic dipole m, of a fixed magnitude, is rotating about a certain axis with angular velocity , with the angle between them staying constant. Calculate the angular distribution and the average power of its radiation (into the free space).
8.26. Solve Problem 12 (also in the low-frequency limit kR << 1), for the case when the sphere’s material has a frequency-independent Ohmic conductivity , and opt = 0, in two limits: (i) of a very large skin depth (s >> R), and
(ii) of a very small skin depth (s << R).
8.27. Complete the solution of the problem started in Sec. 9, by calculating the full power of radiation of the system of two charges oscillating in antiphase along the same straight line – see Fig. 16.
Also, calculate the average radiation power for the case of harmonic oscillations, d( t) = acos t, compare it with the case of a single charge performing similar oscillations, and interpret the difference.
8.28. The system of four alternating charges located at the angles of a square, considered in Problem 3.3(i), is now being rotated around the axis normal to their plane and passing through the square’s center, with a constant angular frequency << a/ v. Calculate the time-averaged angular distribution and the total power of the resulting radiation.
Chapter 8
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Chapter 9. Special Relativity
This chapter starts with a review of special relativity’s basics, including its very convenient 4-vector formalism. This background is then used for the analysis of the relation between the electromagnetic field’s values measured in different inertial reference frames moving relative to each other. The results enable us to discuss relativistic particle dynamics in the electric and magnetic fields, and the analytical mechanics of the particles – and of the electromagnetic field as such.
9.1. Einstein postulates and the Lorentz transform
As was emphasized at the derivation of expressions for the dipole and quadrupole radiation in the last chapter, they are only valid for systems of non-relativistic particles moving with velocities u much lower than c. In order to generalize these results to particles moving with arbitrary u, we need help from the relativity theory. Moreover, an analysis of the motion of charged relativistic particles in electric and magnetic fields is also a natural part of electrodynamics. This is why I will follow the tradition of using this course for a (by necessity, brief) introduction to the special relativity theory. This theory is based on the fundamental idea that measurements of physical variables (including the spatial and even temporal intervals between two events) may give different results in different reference frames, in particular in two inertial frames moving relative to each other translationally (i.e. without rotation), with a certain constant velocity v (Fig. 1).
y
y'
r { x, y, z
}
' r { x' , y' , z' }
v
0
'
0
x
x'
Fig. 9.1. The translational, uniform
mutual motion of two reference frames.
z
z'
In the non-relativistic (Newtonian) mechanics the problem of transfer between such reference frames has a simple solution at least in the limit v << c, because the basic equation of particle dynamics (the 2nd Newton law) 1
m r
U r
r ,
(9.1)
k k
k
(
)
k
k '
k '
where U is the potential energy of inter-particle interactions, is invariant with respect to the so-called Galilean transformation (or just “transform” for short).2 Choosing the coordinates in both frames so that their axes x and x’ are parallel to the vector v (as in Fig. 1), the transform may be represented as 1 Let me hope that the reader does not need a reminder that for Eq. (1) to be valid, the reference frames 0 and 0 ’
have to be inertial – see, e.g., CM Sec. 1.2.
2 It had been first formulated by Galileo Galilei, if only rather informally, as early as in 1638 – four years before Isaac Newton was born! Note also the very unfortunate term “boost”, used sometimes to describe such translational transformations. (It is especially unnatural in the special relativity, not describing accelerations.) In my course, this term is avoided, with the equivalent “transform” used instead.
© K. Likharev
Essential Graduate Physics
EM: Classical Electrodynamics
x x' vt' ,
y y' ,
z z' ,
t t' ,
(9.2a) Galilean
transform
and plugging Eq. (2a) into Eq. (1), we get an absolutely similarly looking equation of motion in the
“moving” reference frame 0 ’. Since the reciprocal transform,
x' x vt,
y y' ,
z' z,
t' t ,
(9.2b)
is similar to the direct one, with the replacement of (+ v) with (– v), we may say that the Galilean invariance means that there is no “master” ( absolute) spatial reference frame in classical mechanics, although the spatial and temporal intervals between different instant events are absolute, i.e. reference-frame invariant: x = x’,…, t = t’.
However, it is straightforward to use Eq. (2) to check that the form of the wave equation 2
2
2
1
2
f 0
,
(9.3)
2
2
2
2
2
x
y
z
c t
describing, in particular, the electromagnetic wave propagation in free space,3 is not Galilean-invariant.4
For the “usual” (say, elastic) waves, which obey a similar equation albeit with a different speed,5 this lack of Galilean invariance is natural and is compatible with the invariance of Eq. (1), from which the wave equation originates. This is because the elastic waves are essentially the oscillations of interacting particles of a certain medium (e.g., an elastic solid), making the reference frame connected to this medium, special. So, if the electromagnetic waves were oscillations of a certain special medium (which was first called the “luminiferous aether”6 and later aether – or just “ether”), similar arguments might be applicable to reconcile Eqs. (2) and (3).
The detection of such a medium was the goal of the measurements carried out between 1881 and 1887 (with better and better precision) by Albert Abraham Michelson and Edward Williams Morley, which are sometimes called “the most famous failed experiments in physics”. Figure 2 shows a crude scheme of these experiments.
mirror
v v
R
E
semi-
expt 1
light
transparent
v E
source
mirror
mirror
expt 2
Fig. 9.2. The Michelson-
Earth
Morley experiment.
detector
3 The discussions in this chapter and most of the next chapter will be restricted to the free-space (and hence dispersion-free) case; some media effects on the radiation by relativistic particles will be discussed in Sec.10.4.
4 It is interesting that the usual (non-relativistic) Schrödinger equation, whose fundamental solution for a free particle is a similar monochromatic wave (albeit with a different dispersion law), is Galilean-invariant, with a certain change of the wavefunction’s phase – see, e.g., QM Chapter 1.
5 See, e.g., CM Secs. 6.5 and 7.7.
6 In ancient Greek mythology, aether is the clean air breathed by the gods residing on Mount Olympus.
Chapter 9
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A nearly-monochromatic wave from a light source is split into two parts (optimally, of equal intensity), using a semi-transparent mirror tilted by angle /4 to the incident wave direction. These two partial waves are reflected back by two fully-reflecting mirrors and arrive at the same semi-transparent mirror again. Here a half of each wave is directed toward the light source (they vanish there without affecting the source), but another half is passed toward an intensity detector, forming, with its counterpart, an interference pattern similar to that in the Young experiment. Thus each of the interfering waves has traveled twice (back and forth) each of two mutually perpendicular “arms” of the interferometer. Assuming that the aether, in which light propagates with speed c, moves with speed v < c along one of the arms, of length ll, it is straightforward (and hence left for the reader’s exercise :-) to get the following expression for the difference between the light roundtrip times:
2
2
l
l
t
l
l v
t
,
(9.4)
c
2
2
1 v / c 1/2
2
2
1 v / c c c
where lt is the length of the second, “transverse” arm of the interferometer (perpendicular to v), and the last, approximate expression is valid at lt ll l and v << c.
Since the Earth moves around the Sun with a speed v E 30 km/s 10-4 c, the arm positions relative to this motion alternate, due to the Earth’s rotation about its axis, every 6 hours – see the right panel of Fig. 2. Hence if we assume that the aether rests in the Sun’s reference frame, then t (and the corresponding shift of the interference fringes), has to change its sign with this half-period as well. The same alternation may be achieved, at a smaller time scale, by a deliberate rotation of the instrument by
/2. In the most precise version of the Michelson-Morley experiment (circa 1887), this shift was expected to be close to 0.4 of the interference pattern period. The results of the search for such a shift were negative, with the error bar about 0.01 of the period.7
The most prominent immediate explanation of this zero result8 was suggested in 1889 by George Francis FitzGerald and (independently and more qualitatively) by H. Lorentz in 1892: as evident from Eq. (4), if the longitudinal arm of the interferometer itself experiences the so-called length contraction, 1/ 2
2
v
l ( v) l ( )
0 1
,
(9.5)
l
l
2
c
while the transverse arm’s length is not affected by its motion through the aether, this effect kills the shift t. This radical idea received strong support from the proof, in 1887-1905, that the Maxwell equations, and hence the wave equation (3), are form-invariant under the so-called Lorentz transform,9
which in particular describes Eq. (5). For the choice of coordinates shown in Fig. 1, the transform reads 7 Through the 20th century, the Michelson-Morley-type experiments were repeated using more and more refined experimental techniques, always with zero results for the apparent aether motion speed. For example, recent experiments using cryogenically cooled optical resonators have reduced the upper limit for such speed to just 310-15 c –see H. Müller et al., Phys. Rev. Lett. 91, 020401 (2003).
8 The zero result of a slightly later experiment, namely a precise measurement of the torque which should be exerted by the moving aether on a charged capacitor, carried out in 1903 by F. Trouton and H. Noble (following G. FitzGerald’s suggestion), seconded the Michelson and Morley’s conclusions.
9 The theoretical work toward this result included important contributions by Woldemart Voigt (in 1887), Hendrik Lorentz (in 1892-1904), Joseph Larmor (in 1897 and 1900), and Henri Poincaré (in 1900 and 1905).
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2
x' vt'
t' ( v / c ) x'
x
Lorentz
, y y' , z z' , t
.
(9.6a) transform
2
2
1 v / c 1/2
2 2
1 v / c 1/2
It is elementary to solve these equations for the primed coordinates to get the reciprocal transform x vt
t ( v / 2
c ) x
x'
y' y
z' z
t'
(9.6b)
1
2
v / 2
c ,
,
,
1/ 2
1 2 2
v c .
/
1/ 2
(I will soon represent Eqs. (6) in a more elegant form – see Eqs. (19) below.)
The Lorentz transform relations (6) are evidently reduced to the Galilean transform formulas (2) at v 2 << c 2. However, all attempts to give a reasonable interpretation of these equalities while keeping the notion of the aether have failed, in particular because of the restrictions imposed by results of earlier experiments carried out in 1851 and 1853 by Hippolyte Fizeau – which were repeated with higher accuracy by the same Michelson and Morley in 1886. These experiments have shown that if one sticks to the aether concept, this hypothetical medium has to be partially “dragged” by any moving dielectric material with a speed proportional to ( – 1). Such local drag would be irreconcilable with the assumed continuity of the aether.
In his famous 1905 paper, Albert Einstein suggested a bold resolution of this contradiction, essentially removing the concept of the aether altogether.10 Moreover, he argued that the Lorentz transform is the general property of time and space, rather than of the electromagnetic field alone. He started with two postulates, the first one essentially repeating the relativity principle formulated a bit earlier (in 1904) by H. Poincaré in the following form:
“… the laws of physical phenomena should be the same, whether for an observer fixed or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion.”11
The second Einstein postulate was that the speed of light c, in free space, should be constant in all reference frames. (This is essentially a denial of the aether’s existence.)
Then, Einstein showed that the Lorenz transform relations (6) naturally follow from his postulates, with a few (very natural) additional assumptions. Let a point source emit a short flash of light, at the moment t = t’ = 0 when the origins of the reference frames shown in Fig. 1 coincide. Then, according to the second of Einstein’s postulates, in each of the frames, the spherical wave propagates with the same speed c, i.e. the coordinates of points of its front, measured in the two frames, have to obey the following equalities:
( ct)2 ( 2
2
2
x y z ) ,
0
(9.7)
( ct' )2 ( 2
2
2
x' y' z' ) 0.
10 In hindsight, this was much relief, because the aether had been a very awkward construct to start with. In particular, according to the basic theory of elasticity (see, e.g., CM Ch. 7), in order to carry such transverse waves as the electromagnetic ones, this medium would need to have a non-zero shear modulus, i.e. behave as an elastic solid – rather than as a rarified gas hypothesized initially by C. Huygens.
11 Note that though the relativity principle excludes the notion of the special (“absolute”) spatial reference frame, its quoted verbal formulation still leaves the possibility of the Galilean “absolute time” t = t’ open. The quantitative relativity theory kills this option – see Eqs. (6) and their discussion below.
Chapter 9
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What may be the general relation between the combinations in the left-hand side of these equations –
not for this particular wave’s front, but in general? A very natural (essentially, the only justifiable) choice is
( ct)2 ( 2
2
2
x y z ) f ( 2
v
) ( ct' )2 ( 2
2
2
x' y' z' ).
(9.8)
Now, according to the first postulate, the same relation should be valid if we swap the reference frames ( x x’, etc.) and replace v with (– v). This is only possible if f 2 = 1, so that excluding the option f = –1
(which is incompatible with the Galilean transform in the limit v/ c 0), we are left with f = +1, i.e.
( ct)2 ( 2
2
2
x y z ) ( ct' )2 ( 2
2
2
x' y' z' ) .
(9.9)
For the line with y = y’ = 0 and z = z’ = 0, Eq. (9) is reduced to 2
2
2
2
( ct) x ( ct' ) x' .
(9.10)
It is very illuminating to interpret this relation as the one resulting from a mutual rotation of the reference frames (that now have to include clocks to measure time) on the plane of the coordinate x and the so-called imaginary time ict – see Fig. 3.
'
x'
Fig. 9.3. The Lorentz transform as a mutual rotation
0
x
of two reference frames on the [ x, ] plane.
Indeed, rewriting Eq. (10) as
2
2
2
2
x ' x' ,
(9.11)
we may consider it as the invariance of the squared radius at the rotation shown in Fig. 3 and described by the following geometric relations:
x x' cos ' sin ,
(9.12a)
x' sin ' cos ,
with the reciprocal relations
x' x cos sin ,
(9.12b)
' x sin cos .
So far, the angle has been arbitrary. In the spirit of Eq. (8), a natural choice is = ( v), with the requirement (0) = 0. To find this function, let us write the definition of the velocity v of frame 0 ’, as measured in frame 0 (which was implied above): for x’ = 0, x = vt. In the variables x and , this means x
x
v
.
(9.13)
x'0
ict x'0
ic
On the other hand, for the same point x’ = 0, Eqs. (12a) yield
Chapter 9
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x
tan .
(9.14)
x'0
These two expressions are compatible only if
iv
tan
,
(9.15)
c
so that
tan
iv / c
1
1
sin
i
(9.16)
1 tan 2 1/2
1 2
v / 2
c
,
cos
1/ 2
1 tan2 1/2 1 2 2
v c
,
/
1/ 2
where and are two very convenient and commonly used dimensionless parameters defined as v
1
1
Relativistic
β ,
.
(9.17) parameters
c
2
2
1 v / c 1/2
2
1 1/2
and
(The vector is called the normalized velocity, while the scalar , the Lorentz factor.)12
Using the above relations for , Eqs. (12) become
x x' i ' ,
i x' ' ,
(9.18a)
x' x i ,
' i x .
(9.18b)
Now returning to the real variables [ x, ct], we get the Lorentz transform relations (6), in a more compact form:
x x' ct' , y y' , z z' , ct ct' x' , (9.19a) Lorentz
transform
x' x ct, y' y, z' z, ct' ct x.
(9.19b) - again
An immediate corollary of Eqs. (19) is that for to stay real, we need v 2 c 2, i.e. that the speed of any physical body (to which we could connect a meaningful reference frame) cannot exceed the speed of light, as measured in any other meaningful reference frame.13
9.2. Relativistic kinematic effects
Before proceeding to other corollaries of Eqs. (19), let us spend a few minutes discussing what these relations actually mean. Evidently, they are trying to tell us that the spatial and temporal intervals are not absolute (as they are in the Newtonian space), but do depend on the reference frame they are measured in. So, we have to understand very clearly what exactly may be measured – and thus may be discussed in a meaningful physics theory. Recognizing this necessity, A. Einstein introduced the notion of numerous imaginary observers that may be distributed all over each reference frame. Each observer 12 Note the following identities: 2 1/(1- 2) and (2 – 1) 2/(1- 2) 22, which are frequently handy in relativity-related algebra. One more function of , the rapidity tanh-1 (so that = i), is also useful for some calculations.
13 All attempts to rationally conjecture particles moving with v > c (called tachyons) have failed – so far, at least.
Possibly the strongest objection against their existence is the fact that the tachyons could be used to communicate back in time, thus violating the causality principle – see, e.g., G. Benford et al., Phys. Rev. D 2, 263 (1970).
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has a clock and may use it to measure the instants of local events, taking place at the observer’s location. He also conjectured, very reasonably, that:
(i) all observers within the same reference frame may agree on a common length measure (“a scale”), i.e. on their relative positions in that frame, and synchronize their clocks,14 and (ii) the observers belonging to different reference frames may agree on the nomenclature of world events (e.g., short flashes of light) to which their respective measurements refer.
Actually, these additional postulates have been already implied in our “derivation” of the Lorentz transform in Sec. 1. For example, by the set { x, y, z, t} we mean the results of space and time measurements of a certain world event, about that all observers belonging to frame 0 agree. Similarly, all observers of frame 0 ’ have to agree about the results { x’, y’, z’, t’}. Finally, when the origin of frame 0 ’ passes by some sequential points xk of frame 0, the observers in the latter frame may measure its passage times tk without a fundamental error, and know that all these times belong to x’ = 0.
Now we can analyze the major corollaries of the Lorentz transform, which are rather striking from the point of view of our everyday (rather non-relativistic) experience.
(i) Length contraction. Let us consider a thin rigid rod oriented along the x-axis, with its length l
x 2 – x 1, where x 1,2 are the coordinates of the rod’s ends, as measured in its rest frame 0, at any instant t (Fig. 4). What would be the rod’s length l’ measured by the Einstein observers in the moving frame 0 ’?
y
y'
x
l
x
1
2
x
x'
0
'
0
v
Fig. 9.4. The relativistic length contraction.
z
z'
At a time instant t’ agreed upon in advance, the observers who find themselves exactly at the rod’s ends, may register that fact, and then subtract their coordinates x’ 1,2 to calculate the apparent rod length l’ x 2 ’ – x 1 ’ in the moving frame. According to Eq. (19a), l may be expressed via this l’ as l x x ( x ' ct' ) ( x ' ct' ) ( x ' x ' ) l' .
(9.20a)
2
1
2
1
2
1
Hence, the rod’s length, as measured in the moving reference frame is
1/ 2
Length
l
v 2
l'
l 1
l
contraction
,
(9.20b)
c 2
in accordance with the FitzGerald-Lorentz hypothesis (5). This is the relativistic length contraction effect: an object is always the longest (has the so-called proper length l) if measured in its rest frame.
14 A posteriori, the Lorenz transform may be used to show that consensus-creating procedures (such as clock synchronization) are indeed possible. The basic idea of the proof is that since at v << c, the relativistic corrections to space and time intervals are of the order of ( v/ c)2, they have negligible effects on clocks being brought together into the same point for synchronization slowly, with a speed u << c. The reader interested in a detailed discussion of this and other fine points of special relativity may be referred to, e.g., either H. Arzeliès, Relativistic Kinematics, Pergamon, 1966, or W. Rindler, Introduction to Special Relativity, 2nd ed., Oxford U. Press, 1991.
Chapter 9
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Note that according to Eqs. (19), the length contraction takes place only in the direction of the relative motion of two reference frames. As was noted in Sec. 1, this result immediately explains the zero result of the Michelson-Morley-type experiments, so that they give very convincing evidence (if not irrefutable proof) of Eqs. (18)-(19).
(ii) Time dilation. Now let us use Eqs. (19a) to find the time interval t, as measured in some reference frame 0, between two world events – say, two ticks of a clock moving with another frame 0 ’
(Fig. 5), i.e. having fixed values of x’, y’, and z’.
y
y'
x
x'
0
v
Fig. 9.5. The relativistic time dilation.
z
z'
Let the time interval between these two events, measured in the clock’s rest frame 0’, be t’ t 2 ’
– t 1 ’. At these two moments, the clock would fly by certain two Einstein’s observers at rest in frame 0, so that they can record the corresponding moments t 1,2 shown by their clocks, and then calculate t as their difference. According to the last of Eqs. (19a),
c Δ t ct ct
,
(9.21a)
2
1
( ct ' x' ) ( ct ' x' )
2
1
c t'
Δ
so that, finally,
t'
Δ
Time
t
Δ t'
Δ
Δ .
(9.21b) dilation
1 v 2 / 2
t'
c 1/ 2
This is the famous relativistic time dilation (or “dilatation”) effect: a time interval is longer if measured in a frame (in our case, frame 0) moving relative to the clock, while that in the clock’s rest frame is the shortest possible – the so-called proper time interval.
This rather counter-intuitive effect is the everyday reality in experiments with high-energy elementary particles. For example, in a typical (and by no means record-breaking) experiment carried out in Fermilab, a beam of charged 200 GeV pions with 1,400 traveled a distance of l = 300 m with the measured loss of only 3% of the initial beam intensity due to the pion decay (mostly, into muon-neutrino pairs) with the proper lifetime t 0 2.5610-8 s. Without the time dilation, only an exp{- l/ ct 0}
~10-17 fraction of the initial pions would survive, while the relativity-corrected number, exp{- l/ ct} =
exp{- l/ c t 0} 0.97, was in full accordance with experimental measurements.
As another example, the global positioning systems (say, the GPS) are designed with the account of the time dilation due to the velocity of their satellites (and also some gravity-induced, i.e. general-relativity corrections, which I would not have time to discuss) and would give large errors without such corrections. So, there is no doubt that time dilation (21) is a reality, though the precision of its experimental tests I am aware of15 has been limited to a few percent, because of the almost unavoidable involvement of less controllable gravity effects – which provide a time interval change of the opposite sign in most experiments near the Earth’s surface.
15 See, e.g., J. Hafele and R. Keating, Science 177, 166 (1972).
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Before the first reliable observation of time dilation (by B. Rossi and D. Hall in 1940), there had been serious doubts about the reality of this effect, the most famous being the twin paradox first posed (together with an immediate suggestion of its resolution) by P. Langevin in 1911. Let us send one of two twins on a long space roundtrip with the maximum speed approaching c. Upon his return to Earth, who of the twins would be older? The naïve approach is to say that due to the relativity principle, not one can be (and hence there is no time dilation) because each twin could claim that their counterpart rather than them, was moving, with the same speed but in the opposite direction. The resolution of the paradox is that one of the twins had to be accelerated to be brought back, and hence the reference frames have to be dissimilar: only one of them may stay inertial all the time. As a result, the twin who had been accelerated (“actually traveling”) would be younger than their sibling when they finally come together.
Constructive proof of this conclusion for the particular case of straight-line travel with a piecewise-constant acceleration, is simple and hence left for the reader’s exercise.
(iii) Velocity transformation. Now let us calculate the velocity u of a moving point, as observed in reference frame 0, provided that its velocity, as measured in frame 0 ’, is u ’ (Fig. 6).
y'
u '
y
u
0
'
0
v
x
x'
Fig. 9.6. The relativistic velocity addition.
z
z'
Keeping the usual definition of velocity, but with due attention to the relativity of not only spatial but also temporal intervals, we may write
dr
d '
r
u
,
u '
.
(9.22)
dt
dt'
Plugging in the differentials of the Lorentz transform relations (6a) into these definitions, we get dx
dx' vdt'
u' v
x
dy
1
dy'
1
u'
u
,
u
y
, (9.23)
x
dt
dt' vdx' / 2
c
1 u' v / 2
c
y
dt
dt' vdx' / 2
c
1 u' v / 2
c
x
x
with a similar formula for uz. In the classical limit v/ c 0, these relations are reduced to u u' v,
u u' ,
u u' ,
(9.24a)
x
x
y
y
z
z
and may be merged into the familiar Galilean form
u u ' v,
v
for c .
(9.24b)
In order to see how unusual the full relativistic rules (23) are at u ~ c, let us first consider a purely longitudinal motion, uy = uz = 0; then16
16 With an account of the identity tanh( a + b) = (tanh a + tanh b)/(1 + tanh a tanh b), which readily follows from MA Eq. (3.5), Eq. (25) shows that rapidities tanh-1 add up exactly as longitudinal velocities at non-relativistic motion, making that notion very convenient for the analysis of transfer between several frames.
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u' v
Longitudinal
u
,
(9.25) velocity
2
1 u'v / c
addition
where u ux and u’ u’x. Figure 7 shows this u as the function of u’, for several values of the reference frames’ relative velocity v.
1
v / c 0.9
0.5
0
u
0.5
0
c
0.9
Fig. 9.7. The addition of longitudinal velocities.
1 1
0
1
u' / c
The first sanity check is that if v = 0, i.e. if the reference frames are at rest relative to each other, then u = u’, as it should be – see the diagonal straight line in Fig. 7. Next, if magnitudes of u’ and v are both below c, so is the magnitude of u. (Also good, because otherwise ordinary particles in one frame would be tachyons in the other one, and the theory would be in big trouble.) Now strange things begin: even as u’ and v are both approaching c, then u is also close to c, but does not exceed it. As an example, if we fired forward a bullet with the relative speed of 0.9 c, from a spaceship moving from the Earth also at 0.9 c, Eq. (25) predicts the speed of the bullet relative to the Earth to be just [(0.9 + 0.9)/(1 +
0.90.9)] c 0.994 c < c, rather than (0.9 + 0.9) c = 1.8 c > c as in the Galilean kinematics. Actually, we could expect this strangeness, because it is necessary to fulfill the 2nd Einstein’s postulate: the independence of the speed of light in any reference frame. Indeed, for u’ = c, Eq. (25) yields u = c, regardless of v.
In the opposite case of a purely transverse motion, when a point moves across the relative motion of the frames (for example, at our choice of coordinates, u’ x = u’ z = 0), Eqs. (23) yield a much less spectacular result
1
u u' u' .
(9.26)
y
y
y
This effect comes purely from the time dilation because the transverse spatial intervals are Lorentz-invariant.
In the case when both u x’ and uy’ are substantial (but uz’ is still zero), we may divide Eqs. (23) by each other to relate the angles of the point’s propagation, as observed in the two reference frames: u
u'
Stellar
y
y
sin '
tan
.
(9.27) aberration
u
cos /
effect
x
u' v
x
' v u'
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Essential Graduate Physics
EM: Classical Electrodynamics
This expression describes, in particular, the so-called stellar aberration effect: the dependence of the observed direction toward a star on the speed v of the telescope’s motion relative to the star – see Fig.
8. (The effect is readily observable experimentally as the annual aberration due to the periodic change of speed v by 2 v E 60 km/s because of the Earth’s rotation about the Sun. Since the aberration’s main part is of the first order in v E/ c ~ 10-4, this effect is very significant and has been known since the early 1700s.)
( u' c)
u '
'
v
Fig. 9.8. The stellar aberration.
For the analysis of this effect, it is sufficient to take, in Eq. (27), u’ = c, i.e. v/ u’ = , and interpret ’ as the “proper” direction to the star, that would be measured at v = 0.17 At << 1, both Eq.
(27) and the Galilean result (which the reader is invited to derive directly from Fig. 8), sin '
tan
,
(9.28)
cos '
may be well approximated by the first-order term
' sin '
.
(9.29)
Unfortunately, it is not easy to use the difference between Eqs. (27) and (28), of the second order in , for special relativity’s confirmation, because other components of the Earth’s motion, such as its rotation, nutation, and torque-induced precession,18 give masking first-order contributions to the aberration.
Finally, for a completely arbitrary direction of the vector u ’, Eqs. (22) may be readily used to calculate the velocity’s magnitude. The most popular form of the resulting expression is the following expression for the square of the relative velocity (or rather the reduced relative velocity ) of two points,
β β β β
2
1
2 2
1
2
.
(9.30)
β β
1
1
1
2
2
where 1,2 v1,2/ c are their normalized velocities as measured in the same reference frame.
17 Strictly speaking, to reconcile the geometries shown in Fig. 1 (for which all our formulas, including Eq. (27), are valid) and Fig. 8 (giving the traditional scheme of the stellar aberration), it is necessary to invert the signs of u (and hence of sin ’ and cos ’) and v, but as it is evident from Eq. (27), all the minus signs cancel, and the formula is valid “as is”.
18 See, e.g., CM Secs. 4.4-4.5.
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(iv) The Doppler effect. Let us consider a monochromatic plane wave of some physical nature, traveling along the x- axis:
f
Re f
i kx t
f
kx
exp (
cos
t arg f f cos
.
(9.31)
Its total phase, kx – t + arg f (in contrast to its amplitude f– see Sec. 5 below) cannot depend on the observer’s reference frame, because the variable f vanishes completely at = ( n + ½) (for all integer n), and such “world events” should be observable in all reference frames. The only way to keep
= ’ at all times is to have19
kx t
k'x' 't'
.
(9.32)
First, let us use this general relation to consider the Doppler effect in the usual non-relativistic mechanical waves, e.g., oscillations of particles of a certain medium. Using the Galilean transform (2), we may rewrite Eq. (32) as
k( x' vt) t k'x' 't
.
(9.33)
Since this transform leaves all space intervals (including the wavelength = 2/ k) intact, we can take k
= k’, so that Eq. (33) yields
'
kv .
(9.34)
For a dispersion-free medium, the wave number k is the ratio of its frequency , as measured in the reference frame bound to the medium, and the wave velocity v w. In particular, if the wave source rests in the medium, we may bind the reference frame 0 to the medium as well, and frame 0 ’ to the wave’s receiver (i.e. v = v r), so that
