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Unformatted text preview: 31 The Origin of the Refractive Index 31—1 The index of refraction We have said before that light goes slower in water than in air, and slower,
slightly, in air than in vacuum. This effect is described by the index of refraction
n. Now we would like to understand how such a slower velocity could come about.
In particular, we should try to see what the relation is to some physical assumptions,
or statements, we made earlier, which were the following: (a) That the total electric ﬁeld in any physical circumstance can always be
represented by the sum of the ﬁelds from all the charges in the universe. (b) That the ﬁeld from a single charge is given by its acceleration evaluated with
a retardation at the speed c, always (for the radiation ﬁeld). But, for a piece of glass, you might think: “Oh, no, you should modify all
this. You should say it is retarded at the speed c/n.” That, however, is not right,
and we have to understand why it is not. It is approximately true that light or any electrical wave does appear to travel
at the speed c/n through a material whose index of refraction is n, but the ﬁelds are
still produced by the motions of all the charges—including the charges moving in
the material—and with these basic contributions of the ﬁeld travelling at the
ultimate velocity c. Our problem is to understand how the apparently slower
velocity comes about. We shall try to understand the effect in a very simple case. A source which
we shall call “the external source” is placed a large distance away from a thin
plate of transparent material, say glass. We inquire about the ﬁeld at a large
distance on the opposite side of the plate. The situation is illustrated by the
diagram of Fig. 31—1, where S and P are imagined to be very far away from the
plate. According to the principles we have stated earlier, an electric ﬁeld anywhere
that is far from all moving charges is the (vector) sum of the ﬁelds produced by the
external source (at S) and the ﬁelds produced by each of the charges in the plate
of glass, every one with its proper retardation at the velocity c. Remember that the
contribution of each charge is not changed by the presence of the other charges.
These are our basic principles. The ﬁeld at P can be written thus: E = Z Eeach charge . all charges
or
E = Es + Z Eeach charge, all other charges where E, is the ﬁeld due to the source alone and would be precisely the ﬁeld at
P if there were no material present. We expect the ﬁeld at P to be different from
E8 if there are any other moving charges. Why should there be charges moving in the glass? We know that all material
consists of atoms which contain electrons. When the electric ﬁeld of the source acts
on these atoms it drives the electrons up and down, because it exerts a force on the
electrons. And moving electrons generate a ﬁeld—they constitute new radiators.
These new radiators are related to the source S, because they are driven by the
ﬁeld of the source. The total ﬁeld is not just the ﬁeld of the source S, but it is
modiﬁed by the additional contribution from the other moving charges. This
means that the ﬁeld is not the same as the one which was there before the glass
was there, but is modiﬁed, and it turns out that it is modiﬁed in such a way that 31—1 31—1 The index of refraction
31—2 The ﬁeld due to the material
31—3 Dispersion 31—4 Absorption 31—5 The energy carried by an
electric wave 31—6 Diffraction of light by a screen Arriving Wave "Transmitted'Wave
 '3
55/: /
79‘ who” 9"”.
I
Source of a: “1
electric wave
"Reflected"
Wave lass plate
Fig. 31—1 . Electric waves passing through a layer of transparent material. VACUUM// / “ /
/ /
/ / /
{7 /\4§t
wave’V
crests\///’ / Fig. 31—2. Relation between refrac
tion and velocity change. the ﬁeld inside the glass appears to be moving at a diﬂerent speed. That is the idea
which we would like to work out quantitatively. Now this is, in the exact case, pretty complicated, because although we have
said that all the other moving charges are driven by the source ﬁeld, that is not
quite true. If we think of a particular charge, it feels not only the source, but like
anything else in the world, it feels all of the charges that are moving. It feels, in
particular, the charges that are moving somewhere else in the glass. So the total
ﬁeld which is acting on a particular charge is a combination of the ﬁelds from the
other charges, whose motions depend on what this particular charge is doing! You
can see that it would take a complicated set of equations to get the complete and
exact formula. It is so complicated that we postpone this problem until next year. Instead we shall work out a very simple case in order to understand all the
physical principles very clearly. We take a circumstance in which the effects from
the other atoms is very small relative to the effects from the source. In other words,
we take a material in which the total ﬁeld is not modiﬁed very much by the motion
of the other charges. That corresponds to a material in which the index of refraction
is very close to 1, which will happen, for example, if the density of the atoms is
very low. Our calculation will be valid for any case in which the index is for any
reason very close to 1. In this way we shall avoid the complications of the most
general, complete solution. Incidentally, you should notice that there is another effect caused by the motion
of the charges in the plate. These charges will also radiate waves back toward the
source S. This backwardgoing ﬁeld is the light we see reﬂected from the surfaces
of transparent materials. It does not come from just the surface. The backward
radiation comes from everywhere in the interior, but it turns out that the total eﬂect
is equivalent to a reﬂection from the surfaces. These reﬂection effects are beyond
our approximation at the moment because we shall be limited to a calculation for a
material with an index so close to 1 that very little light is reﬂected. Before we proceed with our study of how the index of refraction comes about,
we should understand that all that is required to understand refraction is to under
stand why the apparent wave velocity is different in different materials. The
bending of light rays comes about just because the effective speed of the waves is
different in the materials. To remind you how that comes about we have drawn
in Fig. 31—2 several successive crests of an electric wave which arrives from a
vacuum onto the surface of a block of glass. The arrow perpendicular to the wave
crests indicates the direction of travel of the wave. Now all oscillations in the wave
must have the same frequency. (We have seen that driven oscillations have the
same frequency as the driving source.) This means, also, that the wave crests for
the waves on both sides of the surface must have the same spacing along the surface
because they must travel together, so that a charge sitting at the boundary will
feel only one frequency. The shortest distance between crests of the wave, however,
is the wavelength which is the velocity divided by the frequency. On the vacuum
side it is A0 = 27rc/w, and on the other side it is A = 27rv/w or 27rc/wn, if v = c/n
is the velocity of the wave. From the ﬁgure we can see that the only way for the
waves to “ﬁt” properly at the boundary is for the waves in the material to be
travelling at a different angle with respect to the surface. From the geometry of
the ﬁgure you can see that for a “ﬁt” we must have AO/sin 00 = x/sin 0, or
sin 00/sin 0 = n, which is Snell’s law. We shall, for the rest of our discussion,
consider only why light has an effective speed of c/n in material of index n, and
no longer worry, in this chapter, about the bending of the light direction. We go back now to the situation shown in Fig. 31—1. We see that what we
have to do is to calculate the ﬁeld produced at P by all the oscillating charges in
the glass plate. We shall call this part of the ﬁeld Ea, and it is just the sum written
as the second term in Eq. (31.2). When we add it to the term E8, due to the source,
we will have the total ﬁeld at P. 312 This is probably the most complicated thing that we are going to do this year,
but it is complicated only in that there are many pieces that have to be put to
gether; each piece, however, is very simple. Unlike other derivations where we
say, “Forget the derivation, just look at the answer!,” in this case we do not
need the answer so much as the derivation. In other words, the thing to under
stand now is the physical machinery for the production of the index. To see where we are going, let us ﬁrst ﬁnd out what the “correction ﬁeld”
E, would have to be if the total ﬁeld at P is going to look like radiation from the
source that is slowed down while passing through the thin plate. If the plate had
no effect on it, the ﬁeld of a wave travelling to the right (along the zaxis) would be E3 = E0 cos w(t — z/c) (31.3) or, using the exponential notation, E, = Eoeiw“‘z’c>. (31.4) Now what would happen if the wave travelled more slowly in going through
the plate? Let us call the thickness of the plate Az. If the plate were not there the
wave would travel the distance A2 in the time Az/c. But if it appears to travel at
the speed c/n then it should take the longer time n Az/c or the additional time
At = (n — l) Az/c. After that it would continue to travel at the speed c again.
We can take into account the extra delay in getting through the plate by replacing
tin Eq. (31.4) by (t  At) or by [t — (n — l) Az/c]. So the wave after insertion
of the plate should be written Emmam = Eoe”[‘“‘"—““’H’“l. (31.5)
We can also write this equation as
Eaftcr plate = e‘WM—lmzn E09 “(t—Zia), 1 which says that the wave after the plate is obtained from the wave which could
exist without the plate, i, e., from E3, by multiplying by the factor e‘i‘“(”*1)A‘/”.
Now we know that multiplying an oscillating function like eM by a factor e” just
says that we change the phase of the oscillation by the angle 0, which is, of course,
what the extra delay in passing through the thickness Az has done. It has retarded
the phase by the amount w(n — l) Az/c (retarded, because of the minus sign in
the exponent). We have said earlier that the plate should add a ﬁeld E, to the original ﬁeld
E8 = Eoem‘_z’”’, but we have found instead that the effect of the plate is to
multiply the ﬁeld by a factor which shifts its phase. However, that is really all right
because we can get the same result by adding a suitable complex number. It is
particularly easy to ﬁnd the right number to add in the case that A2 is small, for
you will remember that if x is a small number then ex is nearly equal to (1 + x).
We can write, therefore, (rm—1W” : 1 — iw(n — l)Az/c. (31.7)
Using this equality in Eq. (31.6), we have
Emprmm = Eoew‘z’” — w niece/C). (31.8)
W W
E, E, The ﬁrst term is just the ﬁeld from the source, and the second term must just be
equal to Ea, the ﬁeld produced to the right of the plate by the oscillating charges
of the plate—expressed here in terms of the index of refraction n, and depending,
of course, on the strength of the wave from the source. What we have been doing is easily visualized if we look at the complex number
diagram in Fig. 31—3. We ﬁrst draw the number Es (we chose some values for z
and t so that Es comes out horizontal, but this is not necessary). The delay due to 31—3 Imaginary AKIs Angle x w(n  I)Az/c Real Axis Fig. 3l3. Diagram for the trans
mitted wave at a particular f and z. slowing down in the plate would delay the phase of this number, that is, it would
rotate E, through a negative angle. But this is equivalent to adding the small
vector E, at roughly right angles to E8. But that is just what the factor —z' means
in the second term of Eq. (31.8). It says that if E,r is real, then Ea is negative
imaginary or that, in general, E8 and E, make a right angle. 31—2 The ﬁeld due to the material We now have to ask: Is the ﬁeld E, obtained in the second term of Eq. (31.8)
the kind we would expect from oscillating charges in the plate? If we can show
that it is, we will then have calculated what the index n should be! [Since n is the
only nonfundamental number in Eq. (31.8).] We turn now to calculating what
ﬁeld E, the charges in the material will produce. (To help you keep track of the
many symbols we have used up to now, and will be using in the rest of our calcula
tion, we have put them all together in Table 31—1.) Table 31—1
Symbols used in the calculations E8 = ﬁeld from the source
Ea = ﬁeld produced by charges in the plate
A2 = thickness of the plate
z = perpendicular distance from the plate
n = index of refraction
w = frequency (angular) of the radiation
N = number of charges per unit volume in the plate
77 = number of charges per unit area of the plate
q6 = charge on an electron
m = mass of an electron
coo = resonant frequency of an electron bound in an atom If the source S (of Fig. 31—1) is far oﬂ‘ to the left, then the ﬁeld E, will have
the same phase everywhere on the plate, so we can write that in the neighborhood
of the plate E, = EoeWz/c) (31.9)
Right at the plate, where z = 0, we will have
E, = E06“ (at the plate) (31.10) Each of the electrons in the atoms of the plate will feel this electric ﬁeld and
will be driven up and down (we assume the direction of E 0 is vertical) by the electric
force qE. To ﬁnd what motion we expect for the electrons, we will assume that the
atoms are little oscillators, that is, that the electrons are fastened elastically to the
atoms, which means that if a force is applied to an electron its displacement from
its normal position will be proportional to the force. You may think that this is a funny model of an atom if you have heard about
electrons whirling around in orbits. But that is just an oversimpliﬁed picture.
The correct picture of an atom, which is given by the theory of wave mechanics,
says that, so far as problems involving light are concerned, the electrons behave as
though they were held by springs. So we shall suppose that the electrons have a
linear restoring force which, together with their mass m, makes them behave like
little oscillators, with a resonant frequency we. We have already studied such os
cillators, and we know that the equation of their motion is written this way: + (00X) = F, where F is the driving force.
314 For our problem, the driving force comes from the electric ﬁeld of the wave
from the source, so we should use F = 119E. = quoei‘”, (31.12) where qe is the electric charge on the electron and for E, we use the expression
E, = E gel“ from (31.10). Our equation of motion for the electron is then 2 .
m (gr—f + wﬁx) = quoetw‘. (31.13) We have solved this equation before, and we know that the solution is
x = xoei‘”, (31.14) where, by substituting in (31.13), we ﬁnd that x0 = (31.15)
m(w0 — w )
so that x = —————‘;9E° 2 em. (31.16)
m(w0 — w ) We have what we needed to know—the motion of the electrons in the plate. And
it is the same for every electron, except that the mean position (the “zero” of the
motion) is, of course, different for each electron. Now we are ready to ﬁnd the ﬁeld E, that these atoms produce at the point P,
because we have already worked out (at the end of Chapter 30) what ﬁeld is pro
duced by a sheet of charges that all move together. Referring back to Eq. (30.19),
we see that the ﬁeld E, at P is just a negative constant times the velocity of the
charges retarded in time the amount z/c. Differentiating x in Eq. (31.16) to get
the velocity, and sticking in the retardation [or just putting x0 from (31.15)
into (30.18)] yields E. = — "‘19 [1w —‘18E° e”“"’”’] (31.17)
2600 mth — a?) Just as we expected, the driven motion of the electrons produced an extra wave
which travels to the right (that is what the factor ei‘M’Z’”) says), and the amplitude
of this wave is proportional to the number of atoms per unit area in the plate
(the factor n) and also proportional to the strength of the source ﬁeld (the factor
E0). Then there are some factors which depend on the atomic properties (qe, m,
and 0.10), as we should expect. The most important thing, however, is that this formula (31.17) for E, looks
very much like the expression for E, that we got in Eq. (31.8) by saying that the
original wave was delayed in passing through a material with an index of refraction
n. The two expressions will, in fact, be identical if 2
(n — 1)Az = ——"—qe— (31.18) zeom(w% — (1,2). Notice that both sides are proportional to A2, since 11, which is the number of
atoms per unit area, is equal to N AZ, where N is the number of atoms per unit
volume of the plate. Substituting N AZ for n and cancelling the Az, we get our main
result, a formula for the index of refraction in terms of the properties of the atoms
of the material—and of the frequency of the light: N613 ——2—2— (31.19)
260m(w0 — w ) n:1+ This equation gives the “explanation” of the index of refraction that we wished to
obtain. 31—5 31—3 Dispersion Notice that in the above process we have obtained something very interesting.
For we have not only a number for the index of refraction which can be computed
from the basic atomic quantities, but we have also learned how the index of
refraction should vary with the frequency on Of the light. This is something we
would never understand from the simple statement that “light travels slower in a
transparent material.” We still have the problem, of course, of knowing how many
atoms per unit volume there are, and what is their natural frequency we. We do
not know this just yet, because it is different for every different material, and we
cannot get a general theory of that now. Formulation of a general theory of the
properties of different substances—their natural frequencies, and so on—is
possible only with quantum atomic mechanics. Also, different materials have
different properties and different indexes, so we cannot expect, anyway, to get a
general formula for the index which will apply to all substances. However, we shall discuss the formula we have obtained, in various possible
circumstances. First of all, for most ordinary gases (for instance, for air, most
colorless gases, hydrogen, helium, and so on) the natural frequencies of the electron
oscillators correspond to ultraviolet light. These frequencies are higher than the
frequencies of visible light, that is, wo is much larger than to of visible light, and to
a ﬁrst approximation, we can disregard (.02 in comparison with 003. Then we ﬁnd
that the index is nearly constant. So for a gas, the index is nearly constant. This
is also true for most other transparent substances, like glass. If we look at our
expression a little more closely, however, we notice that as w rises, taking a little
bit more away from the denominator, the index also rises. So it rises slowly with
frequency. The index is higher for blue light than for red light. That is the reason
why a prism bends the light more in the blue than in the red. The phenomenon that the index depends upon the frequency is called the
phenomenon of dispersion, because it is the basis of the fact that light is “dispersed”
by a prism into a spectrum. The equation for the index of refraction as a function
of frequency is called a dispersion equation. So we have obtained a dispersion equa
tion. (In the past few years “dispersion equations” have been ﬁnding a new use in
the theory of elementary particles.) Our dispersion equation suggests other interesting effects. If we have a
natural frequency coo which lies in the visible region, or if we measure the index
of refraction of a material like glass in the ultraviolet, where w gets near coo, we
see that at frequencies very close to the natural frequency the index can get enor
mously large, because the denominator can go to zero. Next, suppose that w is
greater than (.00. This would occur, for example, if we take a material like glass,
say, and shine xray radiation on it. In fact, since many materials which are opaque
to visible light, like graphite for instance, are transparent to xrays, we can also
talk about the index of refraction of carbon for xrays. All the natural frequencies
of the carbon atoms would be much lower than the frequency we are using in the
xrays, since xray radiation has a very high frequency. The index of refraction is
that given by our dispersion equation if we set we equal to zero (we neglect <93 in
comparison with wz). A similar situation would occur if we beam radiowaves (or light) on a gas of
free electrons. In the upper atmosphere electrons are liberated from their atoms by
ultraviolet light from the sun and they sit up there as free electrons. For free
electrons wo = 0 (there is no elastic restoring force). Setting coo = O in our disper
sion equation yields the correct formula for the index of refraction for radiowaves
in the stratosphere, where N is now to represent the density of free electrons (num
ber per unit volume) in the stratosphere. But let us look again at the equation, if
we beam xrays on matter, or radiowaves (or any electric waves) on free electrons
the term (to?) — (:02) becomes negative, and we obtain the result that n is less than
one. That means that the effective speed of the waves in the substance is faster
than cl Can that be correct? It is correct. In spite of the fact that it is said that you cannot send signals
any faster than the speed of light, it is nevertheless true that the index of refraction of materials at a particular frequency can be either greater or less than 1. This
31—6 just means that the phase shift which is produced by the scattered light can be
either positive or negative. It can be shown, however, that the speed at which you
can send a signal is not determined by the index at one frequency, but depends on
what the index is at many frequencies. What the index tells us is the speed at which
the nodes (or crests) of the wave travel. The node of a wave is not a signal by itself.
In a perfect wave, which has no modulations of any kind, i.e., which is a steady
oscillation, you cannot really say when it “starts,” so you cannot use it for a timing
signal. In order to send a signal you have to change the wave somehow, make a
notch in it, make it a little bit fatter or thinner. That means that you have to
have more than one frequency in the wave, and it can be shown that the speed at
which signals travel is not dependent upon the index alone, but upon the way that
the index changes with the frequency. This subject we must also delay (until
Chapter 48). Then we will calculate for you the actual speed of signals through
such a piece of glass, and you will see that it will not be faster than the speed of
light, although the nodes, which are mathematical points, do travel faster than
the speed of light. Just to give a slight hint as to how that happens, you will note that the real
difﬁculty has to do with the fact that the responses of the charges are opposite to
the ﬁeld, i.e., the sign has gotten reversed. Thus in our expression for x (Eq. 31.16)
the displacement of the charge is in the direction opposite to the driving ﬁeld,
because (to?) — M) is negative for small coo. The formula says that when the
electric ﬁeld is pulling in one direction, the charge is moving in the opposite direc
tion. How does the charge happen to be going in the opposite direction? It certainly
does not start off in the opposite direction when the ﬁeld is ﬁrst turned on. When
the motion ﬁrst starts there is a transient, which settles down after awhile, and
only then is the phase of the oscillation of the charge opposite to the driving ﬁeld.
And it is then that the phase of the transmitted ﬁeld 'can appear to be advanced
with respect to the source wave. It is this advance in phase which is meant when
we say that the “phase velocity” or velocity of the nodes is greater than c. In
Fig. 31—4 we give a schematic idea of how the waves might look for a case where
the wave is suddenly turned on (to make a signal). You will see from the diagram
that the signal (i.e., the start of the wave) is not earlier for the wave which ends up
with an advance in phase. (0) Wave with no
MONI'ioI 1b) Transmitted VIM
with n>l E
(c) ' I I
Transmltted wave  t
um nel ' I
H iailvonce 0! phase Fig. 31—4. Wave “signals.” Let us now look again at our dispersion equation. We should remark that
our analysis of the refractive index gives a result that is somewhat simpler than you
would actually ﬁnd in nature. To be completely accurate we must add some
reﬁnements. First, we should expect that our model of the atomic oscillator should
have some damping force (otherwise once started it would oscillate forever, and
we do not expect that to happen). We have worked out before (Eq. 23.8) the
motion of a damped oscillator and the result is that the denominator in Eq. (31.16),
and therefore in (31.19), is changed from (mg — M) to (03(2) — (02 + i‘Yw), where
‘Y is the damping coefficient. We need a second modiﬁcation to take into account the fact that there are
several resonant frequencies for a particular kind of atom. It is easy to ﬁx up our 31—7 Fig. 31—5. The index of refraction as
a function of frequency. dispersion equation by imagining that there are several different kinds of oscil
lators, but that each oscillator acts separately, and so we simply add the contri
butions of all the oscillators. Let us say that there are N], electrons per unit of
volume, whose natural frequency is wk and whose damping factor is 7],. We
would then have for our dispersion equation 2
35.. Nk
+ 260"! Z 2 n=1 ——2———
k oak—co +1714» (31.20)
We have, ﬁnally, a complete expression which describes the index of refraction that
is observed for many substances.* The index described by this formula varies with
frequency roughly like the curve shown in Fig. 31—5. You will note that so long as w is not too close to one of the resonant frequen
cies, the slope of the curve is positive. Such a positive slope is called “normal”
dispersion (because it is clearly the most common occurrence). Very near the
resonant frequencies, however, there is a small range of w’s for which the slope is
negative. Such a negative slope is often referred to as “anomalous” (meaning
abnormal) dispersion, because it seemed unusual when it was ﬁrst observed, long
before anyone even knew there were ‘such things as electrons. From our point of
view both slopes are quite “normal”! ‘ 31—4 Absorption Perhaps you have noticed something a little strange about the last form
(Eq. 31.20) we obtained for our dispersion equation. Because of the term i7 we
put in to take account of damping, the index of refraction is now a complex
number! What does that mean? By working out what the real and imaginary parts of n are we could write 72 = n’ — in”, (31.21) where n’ and n” are real numbers. (We use the minus sign in front of the in”
because then n" will turn out to be a positive number, as you can show for yourself.) We can see what such a complex index means by going back to Eq. (31.6),
which is the equation of the wave after it goes through a plate of material with an
index n. If we put our complex n into this equation, and do some rearranging, we
get _ "A _ ,_ A  _
Eafterplate _ e um z/c e mm 1) 2/0 Enema 2/0) W h—‘F__z
A B The last factors, marked B in Eq. (31.22), are just the form we had before, and
again describe a wave whose phase has been delayed by the angle w(n’ — 1) Az/c
in traversing the material. The ﬁrst term (A) is new and is an exponential factor
with a real exponent, because there were two i’s that cancelled. Also, the exponent
is negative, so the factor is a real number less than one. It describes a decrease
in the magnitude of the ﬁeld and, as we should expect, by an amount which is
more the larger A2 is. As the wave goes through the material, it is weakened. The
material is “absorbing” part of the wave. The wave comes out the other side with
less energy. We should not be surprised at this, because the damping we put in
for the oscillators is indeed a friction force and must be expected to cause a loss
of energy. We see that the imaginary part n” of a complex index of refraction
represents an absorption (or “attenuation”) of the wave. In fact, n” is sometimes
referred to as the “absorption index.” We may also point out that an imaginary part to the index n corresponds to
bending the arrow E, in Fig. 31—3 toward the origin. It is clear why the transmitted
ﬁeld is then decreased. (31.22) * Actually, although in quantum mechanics Eq. (31.20) is still valid, its interpretation
is somewhat different. In quantum mechanics even an atom with one electron, like
hydrogen, has several resonant frequencies. Therefore M, is not really the number of
electrons having the frequency wk, but is replaced instead by N19,, where N is the number
of atoms per unit volume and _/‘)r (called the oscillator strength) is a factor that tells how
strongly the atom exhibits each of its resonant frequencies wk. 31—8 Normally, for instance as in glass, the absorption of light is very small.
This is to be expected from our Eq. (31.20), because the imaginary part of the
denominator, ivkw, is much smaller than the term (a): — 032). But if the light fre
quency w is very close to wk then the resonance term (mi — (.02) can become small
compared with ivkw and the index becomes almost completely imaginary. The
absorption of the light becomes the dominant effect. It is just this effect that gives
the dark lines in the spectrum of light which we receive from the sun. The light
from the solar surface has passed through the sun’s atmosphere (as well as the
earth’s), and the light has been strongly absorbed at the resonant frequencies of
the atoms in the solar atmosphere. The observation of such spectral lines in the sunlight allows us to tell the
resonant frequencies of the atoms and hence the chemical composition of the sun’s
atmosphere. The same kind of observations tell us about the materials in the stars.
From such measurements we know that the chemical elements in the sun and in
the stars are the same as those we ﬁnd on the earth. 31—5 The energy carried by an electric wave We have seen that the imaginary part of the index means absorption. We
shall now use this knowledge to ﬁnd out how much energy is carried by a light
wave. We have given earlier an argument that the energy carried by light is
proportional to W, the time average of the square of the electric ﬁeld in the wave.
The decrease in E due to absorption must mean a loss of energy, which would go
into some friction of the electrons and, we might guess, would end up as heat in
the material. If we consider the light arriving on a unit area, say one square centimeter, of
our plate in Fig. 31—1, then we can write the following energy equation (if we assume
that energy is conserved, as we do/): Energy in per sec = energy out per sec + work done per sec. (31.23) For the ﬁrst term we can write 01273—3, where a is the as yet unknown constant of
proportionality which relates the average value of E2 to the energy being carried.
For the second term we must include the part from the radiating atoms of the
material, so we should use a(E, + Ea)2, or (evaluating the square) a(—E? +
275E + F2). All of our calculations have been made for a thin layer of material whose
index is not too far from 1, so that E, would always be much less than E, (just to
make the calculations easier). In keeping with our approximations, we should,
therefore, leave out the term F3, because it is much smaller than m. You may
say: “Then you should leave out m also, because it is much smaller than It is true that ESE“ is much smaller than 173, but we must keep EsEa or our approxi
mation will be the one that would apply if we neglected the presence of the material
completely! One way of checking that our calculations are consistent is to see that
we always keep terms which are proportional to N Az, the area density of atoms
in the material, but we leave out terms which are proportional to (N A2)2 or any
higher power of N Az. Ours is what should be called a “lowdensity approxi
mation.” In the same spirit, we might remark that our energy\equation has neglected
the energy in the reﬂected wave. But that is OK because this 'termggo, is propor
tional to (N Az)2, since the amplitude of the reﬂected wave is proportional to
N Az. For the last term in Eq. (31.23) we wish to compute the rate at which the
incoming wave is doing work on the electrons. We know that work is force times
distance, so the rate of doing work (also called power) is the force times the veloc
ity. It is really F  V, but we do not need to worry about the dot product when the
velocity and force are along the same direction as they are here (except for a
possible minus sign). So for each atom we take m for the average rate of 31—9 "U s
* i=5, E=O Papoun screen (6) l S E=E E=E +5 P * s I s onI .
/hole
.[xwoll (C) S l P ‘X’ pug a Ens, E=ES+ELWH+E$MI=0 wall Fig. 31—6. Diffraction by a screen. doing work. Since there are N Az atoms in a unit area, the last term in Eq. (31.23)
should be N Azqe Our energy equation now looks like £3 = a2? + 2aE,E, + N Az qt E7). (3124) The E2 terms cancel, and we have ZaEsEa = N Az q, ETv. (31.25)
We now go back to Eq. (30.19), which tells us that for large 2 Ea ___ NAzqe
2€00 v(ret by z/c) (31.26) (recalling that n = N Az). Putting Eq. (31.26) into the lefthand side of (31.25), we get N Azqe
260C 2a E,(at z)  v(ret by z/c). However, E,(at z) is _Es (at atoms) retarded by z/c. Since the average is inde
pendent of time, it is the same now as retarded by z/c, or is E, (at atom)  u, the
same average that appears on the righthand side of (31.25). The two sides are
therefore equal if $ = 1, or a = 606‘. (31.27)
We have discovered that if energy is to be conserved, the energy carried in an elec tric wave per unit area and per unit time (or what we have called the intensity)
must be given by eocEZ. If we call the intensity S, we have _ intensity _
S = or = eocE2, (31.28)
energy / area / time where the bar means the time average. We have a nice bonus result from our theory
of the refractive index! 31—6 Diﬂ'raction of light by a screen It is now a good time to take up a somewhat different matter which we can
handle with the machinery of this chapter. In the last chapter we said that when
you have an opaque screen and the light can come through some holes, the distribu
tion of intensity—the diﬁraction pattern—could be obtained by imagining instead
that the holes are replaced by sources (oscillators) uniformly distributed over the
hole. In other words, the diffracted wave is the same as though the hole were a
new source. We have to explain the reason for that, because the hole is, of course,
just where there are no sources, where there are no accelerating charges. Let us ﬁrst ask: “What is an opaque screen?” Suppose we have a completely
opaque screen between a source S and an observer at P, as in Fig. 31—6(a). If the
screen is “opaque” there is no ﬁeld at P. Why is there no ﬁeld there? According
to the basic principles we should obtain the ﬁeld at P as the ﬁeld E, of the source
delayed, plus the ﬁeld from all the other charges around. But, as we have seen
above, the charges in the screen will be set in motion by the ﬁeld E,” and these
motions generate a new ﬁeld which, if the screen is opaque, must exactly cancel
the ﬁeld E, on the back side of the screen. You say: “What a miracle that it bal
ances exactly! Suppose it was not exactly right!” If it were not exactly right (re
member that this opaque screen has some thickness), the ﬁeld toward the rear
part of the screen would not be exactly zero. So, not being zero, it would set
into motion some other charges in the material of the screen, and thus make a little
more ﬁeld, trying to get the total balanced out. So if we make the screen thick
enough, there is no residual ﬁeld, because there is enough opportunity to ﬁnally
get the thing quieted down. In terms of our formulas above we would say that the 31—10 screen has a large and imaginary index, so the wave is absorbed exponentially as it
goes through. You know, of course, that a thin enough sheet of the most opaque
material, even gold, is transparent. Now let us see what happens with an opaque screen which has holes in it, as
in Fig. 31—6(b). What do we expect for the ﬁeld at P? The ﬁeld at P can be repre
sented as a sum of two parts—the ﬁeld due to the source S plus the ﬁeld due to the
wall, i.e., due to the motions of the charges in the walls. We might expect the
motions of the charges in the walls to be complicated, but we can ﬁnd out what
ﬁelds they produce in a rather simple way. Suppose that we were to take the same screen, but plug up the holes, as indi
cated in part (c) of the ﬁgure. We imagine that the plugs are of exactly the same
material as the wall. Mind you, the plugs go where the holes were in case (b).
Now let us calculate the ﬁeld at P. The ﬁeld at P is certainly zero in case (c), but
it is also equal to the ﬁeld from the source plus the ﬁeld due to all the motions of
the atoms in the walls and in the plugs. We can write the following equations: Case (b): Eat p = Es + Ewall:
Case (c): EA, p = 0 = Es + E61311 + E51113, where the primes refer to the case where the plugs are in place, but E, is, of course,
the same in both cases. Now if we subtract the two equations, we get Eat P = (Ewall —' Eivall) _ Eplug Now if the holes are not too small (say many wavelengths across), we would not
expect the presence of the plugs to change the ﬁelds which arrive at the walls except
possibly for a little bit around the edges of the holes. Neglecting this small eﬁect,
we can set Ewan = Elm“ and obtain that Eat P = _E;I)1ug We have the result that the ﬁeld at P when there are holes in a screen (case b) is the
same (except for sign) as the ﬁeld that is produced by that part of a complete opaque
wall which is located where the holes are! (The sign is not too interesting, since we
are usually interested in intensity which is proportional to the square of the ﬁeld.)
It seems like an amazing backwardsforwards argument. It is, however, not only
true (approximately for not too small holes), but useful, and is the justiﬁcation
for the usual theory of diﬁraction. The ﬁeld Efm,g is computed in any particular case by remembering that the
motion of the charges everywhere in the screen is just that which will cancel out
the ﬁeld E, on the back of the screen. Once we know these motions, we add the
radiation ﬁelds at P due just to the charges in the plugs. We remark again that this theory of diffraction is only approximate, and will
be good only if the holes are not too small. For holes which are too small the {,lug term will be small and then the difference between Ev’vau and Ewan (which
difference we have taken to be zero) may be comparable to or larger than the small
E1311,“ term, and our approximation will no longer be valid. 31—11 ...
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This note was uploaded on 06/18/2009 for the course PHYSICS none taught by Professor Leekinohara during the Spring '09 term at Uni. Nottingham  Malaysia.
 Spring '09
 LeeKinohara
 Physics

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