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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 ïŸ you have heard about
electrons whirling around in orbits. But that is just an oversimplifed picture.
'The correct picture of an atom, which is given by the theory of wave mechanics,
says that, so far as problems imuolưing 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 rm, makes them behave
like little oscillators, with a resonant frequency œọ. We have already studied such
oscillators, and we know that the equation of their motion is written this way:
where #' is the driving force.
For our problem, the driving force comes from the electric fñeld of the wave
from the source, so we should use
=q.B, = qeEoe"°°, (31.12)
where q¿ is the electric charge on the electron and for #⁄; we use the expres-
sion = Eoe”f from (31.10). Our equation of motion for the electron is
dŠz 2 In)
m dP + uậ# ] = qeEoe'”“”. (31.13)
W© have solved this equation before, and we know that the solution is
2 = øoc 1°, (31.14)
where, by substituting in (31.13), we fnd that
=————x 31.15
#0 m(u — œ2) ) ( )
so that
I7 -
ca (31.16)
m(uỗ — Ø3)
--- Trang 543 ---
We have what we needed to know——the motion of the electrons in the plate. And
1t is the same for every electron, except that the mean position (the “zero” of
the motion) is, oŸ course, diferent for each electron.
Now we are ready to fnd the fñeld !„ that these atoms produce at the
point , because we have already worked out (at the end of Chapter 30) what
ñeld is produced by a sheet of charges that all move together. Referring back to
Eq. (30.19), we see that the field + at ÐP is just a negative constant times the
velocity of the charges retarded in time by the amount z/c. Diferentiating z in
Eq. (31.16) to get the velocity, and sticking in the retardation |or just putting #o
from (31.15) into (30.18)] yields
Tdqe |. qeEo 2œ(—z/c)
#„=—=—— ———=—— : 31.17
` 2co€ le mÁ(œ8 — (2) , ' )
Just as we expected, the driven motion of the electrons produced an extra wave
which travels to the right (that is what the factor e“2ữ=Z/ says), and the
amplitude of this wave is proportional to the number of atoms per unit area
in the plate (the factor 7?) and also proportional to the strength of the source
fñeld (the factor #o). Then there are some factors which depend on the atomic
properties (qe, rm, and œạ), as we should expect.
The most important thing, however, is that this formula (31.17) for 2+ looks
very much like the expression for ⁄„ that we got in Bq. (31.8) by saying that
the original wave was delayed in passing through a material with an index of
refraction n. 'Phe bwo expressions will, in fact, be identical if
—1)Az=—————.. 31.18
(x— À2 2com(uậ — œ2) ' )
Notice that both sides are proportional to Az, since ?, which is the number of
atoms øer ni œrea, is equal to N Az, where is the number of atoms per unit
0olume of the plate. Substituting Az for 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:
=l+———.-. 31.19
⁄ " 2cogm(„8 — œ2) ' )
'This equation gives the “explanation” of the index of refraction that we wished
to obtain.
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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 œ 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 frequenecy œọ.
W©e do not know this just yet, because it is diferent for every diferent material,
and we cannot get a general theory of that now. Formulation oŸ a general theory
of the properties of diferent substances—their natural frequencies, and so on——1s
possible only with quantum atomic mechanics. Also, diferent materials have
diÑferent properties and diferent indexes, so we cannot expect, anyway, to get a
general formula for the index which will apply to all substanees.
However, we shall discuss the formula we have obtained, in various possible
circumstances. Pirst 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 requencies are higher
than the frequencies of visible light, that is, œọ is mụuch larger than œ of visible
light, and to a frst approximation, we can disregard œŠ in comparison with øÿ.
Then we fñnd 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 œ Tis©s,
taking a little bit more away from the denominator, the index also rises. 5o ?t
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 đ/sperszon, 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 frequeney 1s called a d¿spersion equation. So we have obtained a dispersion
cquation. (In the past few years “dispersion equations” have been fnding a new
use in the theory of elementary particles.)
Our dispersion equation suggests other interesting efects. If we have a
natural requency œọ which lies in the visible region, or if we measure the index
of refraction of a material like glass in the ultraviolet, where œ gets near œọ, we
see that at Írequencies very close to the natural frequency the index can get
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