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W© go back now to the situation shown in Fig. 3Í-1. We see that what we
hawve to do 1s to calculate the fñeld produced at by all the oscillating charges
in the glass plate. We shall call this part of the feld 4, and ït is just the sum
written as the second term in Bq. (31.2). When we add it to the term #⁄¿, due to
the source, we will have the total feld at P.
--- Trang 539 ---
Thịs is probably the most complicated thing that we are going to do this
year, but i is complicated only in that there are many pieces that have to be put
together; each piece, however, is very simple. nlike other derivations where we
say, “Forget the derivation, just look at the answerl,” in this case we do not need
the answer so mụch as the derivation. In other words, the thing to understand
now is the physical machinery for the produection of the index.
To see where we are going, let us fñrst ñnd out what the “correction fñeld” „
would have to be if the total fñeld at ? 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 feld of a wave travelling to the right (along the z-axis) would
12; —= Eo cosu(É — z/c) (31.3)
or, using the exponential notation,
E, = Eoele~z/©), (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 Az in the time Az/e. But ifit appears to travel
at the speed c/n then it should take the longer tìme œ Az/c or the add¿Hional
time A£ = (m — 1) Az/c. After that it would continue to travel at the speed e
again. We can take into account the extra delay in getting through the plate by
replacing £ in Eq. (31.4) by (# — Af) or by [£ — (m — 1) Az/c|. 5o the wave after
Insertion of the plate should be written
J2after plate —— EocfelF-Œ=1) Az/e—z/s * (31.5)
W© can also write this equation as
DÀNG plate — eTie(n=1) Az/e Enel2—=z/©), (31.6)
which says that the wave after the plate is obtained from the wave which could
exist without the plate, i.e., from #;, by multiplying by the factor e~?#Œ=1)Az/€,
Now we know that multiplying an oscillating funetion like e?“f by a factor c?#
Just says that we change the phase of the oscillation by the angle Ø, which is, of
course, what the extra delay in passing through the thickness Az has done. lt
has retarded the phase by the amount œ(nø — 1) Az/e (retarded, because of the
mỉnus sign in the exponent).
--- Trang 540 ---
W©e have said earlier that the plate should adđ a fñeld 4 to the original
ñeld #„ = Ese!2Œ~Z/*), bụt we have found instead that the efect of the plate is
to rmmuliiplụ the ñeld 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.
lt is particularly easy to find the right number to add in the case that Az is
small, for you will remember that IÝ z is a small number then e” is nearly equal
to (1+z). We can write, therefore,
e~s(@=1)AZ/€ = 1 — ju(n — 1) AZ/e. (31.7)
Using this equality in Eq. (31.6), we have
ö(m„— 1A Ố
đ2atter plate — Epge20=z/2 — o§n ) : Ege-z/©) h (31.8)
"¬—— —“_——~Ö
The first term is just the fñeld om the source, and the second term must just be
cqual to !4, the fñeld produced to the right of the plate by the oscillating charges
of the plate—expressed here in terms of the index of refraction ø, and depending,
of course, on the strength of the wave from the source.
'What we have been doïng is easily visualized if we look at the complex number
diagram in Eig. 31-3. We first draw the number #; (we chose some values for
z and £ so that E2, comes out horizontal, but this is not necessary). The delay
due to slowing down in the plate would delay the phase of this number, that 1s,
it would rotate ; through a negative angle. But this is equivalent to adding
the small vector „ at roughly right angles to !2¿. But that is just what the
factor —¿ means in the second term of Eq. (31.8). It says that if #2; is real, then
„4 is negative imaginary or that, in general, #⁄⁄; and !4 make a right angle.
lmaginary Axis
Angle = u(n — 1)Az/c
= Real Axis
_¬ ` Ea
Fig. 31-3. Diagram for the transmitted wave at a particular £ and z.
--- Trang 541 ---
31-2 The feld due to the material
W©e now have to ask: Is the field #„ 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 ø should bel [Since øœ
is the only nonfundamental number in Eq. (31.8).| We turn now to calculating
what field F„ the charges in the material will produee. (To help you keep track
of the many symbols we have used up to now, and will be using in the rest of
our calculation, we have put them all together in Table 31-1.)
Table 31-1
Symbols used in the calculations
FZ„ = field from the source
tạ = field produced by charges in the plate
Az = thickness of the plate
z = perpendicular distance from the plate
m = index of refraction
œ = frequency (angular) of the radiation
NÑ = number of charges per unit volume in the plate
rạ = number of charges per unit area of the plate
qe — charge on an electron
mm —= mass of an electron
œg — resonant Írequency of an electron bound in an atom
Tf the source Š (of Eig. 31-1) is far off to the left, then the field #⁄; will have
the same phase everywhere on the plate, so we can write that in the neighborhood
of the plate
E, = Eoele~z/©), (31.9)
Right at the plate, where z = 0, we will have
= Eoe"°f (at the plate). (31.10)
Bach 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 oŸ 2o is vertical) by
the electrie force g#. To fnd what motion we expect for the electrons, we will
--- Trang 542 ---
assume that the atoms are little oscillators, that is, that the electrons are fastened