text stringlengths 0 6.73k |
|---|
moving in the material—and with these basic contributions of the fñeld travelling |
at the ultimate velocity c. Our problem ¡is to understand how the apparenth |
slower velocity comes about. |
We shall try to understand the efect in a very simple case. A source which we |
shall call “the ez#ernal source” is placed a large distance away om a thin plate |
of transparent material, say glass. We inquire about the fñeld at a large distance |
on the opposite side of the plate. The situation is illustrated by the diagram |
of FEig. 31-1, where Š and ? are imagined to be very far away from the plate. |
According to the principles we have stated earlier, an electric ñeld anywhere |
--- Trang 536 --- |
Arriving wave ; “Transmitted” wave |
s8) =/ |
ì É What is me |
SE Han th |
“Reflected” |
Wave / Glass plate |
Fig. 31-1. Electric waves passing through a layer of transparent |
material. |
that is far from all moving charges is the (vector) sum of the felds produced by |
the external source (at ,S) ønd the fields produced by cách of the charges in the |
plate of glass, cuer one tuith is proper retardation at the 0elocitu c. Remember |
that the contribution of each charge is not changed by the presence of the other |
charges. These are our basic principles. The fñield at can be written thus: |
Eb—= » J2cách charge (31.1) |
all charges |
b—= +1. + » đ2cách charge› (31.2) |
all other charges |
where #2, ¡is the feld due to the source alone and would be precisely the fñeld |
at ÐP 7 there tuere no rmmatertal present. We expect the field at P to be diferent |
trom #⁄, ïf 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 fñeld oøƒ the source |
acts on these atoms it drives the electrons up and down, because I1 exerts a |
force on the electrons. And moving electrons generate a field—they constitute |
new radiators. These new radiators are related to the source Š, because they |
are driven by the fñeld of the source. "The total fñeld is not just the feld of the |
source Š, but it is modifñed by the additional contribution from the other moving |
charges. This means that the fñeld 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 the field inside the glass appears to be moving at a diferent |
speed. 'Phat is the idea which we would like to work out quantitatively. |
--- Trang 537 --- |
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 øi/ of the charges that are moving. I§ feels, |
in particular, the charges that are moving somewhere else in the glass. So the |
total feld which is acting on a particular chorge is a combination of the fields |
from the other charges, +0„ose rmmotions depend on tuhat this particular charge 1s |
đo”ng! You can see that it would take a complicated set of equations to get the |
complete and exact formula. Ït 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 cireumstance in which the efects from |
the other atoms are very small relative to the efects from the source. In other |
words, we take a material in which the total fñeld is not modifed 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, 1f the |
density of the atoms 1s 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 efect caused by the |
motion of the charges in the plate. These charges will also radiate waves back |
toward the source Š. 'Phis backward-going fñeld is the light we see relected 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 efect is equivalent to a refection from the surfaces. These refection |
efects 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 |
refected. |
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 |
understand why the apparent wave 0elocitu is diferent in diferent materials. Thhe |
bending of light rays comes about just 0ecause the efective speed of the waves is |
difÑferent in the materials. To remind you how that comes about we have drawn |
in Eig. 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 |
--- Trang 538 --- |
⁄⁄ ⁄ / |
VACUUM „⁄ˆ ,⁄ ˆ/⁄ GILASS |
⁄ ⁄ „ ,⁄ |
"4 ⁄ XS |
crests `\⁄ ⁄ |
Fig. 31-2. Relation between refraction and velocity change. |
wave crests indicates the direction of travel of the wave. Now all oscillations in |
the wave must have the same ƒreqguenec. (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 spacïng |
qlong the surƒace because they must travel together, so that a charge sitting |
at the boundary will feel only one frequency. The shorfes‡ distance bebween |
crests of the wave, however, ¡is the wavelength which is the velocity divided by |
the requency. Ôn the vacuum side it is Ào = 2zc/œ, and on the other side it is |
À = 270/u or 2#c/(œn, 1Ÿ 0 = cƒn is the velocity of the wave. From the fgure we |
can see that the only way for the waves to “ft” properly at the boundary is for |
the waves in the material to be travelling at a dilferent angle with respect to the |
surface. From the geometry of the fñgure you can see that for a “ñt” we must |
have Ào/sin Øo = À/sin 0, or sin Øo/sỉn Ø = ø=, which is Snell's law. We shall, for |
the resi of our discussion, consider only why light has an efective speed oŸ c/n |
in material of index m=, and no longer worry, in this chapter, about the bending |
of the light direction. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.