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and on the right i6 is 1/n, where ø is the index of refraction. The light travels |
more slowly in the glass by a facbOr n. |
Now suppose that we have a point at Ó, at a distance s from the front surface |
of the glass, and another point Ó“ at a distance sf inside the glass, and we desire |
to arrange the curved surface in such a manner that every ray from @ which hits |
the surface, at any point , will be bent so as to proceed toward the point Ó”. |
For that to be true, we have to shape the surface in such a way that the time it |
takes for the light to go from Ó to Ð, that is, the distance ÓØ?P divided by the |
speed of light (the speed here is unity), plus œ - Ó“P, which is the time it takes |
to go from P to Ớƒ, is equal to a constant independent of the point P. Thịs |
condition supplies us with an equation for determining the surface. The answer |
1s that the surface is a very complicated fourth-degree curve, and the student |
may entertain himself by trying to calculate it by analytic geometry. It is simpler |
to try a special case that corresponds to s —> œo, because then the curve 1s a |
second-degree curve and is more recognizable. lt is interesting to compare this |
curve with the parabolic curve we found for a focusing mirror when the light is |
coming from infnity. |
So the proper surface cannot easily be made——to focus the light from one |
point to another requires a rather complicated surface. Ït turns out in practice |
that we do not try to make such complicated surfaces ordinarily, but instead we |
make a compromise. Instead of trying to get aøÏl the rays to come to a Íocus, we |
arrange it so that only the rays fairly close to the axis Ó” come to a focus. The |
farther ones may deviate if they want to, unfortunately, because the ideal surface |
1s complicated, and we use instead a spherical surface with the right curvature at |
the axis. Ït is so much easier to fabricate a sphere than other surfaces that it 1s |
proftable for us to fnd out what happens to rays striking a spherical surface, |
supposing that only the rays near the axis are going to be focused perfectly. |
'Those rays which are near the axis are sometimes called parazial røs, and what |
we are analyzing are the conditions for the focusing of paraxial rays. We shall |
discuss later the errors that are introduced by the fact that all rays are not aÌways |
close to the axis. |
'Thus, supposing ? is close to the axis, we drop a perpendicular P@) such that |
the height P@) is h. For a moment, we imagine that the surface is a plane passing |
through ?. In that case, the time needed to go from @ to would exceed the |
time from Ó to @, and also, the time from to Ó” would exceed the time from |
--- Trang 473 --- |
Q to Ở But that is why the glass must be curved, because the total excess |
time must be compensated by the delay in passing from V to Q! Ñow the ezcess |
time along route ÓP is h2/2s, and the excess time on the other route is nh2/2s”. |
This excess time, which must be matched by the delay in going along VQ, differs |
from what it would have been in a vacuum, because there is a medium present. |
In other words, the time to go from V to Q is not as 1Í it were straight in the |
air, but 1E is slower by the factor ø%œ, so that the excess delay in this distance 1s |
then (wT— 1)VQ. And now, how large is VQ? TÍ the point Ở is the center of the |
sphere and if its radius is #, we see by the same formula that the distance V) is |
cqual to h”/2ÿ. Therefore we discover that the law that connects the distances s |
and s/, and that gives us the radius of curvature ? of the surface that we need, is |
(h2/2s) + (nh2/2s!) = (n — 1)h”/2R (27.2) |
(1/s) + (n/s) = (n — 1)/R. (27.3) |
If we have a position Ó and another position ÓØ”, and want to focus light from @ |
to Ø7, then we can calculate the required radius of curvature ## of the surface by |
this formula. |
Now it turns out, interestingly, that the same lens, with the same curvature †, |
will focus for other distances, namely, for any pair of distances such that the sum |
of the two reciprocals, one multiplied by m, is a constant. 'Phus a given lens will |
(so long as we limit ourselves to paraxial rays) focus not only from Ó to Ở', but |
between an infinite number of other pairs of points, so long as those pairs of |
points bear the relationship that 1/s + œ/s” is a constant, characteristic of the |
In particular, an interesting case is that in which s —> oo. W© can see from the |
formula that as one s increases, the other decreases. In other words, if point Ó |
goes out, poini Ó“ comes in, and vice versa. As point Ó goes toward infinity, |
point @“ keeps moving in until it reaches a certain distance, called the ƒocal |
length ƒ', inside the material. If parallel rays come in, they will meet the axis |
at a disbance ƒ7. Likewise, we could imagine it the other way. (Remember the |
reciprocity rule: if light will go rom Ó to Ó”, of course it will also go from Ớ/ |
to Ó.) Therefore, if we had a light source inside the gÌass, we might want to know |
where the focus is. In particular, if the light in the glass were at infinity (same |
problem) where would it come to a focus outside? Thịis distance is called ƒ. Of |
course, we can also put it the other way. If we had a light source at ƒ and the |
--- Trang 474 --- |
light went through the surface, then ¡% would go out as a parallel beam. We can |
easily fnd out what ƒ and 7 are: |
m/Ƒ) =(n— 1)/R Or ƒƑ = Rn/(n— 1), (27.4) |
1/ƒ =(n—1)/R OF ƒ =R/(n- ]). (27.5) |
W© see an interesting thing: iƒwe divide each focal length by the corresponding |
index of refraction we get the same resultl 'Phis theorem, in fact, is general. lt is |
true of any system oŸ lenses, no matter how complicated, so it is interesting to |
remember. We did not prove here that it is general—we merely noted i% for a |
single surface, but it happens to be true in general that the two focal lengths of |
a system are related in this way. Sometimes q. (27.3) is written in the form |
1/s-+m/s = 1/ƒ. (27.6) |
Thịis is more useful than (27.3) because we can measure ƒ more easily than we |
can measure the curvature and index of refraction of the lens: if we are not |
interested in designing a lens or in knowing how it got that way, but simply liÑt |
1t of a shelf, the interesting quantity is ƒ, not the œ and the 1 and the #l |
Now an interesting situation occurs If s becomes less than ƒ. What happens |
then? IÝ s < ƒ, then (1/s) > (1/ƒ), and therefore s” is negative; our equation says |
that the light will focus only with a negative value of s”, whatever that meansl |
It does mean something very interesting and very defñnite. It is still a useful |
formula, in other words, even when the numbers are negative. What it means is |
shown in Fig. 27-3. If we draw the rays which are diverging from Ó, they will be |
bent, it is true, at the surface, and they will not come to a focus, because ) is so |
close in that they are “beyond parallel” However, they diverge as if they had |
come from a point Ó“ ou£side the glass. This is an apparent image, sometimes |
— ===- |
Fig. 27-3. A virtual image. |
--- Trang 475 --- |
called a ơrtual image. The image @' in Fig. 27-2 is called a real zœmage. TỶ the |
light really comes to a point, it is a real image. But if the light appears to be |
coming ƒrom a point, a fictitious point diferent from the original point, it is a |
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