Datasets:
vnpthack
/
vnpt_data

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
0
6.73k
virtual image. So when s” comes out negative, it means that Ó” is on the other
side of the surface, and everything is all right.
| — ===> =>
AIR GLASS
Fig. 27-4. A plane surface re-images the light from ' to OÓ.
Now consider the interesting case where ?## is equal to infnity; then we have
(1/5) + (n/s) =0. In other words, s” = —?ws, which means that if we look from a
dense medium into a rare medium and see a poïnt in the rare medium, it appears
to be deeper by a factor ø. Likewise, we can use the same equation backwards, so
that iƒ we look into a plane surface at an object that is at a certain distance inside
the dense medium, it will appear as though the light is coming from not as far
back (Fig. 27-4). When we look at the bottom of a swimming pool from above,
it does not look as deep as iÈ really is, by a factor 3/4, which is the reciprocal of
the index of refraction of water.
W© could go on, of course, to discuss the spherical mirror. But if one appreci-
ates the ideas involved, he should be able to work it out for himself. Therefore
we leave it to the student to work out the formula for the spherical mirror, but
we mention that it is well to adopt certain conventions concerning the distances
involved:
(1) The object distance s is positive if the point Ó is to the left of the surface.
(2) The image distance s” is positive if the point Ớf is to the right of the surface.
(3) The radius of curvature of the surface is positive iƒ the center is to the right
of the surface.
In Fig. 27-2, for example, s, s/, and ## are all positive; in Pig. 27-3, s and ## are
--- Trang 476 ---
positive, but s” is negative. IÝ we had used a concave surface, our formula (27.3)
would still give the correct result if we merely make a negative quantity.
In working out the corresponding formula for a mirror, using the above
conventions, you will fnd that if you put ø = —1 throughout the formula (27.3)
(as though the material behind the mirror had an index —1), the right formula
for a mirror resultsl
Although the derivation of formula (27.3) is simple and elegamt, using least
time, one can of course work out the same formula using 5nell's law, remembering
that the angles are so small that the sines of angles can be replaced by the angles
themselves.
27-3 The focal length of a lens
Now we go on to consider another situation, a very practical one. Most of the
lenses that we use have two surfaces, not just one. How does this affect matters?
Suppose that we have ÿwo surfaces of diferent curvature, with glass filling the
space bebween them (Eig. 27-5). We want to study the problem of focusing from
a point Ó to an alternate point Ớ”. How can we do that? The answer is this:
Eirst, use formula (27.3) for the first surface, forgetting about the second surface.
This will tell us that the light which was diverging from @Ø will appear to be
converging or diverging, depending on the sign, from some other point, say Ó”.
Now we consider a new problem. We have a diferent surface, between glass and
air, in which rays are converging toward a certain point Ó'. Where will they
actually converge? We use the same formula again!l We fnd that they converge
at Ø“”. Thus, if necessary, we can go through 7ð surfaces by just using the same
formula in succession, from one to the nextl
x —m `...
Fig. 27-5. lmage formation by a two-surface lens.
--- Trang 477 ---
There are some rather high-class formulas that would save us considerable
energy in the few times in our lives that we might have to chase the light through
fve surfaces, but it is easier just to chase it through fve surfaces when the
problem arises than it is to memorize a lot of formulas, because it may be we
will never have to chase it through any surfaces at alll
In any case, the principle is that when we go through one surface we find a
new position, a new focal point, and then take that point as the starting poiïnt
for the next surface, and so on. In order to actually do this, since on the second
surface we are going from %øw to 1 rather than from 1 to øœ, and since in many
systems there is more than one kind of glass, so that there are indices 0ø, na,
..., we really need a generalization of formula (27.3) for a case where there are
two diferent indices, + and nạ, rather than only nø. 'Phen ït is not dificult to
prove that the general form of (27.3) is
(m1/5) + (na/s)) = (na — mì)/R. (27.7)
Particularly simple is the special case in which the two surfaces are very close
together——so close that we may ignore small errors due to the thickness. lÝ we
draw the lens as shown in Fig. 27-6, we may ask this question: How must the
lens be built so as to focus light from Ó to Ø7? Suppose the light comes exactÌy
to the edge of the lens, at point P. Then the excess tỉme in going from Ó to
is (mịh2/2s) + (nịh2/25/), ignoring for a moment the presence of the thickness 7”
of glass of index nạ. Now, to make the time for the direct path equal to that for
the path ÓPŒỚ", we have to use a piece of glass whose thickness 7' at the center
1s such that the delay introduced in going through this thickness is enough to
compensate for the excess time above. Therefore the thickness of the lens at the
O le) œ
HỊ HỊ
Fig. 27-6. A thin lens with two positive radii.
--- Trang 478 ---
center must be given by the relationship
(nìh2/2s) + (nìh?/25)) = (nạ — mì)T. (27.8)
W© can also express 7 in terms of the radii f¡ and ñ¿ of the two surfaces. Paying
attention to our convention (3), we thus fñnd, for ?#ị < #2 (a convex lens),
7 = (hˆ/2RI) - (h°/2Ã). (27.9)
'Therefore, we fnally get
(mì/s) + (mị/s) — (mạ — mị)(1/ị — 1/R). (27.10)
Now we note again that if one of the points is at infinity, the other will be at a
point which we will call the focal length ƒ. The focal length ƒ is given by
1/ƒ =(nm— 1)(1/Tị — 1/1). (27.11)
where ø = nạ/m.
Now, 1ƒ we take the opposite case, where s goes to infinity, we see that sf 1s
at the focal length ƒ7. This time the focal lengths are equal. (This is another
special case of the general rule that the ratio of the two focal lengths is the ratio
of the indices of refraction in the two media in which the rays focus. In this
particular optical system, the initial and fnal indices are the same, so the two
focal lengths are equal.)
Forgetting for a moment about the actual formula for the focal length, if
we bought a lens that somebody designed with certain radii of curvature and a
certain index, we could measure the focal length, say, by seeing where a point at