Unformatted text preview: 18. Lecture 18
18.1 Lens equation
Going back to the convergent lens, we can once again ﬁnd an equation that determines
the position and size of the image given the position and size of the object. The sign
convention for the image’s position is now that s > 0 if the image is on opposite sides
of the lens as the object and s < 0 if the image is on the same side of the object. This
makes sense because we look at the lens from behind whereas we looked at the mirror
from the front. That is a real image has s > 0 in both cases. With these conventions
and looking at ﬁgure 101 we can derive, similarly as for the mirror:
h
−h
=
s−f
f (18.1) − h
h
=
−f
s
f (18.2) and also Since the equations are the same as for the mirror we can immediately derive
sf
s−f
hf
h = −
s−f
s = (18.3)
(18.4) or equivalently:
1
1
1
+ =
ss
f
−h
h
=
s
s (18.5)
(18.6) It is easy to see that once again these equations are valid if s > f or s < f .
18.2 Divergent Lens
When rays arrive parallel to the axis into a divergent lens they become divergent and
seem to originate from a focal point situated on the same side of the lens from where
the rays arrived. As seen in ﬁg. 104 the image is always virtual and upright. The
equation for the lens can be applied to this case if we take f < 0. – 118 – Figure 104: Image of an object by a divergent lens. It is always virtual and upright. 18.3 Camera, microscope, telescope
There are numerous application of lenses and mirrors. The photographic camera is
one of them. It actually works on the same principle as our eye. It creates an image
on a surface sensitive to light. The image is then read electronically or chemically by
producing a reaction on a ﬁlm. The minimal setup is a convergent lens which creates
a real image as in ﬁg.101. The screen is positioned at the point where the image is
formed.
Another application is the microscope. The minimal setup in this case is a convergent lens as in ﬁg.102. Actually this is also the usual magnifying glass. Actual
microscopes have more lenses that increase the magniﬁcation but work on the same
principle.
Finally we have the telescope which is used to look at objects far away. When
a point is far away, the rays coming from such point, for all practical purposes are
parallel to each other. On the other hand if we have an extended object far away, and
the object is large enough, the rays originating from diﬀerent points of the object arrive
with diﬀerent angles. In that way we can talk about the angular size of an object which
is the diﬀerence in angles between rays arriving from opposite edges of the object. As
seen in ﬁgure 105, out of parallel rays, a convergent lens forms an image on its focal
plane. This is how our eyes form images for objects at inﬁnity. The angular size of
the object is determines the size of the image on our retina. In that way we see for – 119 – example the Moon as an object which has an apparent size which does not seem too
big compared to everyday objects even if we know it is much larger than any object we
can handle. The reason of course being that the Moon is much further away that any
everyday object.
If, as in ﬁgure 105 we introduce another lens whose focal point coincides with the
one of the ﬁrst lens, the rays will once again be parallel but now the angular size will
be diﬀerent. We want the angular size to be larger so the object will appear larger and
we will be able to see more details. From ﬁg.105 we can derive that
tan θ = h
,
f tan θ = h
f (18.7) where h is the height of the image created by the ﬁrst lens and f , f are the focal
distances of both lenses. The angles θ and θ are the angles at which a given ray arrives
and then emerges from the telescope. Since we always consider small angles we have
tan θ θ, tan θ θ (18.8) which implies θ
f
=
(18.9)
θ
f
which is deﬁned as the magniﬁcation of the telescope. If we have f < f then we also
have θ > θ as we wanted.
magniﬁcation = 18.4 Aberrations
We insist once again that all the equations and image formation we nave studied for
mirrors and lenses are approximations for small angles. Sometimes these are called the
laws of ideal mirrors and lenses whereas actual ones do not behave exactly the same.
The diﬀerences between the ideal and actual behavior are called aberrations. Let us
discuss two simple ones. First, we said that a spherical mirror forms a sharp image
only if the rays arrive close to the axis. namely the mirror is small compared to the
radius of curvature. This is illustrated in ﬁg.106 where we see that the rays emanating
from the object do not all cross at the same point. For rays reaching the mirror further
from the axis this is evident. This is called spherical aberration. If we put a screen or
photographic plate at the position of the image the image will be blurred for such a big
mirror. On the other hand, if we just take a look from the left with our own eyes this
might not be so evident because our eyes are quite small and will only capture a few
of the rays. Such rays would intersect at the image point and the image will appear
sharp essentially because our eyes are taking advantage only of the central part of the
mirror. – 120 – Figure 105: Simple refractive telescope. Two lenses increase the angular size of objects
situated at inﬁnity Another kind of aberration is called color or chromatic aberration. It applies only
to lenses. Although we did not discuss this in detail it turns out that the index of
refraction is diﬀerent for diﬀerent colors as seen in a prism which separate the diﬀerent
colors from white light. A lens works similarly and therefore the focal point is slightly
diﬀerent for diﬀerent colors. This distorts the images produced by lenses and needs
to be corrected in some applications by putting several lenses such that their color
aberrations cancel each other. – 121 – Figure 106: Spherical mirror. Images formed by a big mirror (compared with its curvature
radius) are not sharp. – 122 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue.
 Fall '11
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