Topic 38. The simple magnifying glass
In
this section we describe how a single positive lens can be used as a simple
magnifying glass. Recall that the
magnification of a lens is the ratio of the size of the image, si to
the size of the object, so. As we have shown before (Topic
31. Camera Lenses, Part 3. The Effect of Focal Length), the magnification
of a simple lens is the ratio of the distance from the lens to the image, di
divided by the distance from the lens to the object, do:
The
distance, di , between the eye-lens and the retina is fixed by the
size and shape of the eye. Therefore, the magnification can be changed only by
changing the distance between the object and the eye, do. Since this distance is in the denominator of
the fraction above, the magnification increases as an object is brought closer
to the eye. The maximum magnification
will be realized when the object is brought to the near point. The
magnification could be increased by bringing the object still closer, but the
strength of the eye-lens can not be made great enough to bring the object into
focus at these short distances. Since the distance between the eye-lens and the
retina is about 20 mm and the near-point is about 25 cm, the maximum
magnification of the eye is about 2/25= 0.08. (Note that this magnification
describes the relationship between physical size of the image on the retina and
the physical size of the object. All animals learn to relate the physical size
of an image on the retina back to the physical size of the object that produced
it.)
If a positive lens is placed in front of the eye, the combination of this lens and the eye itself result in a lens whose strength is the sum of the strengths of the two lenses. (See Topic 27. Compound Lenses). In other words, the focal length of the combination is smaller than the focal length of the eye itself. This compound lens produces an image which is in front of the retina and is therefore out of focus. However, since the image is now in front of the retina, the image can be moved back onto the retina by bringing the object still closer to the eye. This will bring the object back into focus on the retina and simultaneously increase the magnification. Thus the simple magnifying glass works by allowing the object to be brought closer to the eye while still being kept in focus.
Since the focal length of the eye-lens can be changed by the muscles in the eye, a calculation of the magnification of the additional lens in front of the eye depends on how the eye-lens is configured. These results are easy to calculate using the algebraic form of the ray-tracing rules.
There are two extreme situations: (1) the eye lens is configured with as short a focal length as possible so that it is still focused on the object at the near-point. If a person then puts an additional positive lens whose focal length is f (in cm) in front of the eye, and if the eye-lens remains set to its shortest possible focal length, then the magnification (relative to the magnification of the eye by itself as described above) is
In order to be in sharp focus in this situation, the object must be placed at a distance of
Since the denominator of the fraction is always greater than the numerator, the value of the fraction is always less than 1, so that the object distance is less than the focal length of the positive lens that has been added. For example, if the focal length of the additional lens is 5 cm, the magnification is 1+5=6, and the object will be in focus when it is about 4.2 cm in front of the eye.
The
second situation is when the eye lens is configured to be as relaxed as
possible so that it is focused on a very distant object. If a person then puts
an additional positive lens whose focal length is f (in cm) in front of the
eye, and if the eye-lens remains set to its longest possible focal length, then
the magnification (relative to the magnification of the eye itself as described
above) is
In order to be in sharp focus in this situation, the object must be located at a distance equal to the focal length of the additional lens. That is,
Since this distance is somewhat greater than the distance in the first situation, it should not be surprising that the magnification is somewhat smaller. Using the same additional lens as above, the magnification would be 5 and the object would be in clear focus when it was 5 cm in front of the eye.