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Angular Magnification by a Simple Microscope
The Apparent Size of an Object
When an object is observed, the light entering the eye forms a real (and therefore inverted) image on the retina, the "screen" at the back of the eye.
Without the need for a detailed ray diagram, we can predict the size of the image on the retina as shown below, simply remembering that a ray through the optical center of a lens (or system of lenses) passes through undeviated.

The size of the image on the retina of the observer depends on two rather obvious factors:
1. the real size of the object, h
2. the distance of the object from the observer, u.

However, as also shown in the diagrams above, these two factors together determine the size of the angle subtended by the object at the eye, the angle a in the diagrams.
In other words, we can say that the perceived size of the object will depend on the angle α.
The apparent size of an object is directly proportional to this angle.

Optical instruments like microscopes and telescopes provide magnification by increasing the size of the angle subtended at the eye.

Angular Magnification by a Simple Microscope
The angular magnification (sometimes called magnifying power) is defined as follows

In order to see an object in as much detail as possible the person would be expected to hold it as close to the eye as possible.
The closest that a person can hold an object and still see it clearly is called the least distance of distinct vision, D.
An object placed at this distance from the eye is said to be at the near point of the eye.
This distance varies from person to person but is taken to be 25cm on average.

Consider first an observer looking at a small object, placed at the near point.
Assuming α is a small angle then
Now consider the person to be using a simple microscope, with the object distance, u, chosen to produce an image at distance D from the lens.
If the eye is placed close to the lens, the angle subtended at the eye by the image is approximately given by
So the angular magnification, in this situation, is given by

Therefore, when the image distance is D, the angular magnification is (not surprisingly) equal to the linear magnification

Rearranging the lens equation gives

In this case, M = v/u and v = D, therefore
 Equation 1

for a simple microscope with the image at the near point of the observer.

We will now consider the situation in which the object is moved so that the image is at infinity.
Obviously, as the object distance has increased, β is now a smaller angle than in the previous case.
Again, remember that β is a small angle, so we can write

so the angular magnification is now given by

which means that
 Equation 2
If we compare equations 1 and 2 we might, at first sight, be tempted to think there's a mistake somewhere...
The magnification given by equation 2 must be less than that given by equation 1.
If you can't see why this is the case... see here

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