Microscope Magnification
Refer to Fig. 1914. Suppose that f _{o} is the focal length (in meters) of the objective lens and f _{e} is the focal length (in meters) of the eyepiece. Assume that the objective and the eyepiece are placed along a common axis and that the distance between their centers is adjusted for proper focus. Let s represent the distance (in meters) from the objective to the real image it forms of the object under examination. The microscopic magnification (a dimensionless quantity denoted m in this context) is given by
m = [( s − f _{o} )/ f _{o} )] [( f _{e} + 0.25)/ f _{e} ]
Fig. 1914 . Calculation of the magnification factor in a compound microscope. See text for details.
The quantity 0.25 represents the average near point of the human eye, which is the closest distance over which the eye can focus on an object: approximately 0.25 m.
A less formal method of calculating the magnification of a microscope is to multiply the magnification of the objective by the magnification of the eyepiece. These numbers are provided with objectives and eyepieces and are based on the use of an air medium between the objective and the specimen, as well as on the standard distance between the objective and the eyepiece. If m _{e} is the power of the eyepiece and m _{o} is the power of the objective, then the power m of the microscope as a whole is
m = m _{e} m _{o}
Numerical Aperture And Resolution
In an optical microscope, the numerical aperture of the objective is an important specification in determining the resolution, or the amount of detail the microscope can render. This is defined as shown in Fig. 1915.
Let L be a line passing through a point P in the specimen to be examined, as well as through the center of the objective. Let K be a line passing through P and intersecting the outer edge of the objective lens opening. (It is assumed that this outer edge is circular.) Let q be the measure of the angle between lines L and K . Let M be the medium between the objective and the sample under examination. This medium M is usually air, but not always. Let r _{m} be the refractive index of M . Then the numerical aperture of the objective A _{o} is given by
A _{o} = r _{m} sin q
In general, the greater the value of A _{o} , the better is the resolution. There are three ways to increase the A _{o} of a microscope objective of a given focal length:
 The diameter of the objective can be increased.
 The value of r _{m} can be increased.
 The wavelength of the illuminating light can be decreased.
Largediameter objectives having short focal lengths, thereby providing high magnification, are difficult to construct. Thus, when scientists want to examine an object in high detail, they can use blue light, which has a relatively short wavelength. Alternatively, or in addition, the medium M between the objective and the specimen can be changed to something with a high index of refraction, such as clear oil. This shortens the wavelength of the illuminating beam that strikes the objective because it slows down the speed of light in M . (Remember the relation between the speed of an electromagnetic disturbance, the wavelength, and the frequency!) A side effect of this tactic is a reduction in the effective magnification of the objective lens, but this can be compensated for by using an objective with a smaller radius of surface curvature or by increasing the distance between the objective and the eyepiece.
The use of monochromatic light rather than white light offers another advantage. Chromatic aberration affects the light passing through a microscope in the same way that it affects the light passing through a telescope. If the light has only one wavelength, chromatic aberration does not occur. In addition, the use of various colors of monochromatic light (red, orange, yellow, green, or blue) can accentuate structural or anatomic features in a specimen that do not always show up well in white light.
The Compound Microscope Practice Problem
Problem
A compound microscope objective is specified as 10×, whereas the eyepiece is rated at 5×. What is the power m of this instrument?
Solution
Multiply the magnification factor of the objective by that of the eyepiece:
m = (5 × 10) × = 50×
Practice problems of these concepts can be found at: Optics Practice Test
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