Magnification

From Wikipedia, the free encyclopedia

For the 2001 album of progressive rock band Yes, see Magnification (album).

Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called magnification. When this number is less than one it refers to a reduction in size, sometimes called minification.

Typically magnification is related to scaling up visuals or images to be able to see more detail, increasing resolution, using optics, printing techniques, or digital processing. In all cases, the magnification of the image does not change the perspective of the image.

Contents

Magnification as a number (optical magnification)

Optical magnification is the ratio between the apparent size of an object (or its size in an image) and its true size, and thus it is a dimensionless number.

\mathrm{MA}=\frac{\tan \varepsilon}{\tan \varepsilon_0}
{\varepsilon_0}
By convention, for magnifying glasses and optical microscopes, where the size of the object is a linear dimension and the apparent size is an angle, the magnification is the ratio between the apparent (angular) size as seen in the eyepiece and the angular size of the object when placed at the conventional closest distance of distinct vision of 25 cm from the eye.

Optical magnification is sometimes referred to as "power" (for example "10× power"), although this can lead to confusion with optical power.

Calculating the magnification of optical systems

M = {f \over f-d_o}
where f is the focal length and do is the distance from the lens to the object and ho is the height of the object. Note that for real images, M is negative and the image is inverted. For virtual images, M is positive and the image is upright. With di the distance from the lens to the image and hi being the height of the image, the magnification can also be written as:
M = -{d_i \over d_o} = {h_i \over h_o}
Note again that a negative magnification implies an inverted image.
M = +{d_i \over d_o}
M= {f_o \over f_e}
where fo is the focal length of the objective lens and fe is the focal length of the eyepiece. The angular magnification is given by
\mathrm{MA}= {f_o \over f_e}
\mathrm{MA}={25\ \mathrm{cm}\over f}\quad.
If instead the lens is held very close to the eye, and the object is placed close to the lens, a larger angular magnification can be obtained, approaching
\mathrm{MA}={25\ \mathrm{cm}\over f}+1\quad .
Here, f is the focal length of the lens in centimeters. The constant 25 cm is an estimate of the "near point" distance of the eye—the closest distance at which the eye can focus.
\mathrm{MA}=M_o \times M_e
where Mo is the magnification of the objective and Me the magnification of the eyepiece. The magnification of the objective depends on its focal length fo and on the distance d between objective back focal plane and the focal plane of the eyepiece (called the tube length):
M_o={d \over f_o}
The magnification of the eyepiece depends upon its focal length fe and calculated by the same equation as that of a magnifying glass (above).

Note that both astronomical telescopes as well as simple microscopes produce an inverted image, thus the equation for the magnification of a telescope or microscope is often given with a minus sign.

Measurement of telescope magnification

Measuring the actual angular magnification of a telescope is difficult, but it is possible to use the reciprocal relationship between the linear magnification and the angular magnification, since the linear magnification is constant for all objects.

The telescope is focussed correctly for viewing objects at the distance for which the angular magnification is to be determined and then the object glass is used as an object the image of which is known as the Ramsden disc. The diameter of this may be measured using an instrument known as a Ramsden dynamometer which consists of a Ramsden eyepiece with micrometer cross hairs in the back focal plane. This is mounted in front of the telescope eyepiece and used to measure the diameter of the Ramsden disc. This will be much smaller than the object glass diameter, which gives the linear magnification (actually a reduction), the angular magnification can be determined from

MA = 1 / M = DObjective / DRamsden

Other uses