How To Calculate Linear Magnification

Magnification is the process of appearing to enlarge an object for purposes of visual inspection and analysis. Microscopes, binoculars and telescopes all magnify things using the special tricks embedded in the nature of light-transducing lenses in a variety of shapes.

Linear magnification refers to one of the properties of convex lenses, or those that show an outward curvature, like a sphere that has been severely flattened. Their counterparts in the optical world are concave lenses, or those that are curved inward and bend light rays differently than convex lenses.

When light rays traveling in parallel are bent as they pass through a convex lens, they are bent toward, and thus become focused on, a common point on the opposite side of the lens. This point, F, is called the focal point, and the distance to F from the center of the lens, denoted f, is called the focal length.

The power of a magnifying lens is just the inverse of its focal length: P = 1 / f. This means that lenses that have short focal lengths have strong magnification capabilities, whereas a higher value of f implies lower magnifying power.

Linear magnification, also called lateral magnification or transverse magnification, is just the ratio of size of the image of an object created by a lens to the object's true size. If the image and the object are both in the same physical medium (e.g., water, air or outer space), then the lateral magnification formula is the size of the image divided by the size of the object:

\(M = \frac{-i}{o}\)

Here M is the magnification, i is the image height and o is the object height. The minus sign (sometimes omitted) is a reminder that images of objects formed by convex mirrors appear inverted, or upside-down.

The lens formula in physics relates the focal length of an image formed by a thin lens, the distance of the image from the center of the lens, and the distance of the object from the center of the lens. The equation is

\(\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}\)

Say you position a tube of lipstick 10 cm from a convex lens with a focal length of 6 cm. How far away will the image appear on the other side of the lens?

For d_o= 10 and f = 4, you have:

\(\begin{aligned}
&\frac{1}{10}+\frac{1}{d_i}=\frac{1}{4} \
&\frac{1}{d_i}=0.15 \
&d_i=6.7
\end{aligned}\)

You can experiment with different numbers here to gain a sense of how altering the physical set-up affects the optical results in this type of problem.

Note that this is another way to express the concept of linear magnification. The ratio d_i to d_o is the same as the ratio of i to o. That is, the ratio of the height of the object to the height of its image is the same as the ratio of the length of the object to the length of its image.

The negative sign as applied to an image that appears on the opposite side of the lens from the object indicates that the image is "real," i.e., that it can be projected onto a screen or some other medium. A virtual image, on the other hand, appears on the same side of the lens as the object and is not associated with a negative sign in pertinent equations.

Although such topics lie beyond the scope of the present discussion, a variety of lens equations pertaining to a host of real-life situations, many of them involving changes in media (e.g., from air to water), can be uncovered with ease on the internet.