6.1.11. Magnification

Table of Contents

What Magnification Means

Magnification tells us how large or small an image appears compared with the object. In geometrical optics, it is a ratio of image size to object size. If an image is larger than the object, the magnification is greater than 1. If the image is smaller, the magnification is less than 1.

If the object height is $h_o$ and the image height is $h_i$, the magnification $m$ is

$$
m = \frac{h_i}{h_o}
$$

This is the basic size ratio. It helps us describe what a mirror or lens does to an object.

Important definition:
$$
m = \frac{h_i}{h_o}
$$
Magnification is a ratio, so it has no unit.

Sign of Magnification

Magnification is not only about size, it also tells us about image orientation. In geometrical optics, the sign matters.

If $m$ is positive, the image is upright relative to the object. If $m$ is negative, the image is inverted.

This gives a very useful interpretation:

Magnification $m$	Meaning
$m > 1$	Image is larger than the object
$0 < m < 1$	Image is smaller and upright
$m < -1$	Image is larger and inverted
$-1 < m < 0$	Image is smaller and inverted

An upright image has the same vertical orientation as the object. An inverted image appears turned upside down.

Sign rule for magnification:
Positive magnification means upright image.
Negative magnification means inverted image.

Magnification and Distances

For mirrors and lenses, magnification can also be written using image distance and object distance. If $d_i$ is the image distance and $d_o$ is the object distance, then

$$
m = -\frac{d_i}{d_o}
$$

This formula connects image size to image position.

A negative sign appears because, under the usual sign convention, real inverted images and virtual upright images follow different distance signs.

Key relation:
$$
m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}
$$
This is one of the most important formulas in geometrical optics.

Physical Interpretation

This formula gives immediate information. If the image distance has a larger magnitude than the object distance, the image tends to be larger in size. If the image distance has a smaller magnitude, the image tends to be smaller.

For example, if

$$
d_o = 20 \text{ cm}, \qquad d_i = 40 \text{ cm}
$$

then

$$
m = -\frac{40}{20} = -2
$$

So the image is inverted and twice as tall as the object.

If instead

$$
d_o = 30 \text{ cm}, \qquad d_i = -10 \text{ cm}
$$

then

$$
m = -\frac{-10}{30} = \frac{1}{3}
$$

So the image is upright and one third the height of the object.

Magnification in Mirrors and Lenses

Magnification works the same general way for both mirrors and lenses. The image may be real or virtual, upright or inverted, enlarged or reduced. Magnification summarizes all of that in one number.

A plane mirror is a simple case. It forms an image that is the same size as the object, so

$$
m = 1
$$

The image is upright.

Concave mirrors and convex lenses can produce magnifications greater than 1 or less than 1, depending on object position. Convex mirrors and concave lenses usually produce upright reduced images, so their magnification is positive but less than 1.

Visual Meaning

The idea of magnification can be understood from similar triangles in ray geometry. The rays from the top of the object and the top of the image form triangles with the optical axis. These triangles lead to the relation between heights and distances.

Rendering scientific sketch…

Object and image heights in a lens system

In this sketch, the image is below the axis, so it is inverted. That means the magnification is negative.

Examples

Suppose an object has height

$$
h_o = 4 \text{ cm}
$$

and the image height is

$$
h_i = -12 \text{ cm}
$$

Then

$$
m = \frac{h_i}{h_o} = \frac{-12}{4} = -3
$$

The image is inverted and three times larger than the object.

Now suppose

$$
h_o = 5 \text{ cm}, \qquad h_i = 2 \text{ cm}
$$

Then

$$
m = \frac{2}{5} = 0.4
$$

The image is upright and smaller than the object.

Magnification and What You See

Magnification describes geometric image size, not necessarily how large something seems to your eye in everyday life. Apparent visual size can depend on distance from the eye, but in geometrical optics, magnification means the actual ratio of image height to object height.

This makes magnification a precise mathematical idea, useful for mirrors, lenses, and optical instruments.

Do not confuse image size with image distance.
A large image is not always far away, and a small image is not always close.
Magnification depends on the ratio of heights, or equivalently on $-\frac{d_i}{d_o}$.

Summary

Magnification tells how image size compares with object size. It is given by

$$
m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}
$$

A positive value means the image is upright. A negative value means the image is inverted. A magnitude greater than 1 means enlarged, and a magnitude less than 1 means reduced.

With one number, magnification gives both size information and orientation information, which makes it one of the most useful ideas in geometrical optics.

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