KAHIBARO
Table of Contents
Image formation by spherical mirrors
The mirror equation describes how an object and its image are related for a spherical mirror. It is used for concave and convex mirrors, and it connects three quantities, the object distance, the image distance, and the focal length.
If the object distance is called $d_o$, the image distance is called $d_i$, and the focal length is called $f$, then the mirror equation is
$$
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
$$
This compact formula is one of the most important results in geometrical optics.
For a spherical mirror,
$$
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
$$
Always use a consistent sign convention when applying it.
Meaning of the quantities
The object distance $d_o$ is the distance from the mirror to the object. The image distance $d_i$ is the distance from the mirror to the image. The focal length $f$ is the distance from the mirror to the focal point.
For a concave mirror, the focal point is in front of the mirror, so its focal length is positive in the usual sign convention. For a convex mirror, the focal point is behind the mirror, so its focal length is negative.
The radius of curvature $R$ of the spherical mirror is related to the focal length by
$$
f = \frac{R}{2}
$$
This relation is valid for spherical mirrors in the paraxial approximation, which means rays are close to the principal axis.
For spherical mirrors,
$$
f = \frac{R}{2}
$$
This is true only for the usual small angle, paraxial case.
Sign convention
To use the mirror equation correctly, signs matter. A common sign convention is shown below.
| Quantity | Positive when | Negative when |
|---|---|---|
| $d_o$ | object is in front of the mirror | object is behind the mirror |
| $d_i$ | image is in front of the mirror, real image | image is behind the mirror, virtual image |
| $f$ | concave mirror | convex mirror |
In most beginner problems, the object is in front of the mirror, so $d_o > 0$.
A real image forms in front of the mirror and has $d_i > 0$. A virtual image forms behind the mirror and has $d_i < 0$.
Concave mirror cases
For a concave mirror, $f > 0$. Depending on where the object is placed, the image can be real or virtual.
If the object is farther than the focal point, the reflected rays actually meet, and the image is real. Then $d_i$ is positive.
If the object is placed between the mirror and the focal point, the reflected rays spread out, but their backward extensions meet behind the mirror. Then the image is virtual, and $d_i$ is negative.
This is exactly what the mirror equation predicts.
Example with a concave mirror
Suppose a concave mirror has focal length
$$
f = 10 \text{ cm}
$$
and the object is placed at
$$
d_o = 30 \text{ cm}
$$
Then
$$
\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}
= \frac{1}{10} - \frac{1}{30}
= \frac{3}{30} - \frac{1}{30}
= \frac{2}{30}
= \frac{1}{15}
$$
so
$$
d_i = 15 \text{ cm}
$$
The positive value means the image is real and forms in front of the mirror.
Convex mirror cases
For a convex mirror, $f < 0$. A convex mirror always produces a virtual image for a real object in front of the mirror. That means $d_i$ comes out negative.
Example with a convex mirror
Suppose a convex mirror has focal length
$$
f = -20 \text{ cm}
$$
and the object distance is
$$
d_o = 40 \text{ cm}
$$
Then
$$
\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}
= -\frac{1}{20} - \frac{1}{40}
= -\frac{2}{40} - \frac{1}{40}
= -\frac{3}{40}
$$
so
$$
d_i = -\frac{40}{3} \text{ cm} \approx -13.3 \text{ cm}
$$
The negative sign shows that the image is behind the mirror, so it is virtual.
Rearranging the mirror equation
Sometimes you know the image distance and need the object distance, or you know the object distance and need the focal length. The equation can be rearranged.
To solve for image distance,
$$
\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}
$$
To solve for object distance,
$$
\frac{1}{d_o} = \frac{1}{f} - \frac{1}{d_i}
$$
To solve for focal length,
$$
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
$$
You can then invert the result to get the required quantity.
Special cases
There are a few important limiting situations.
If the object is very far away, then $d_o \to \infty$. Since
$$
\frac{1}{d_o} \to 0
$$
the mirror equation becomes
$$
\frac{1}{f} = \frac{1}{d_i}
$$
so
$$
d_i = f
$$
This means light from a very distant object forms an image at the focal point.
If the object is at the focal point of a concave mirror, then $d_o = f$. The equation gives
$$
\frac{1}{d_i} = \frac{1}{f} - \frac{1}{f} = 0
$$
so
$$
d_i \to \infty
$$
The reflected rays are parallel, and no finite image distance exists.
Important special results:
$$
d_o \to \infty \Rightarrow d_i = f
$$
and for a concave mirror,
$$
d_o = f \Rightarrow d_i \to \infty
$$
Geometrical basis
The mirror equation comes from ray geometry for spherical mirrors. Using rays close to the principal axis and similar triangles, one can derive the relation between object distance, image distance, and focal length.
The result is an approximation that works well when rays are not too far from the principal axis. For rays far from the axis, spherical aberration appears, and the image is not perfectly sharp.
Practical use
In problems, the usual steps are simple. First identify the mirror type. Then assign the correct sign to $f$. Next write down $d_o$ with the proper sign. Then solve the mirror equation for the unknown quantity. Finally check whether the sign of $d_i$ makes physical sense.
| Mirror type | Sign of $f$ | Typical image for a real object |
|---|---|---|
| Concave | positive | real or virtual |
| Convex | negative | always virtual |
A positive image distance means the image forms in front of the mirror. A negative image distance means it forms behind the mirror.
Connection to image location
The mirror equation tells where the image is, but not by itself whether the image is upright or inverted, or larger or smaller. Those questions are handled by magnification, which is a separate topic. Still, the sign and size of $d_i$ already tell you whether the image is real or virtual.
Summary formula set
The mirror equation and its related radius formula are the main results of this chapter.
Main formulas for spherical mirrors:
$$
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
$$
$$
f = \frac{R}{2}
$$
With the common sign convention:
real image $\Rightarrow d_i > 0$, virtual image $\Rightarrow d_i < 0$
concave mirror $\Rightarrow f > 0$, convex mirror $\Rightarrow f < 0$
These formulas let you calculate the image position for spherical mirrors quickly and consistently.