## Summary of Mirrors and Lenses

We reiterate some of the important points made in this section of our work. Our
results are valid for spherical mirrors and lenses, for sources close to the optical
axis.

### Sign Conventions

Let Side A of an optical component be the side from which light starts, and let Side B be
the side to which light travels. With mirrors, Sides A and B are identical. If
*s* is the source distance, *i* is the image distance, *R* is the radius of
curvature, and *f* is the focal length, then, our sign conventions are as follows.

- The sign of
*s* is determined by Side A. If the source is on Side A, *s*
is positive; if it is on the side opposite to Side A, *s* is negative.
- The signs of
*i*, *R*, *f* are determined by Side B. For the image and
the focal point, their measurements are positive if they are on Side B, and negative if
they are on the side opposite to Side B. *R* is positive provided the centre of
curvature is on Side B; it will be negative if it is on the side opposite Side B.

### Image Position

For mirrors, the source distance *s*, image distance *i*, and focal length
*f* satisfy the equation

1/*s* + 1/*i* = 1/*f*
For lenses, they satisfy the equation

where *n*_{1} is the index of refraction for the region surrounding the lens,
and *n*_{2} is the index for the lens itself. *R*_{1}
and *R*_{2} are the radii of curvature for the first and second surfaces of the
lens respectively. In air, this equation reduces to the lens-makers' equation,

where *n* is the index of the lens.

### Focus

The focal length of a mirror is very nearly half of its radius,

*f* = *R*/2
For a thin lens, the focal length is found using the image-source equation, taking
*s* to approach infinity. In air, the symmetric foci of a lens are *f* units
away from the lens, where *f* satisfies

### Magnification

The magnification of a mirror or a lens is the ratio of the image distance to the
source distance,

*M* = -*i*/*s*
*M* is negative if the image is upside down, and positive if it is right side up.

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