Spherical Mirrors

"A modern compter hovers between the obsolescent and the non existent"
Sydney Brenner

There are two kinds of spherical mirrors, concave and convex.
The focal point (F) of a concave mirror is the point at which a parallel beam of light is "focussed" after reflection in the mirror. For a convex mirror the focal point is the point from which light appears to have originated after reflection from the mirror. The centre of curvature (C) is the centre of the circle (sphere) of which the mirror is an arc.
The focal length (f) and radius of curvature (R) are defined in the diagram at the right. It can be shown that R = 2f. "A" in the diagram is known as the "vertex" (often labeled V).

Image formation in spherical mirrors is defined by certain "characteristic" rays whose behaviour is governed by the law of reflection.

Rays parallel to the principal axis are reflected through the focal point - concave (or as if they came from the focal point - convex ).
Rays passing through the focal point are reflected parallel to the principal axis - concave (for convex mirrors a ray that would have passed through the focal point is reflected parallel to the axis).
Rays passing through the centre of curvature are reflected back along their original path -concave (or a ray which would have passed through the centre of curvature is reflected back along itself - convex )

Concave mirror. In the animation the first two rays from the object are examples of the first two characteristic rays described above. Only two rays are needed to define the position of the image. The paths of the other rays in the animation are defined since the image position is already known.
The general characteristics of the image depend on the location of the object with respect to the centre of curvature and the focal point. Animations of image formation in a concave mirror for the five possible object positions can be observed by choosing from the following options.
Convex mirror. The general characteristics of images in convex mirrors are independent of the location of the object. Three examples are shown below.

Note that for the convex mirror the reflected rays DIVERGE (this is also the case for the concave mirror when the object is closer than the focal point to the mirror). In these cases the image formed is virtual - light rays do not pass though it, but to an observer " appear" to come from it.

Perhaps the most common everyday experience of a convex mirror is the passenger side rear-view mirror in your car. As can be seen from the above ray diagrams, the image you see in a convex mirror is always smaller than the object. You "know" the typical size of a car or truck, so much so that the "size" it appears tells you how far away it is (if you see a tiny car in the mirror you "know" it is far away). Thus, seeing a vehicle "looking smaller" due to the convex mirror, will make you think the vehicle is further away from the mirror than it actually is, hence the warning, "objects mirror are closer than they appear".

Mirror equation. For objects placed close to the principle axis the distance of the object from the vertex of the mirror (p), the distance of the image from the vertex of the mirror (q) and the focal length (f) are related by the following equation,
Magnification. The magnification (m) is defined by,

Sign Conventions. In order to make use of the above formulae the following sign conventions must be followed.

SIGN	+	-
f - focal length	Concave	Convex
p - object distance	Real	Virtual
q - image distance	Real	Virtual
m - magnification	Upright image	Inverted image

Image properties. There are four basic properties, dependent on the position of the object, as indicated in the table below. These properties can be verified either graphically or by using the mirror equation and the definition of magnification.

Mirror	Object location	Image location	Type	Orientation	Relative size
CONCAVE	At infinity	At F	Real	Inverted	Smaller
CONCAVE	Beyond C	Between F and C	Real	Inverted	Smaller
CONCAVE	At C	At C	Real	Inverted	Same size
CONCAVE	Between C and F	Beyond C	Real	Inverted	Larger
CONCAVE	At F	At infinity	No image	No image	No image
CONCAVE	Closer than F	Behind the mirror	Virtual	Upright	Larger

CONVEX	Anywhere	Behind the mirror	Virtual	Upright	Smaller

What's a light-year?
One-third less calories than a regular year.
(Very Punny)

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu