Friday 8-2-96
The relevant section in the book is 22.2
A spherical mirror is basically a mirror that has been cut from a sphere. It has a center of curvature, C, which corresponds to the center of the sphere it was cut from, a radius of curvature, R, which corresponds to the radius of the sphere, and a focal point (the point where parallel light rays are focused to) which is located half the distance from the mirror to the center of curvature. The focal length, f, is therefore:
f=R / 2
Spherical mirrors are either concave (converging) mirrors or convex (diverging) mirrors, depending on which side of the spherical surface is reflective. If the inside surface is (the side towards the center of curvature), the mirror is concave; if the outside is reflective, it's a convex mirror.
Concave mirrors can form either real or virtual images, depending on where the object is relative to the focal point. A convex mirror can only form virtual images.
There are three steps to follow to dtermine where the image of an object is located, and to determine what kind of image it is (real or virtual, upright or inverted).
Step 1 - Draw a ray diagram. The more careful you are in constructing this, the better idea you'll have where the image is.
Step 2 - Apply the spherical mirror equation to determine the image distance, di. (Or find do, the object distance, or f, the focal length, depending on what is given.)
Step 3 - Make sure steps 1 and 2 agree.
A ray diagram is very helpful in determining where the image of an object is. Two rays are usually drawn for this, although a third is useful to add as a check. The first is known as the parallel ray; it is drawn from the tip of the object parallel to the optic axis (the line drawn through C, f, and the center of the mirror). A parallel ray reflects off the mirror and either passes through the focal point, or can be extended back to pass through the focal point.
The second ray is the chief ray. This is drawn from the tip of the object through the center of curvature, so it hits, and reflects, off the mirror at a 90 degree angle, coming back the way it came. Where the chief ray and parallel ray meet is the tip of the image.
A third ray, the focal ray, can be added as a check. This is a mirror image (reflected in the optic axis) of the parallel ray. The focal ray is drawn from the tip of the object through the focal point, reflecting off the mirror to follow a line parallel to the optic axis. All three rays should meet at the same point.
The spherical mirror equation relates the object distance (do), the image distance (di) and the focal length, f. Knowing two of those, you can determine the third, using:
1 / do + 1 / di = 1 / f
The magnification factor, M, is still the ratio of the image height to the object height. This ratio is equal to the image distance over the object distance, and, by convention, it has a negative sign out front:
M = - di / do
If you take the absolute value of this (i.e., drop the sign), a number less than one means the image is smaller than the object. A magnification factor greater than 1 means that the image is larger than the object.
Sign conventions: