Light is of fundamental theoretical and practical importance to us. Through it we are able to sense the extremely small and large, we are able to sense and analyze objects and phenomena at the greatest of distances, even through the vacuum of space. Our ability to use the phenomenon of light depends on the design of our bodies and the construction of devices that, to serve our ends, alter the paths of light rays.

This chapter will concentrate on the practical calculation of how light paths are reversed and bent by different surfaces.

The first principle of geometrical optics is that light, in undisturbed travel, moves in straight lines. In fact, for most purposes, the path of a light ray defines a straight line. Thus we draw light from a point source like the image to the right. Light in a beam is approximately parallel:

When light strikes a polished surface, it is reflected such that its "angle of incidence" (Ø_i) equals its "angle of reflection" (Ø_r).

1. Find the direction of the light ray after is completes its reflection off the second mirror:

This takes a little geometry: Since 18° and 100° are two angles of a triangle, the third must be 62°. Ør = 28° for the 2nd mirror

2. Find the angle of reflection a in terms of Ø and Ø:

a = 90 - [180 - (90 - Ø) - Ø] a = 90 - [90 + Ø - Ø] a = -Ø + Ø

The geometry starts to get sticky on the next problem.

3. A small mirror has the shape of a section of a sphere, 100 cm in radius. Light passes parallel to the axis of the mirror, 3 cm from the axis. After it reflects off the mirror, where does it cross the axis?

First, recognize that the radius line is everywhere normal to the spherical surface. This means that the incident ray and reflected ray make equal angles with the radius.

In general, for small curved mirrors, the focal length of the mirror will be 1/2 the radius of curvature of the mirror.

4. A point source is located at a distance s from a flat mirror. Prove that any ray reflected off the mirror will appear to have come from a source located at -s behind the mirror.

Consider one ray: Note that: Ø1 = Ø2 (alternate interior angles) Ø2 = Ø3 (incidence = reflection) Ø3 = Ø4 (corresponding angles) Since they share a common side, the two triangles are congruent. So s = x. Therefore all rays will appear to originate from the image. note: this explains why, when you look in a mirror, there seems to be a little world, exactly like your own, behind the mirror.

Here's a good question to ponder: Why is it that when you look into a mirror you see things reversed left-to-right, but not top-to-bottom? Shouldn't symmetry demand that you not only see your sides reversed, but your head and feet as well? (This is the kind of thing that would stump a philosopher for hours. If you have trouble with it, perhaps you should consider a career in philosophy.)

5. An object is 2 meters from a mirror. How far is it from its own image?

The object and its image must be 4 m apart: distance = 2 m * 2 distance = 4 m

6. Two mirrors are placed facing each other, separated by 3 meters. Claire deAiles sits 1 meter from one, facing it. How far away are her 1st and 2nd images?

The images are 2 m and 6 m away

7. Two mirrors are laced at right angles. Sketch the location of all the images formed by these two mirrors.

8. Locate the image of the object in the mirror when the object and mirror are positioned as shown.

9. Two mirrors are set at an acute angle as shown. Sketch the location of all the images.

The speed of light in a vacuum is 3.00 * 10⁸ m/sec, but when light passes into a transparent medium, it is slowed down. In glass it travels at 1.97 * 10⁸ m/sec; in water, 2.26 * 10⁸ m/sec; in diamond, 1.24 * 10⁸ m/sec.

We define a new property of transparent materials, the "index of refraction", to be the ratio of light speeds in a vacuum and in the material.

	speed of light in a vacuum
n =	------------------------------
	speed of light in a material

10. What is the index of refraction for glass?

n = vvacuum/vglass n = (3.00 * 108 m/sec)/(1.97 * 108 m/sec) n = 1.52 note: I hope you mentally canceled the 108s before calculating

11. What is the index of refraction for diamond?

n = vvacuum/vdiamond n = 3.00/1.24 n = 2.42

12. The index of refraction for quartz is 1.46. What is the speed of light in quartz?

n = vvacuum/vguartz vquartz = vvacuum/n vquartz = (3.00 * 108)/(1.46) vquartz = 2.05 * 108 m/sec

13. In a medium whose index of refraction is n, light travels with a velocity v. In a vacuum it travels with velocity c. Express v in terms of c and n.

n = c/v v = c/n