## L-1 GEOMETRY OF RAYS
To study the laws reflection of refraction of light, and to use geometric methods for applying these laws.
Cork board and pins (or small electric lamps) Compass and straightedge.
The law of refraction has been known since ancient times. Ptolemy
(about 75 CE) described it in his book on optics. When a stick, say
a pencil, is partly submerged in a glass of water, the stick seems to
be bent. These early scientists did not believe that the stick was
We now know the law to be:
n where the angles θ
Lenses. By the 17th century scientists had found out how
to exploit refraction to make lenses, and knew how to
grind glass lenses with spherical surfaces. All of the
results which apply to lenses and lens combination
follow from the refraction law, Eq. 1. Only high
school mathematics is required to show the logic of
this, but it does get lengthy and involved. Rather than
do it all in detail, we will instead show you how to go
about it using simple geometry, in order to convince
you that you could do it if you had to. Once one
obtains the formula for a simple thin lens, the rest is very easy to do.
Consider a single spherically curved surface S separating two media of different refractive index. Our problem is to find the direction of the emergent ray, when given the direction of the incident ray. Or, given the emergent ray, we must find the incident ray. This type of problem may be solved with ruler and compass by following the steps outlined below.
_{1}
and n_{2}. Let a ray strike this surface at some
point, P. The refraction law holds even
though the surface is curved, with angles
measured to the normal (perpendicular) line
drawn at point P on the surface. In the
special case of a spherical surface, the
normal also happens to be collinear with a
radial line (passing through the sphere
center, C).
(1) Choose an arbitrary scale of lengths to
represent refractive index. Suppose we had n_{1} =
1 (air) and n_{2} = 1.5 (glass), then we might
choose a length of 3 cm to represent n1 and a
length of 4.5 cm to represent n_{2}. [In our reduced
size figures the lengths are different from these
values, but the proportions are the same. The
line PB is 1.5 times the length of the line AP.]
(2) Mark off the length AP representing n1
along the radial line. Likewise mark off the length PB representing n
_{1}sin(θ_{1})
(4) With a compass, draw an arc of a circle
of radius R centered on B. Draw the line
PE tangent to the circle. This is the
emergent ray, since it forms a leg of a right
triangle satisfying R = n
_{1} =
1.4 and n_{2} = 1. The line AP is 1.4 times as
long as the line PB.
Notice that in all of the constructions the direction of the surface curvature plays no role at all. The angle of refraction at a particular point is independent of the amount or sense of the curvature at that point.
n so, sin(θ So we see 2 can be 90° only when n
So, if you know the radius of curvature of a surface, the refractive indices, and the angle of the incident ray and where it strikes the surface; you can then always find the angle of the emergent ray. And if the emergent ray goes on to strike another surface, you could repeat the process, to find the emergent direction from that surface, and so on. It might become tedious, but the principles are straightforward. In fact, when high-speed computers are employed for lens design, this is exactly the mathematical method used. Each curved surface bends (deviates) the ray. Fig. 10 shows the refraction at the surfaces of
a compound lens. After passing through the first two surfaces the total angle of deviation, δ, is given by δ = α + β.
(1) Use the geometric method to find how each of these three rays in Fig. 9 is bent at the air-to-glass surface. Take the index of refraction of the glass to be 1.5. You may do this directly on this page, or on a copy or tracing of the diagram.
(2) Use geometric construction and the refraction law to find the critical angle for a ray passing from glass (n = 1.5) to air (n = 1).
An assortment of lens and mirror shapes are available (specimens), made of glass or plastic. A corkboard and pins may be used in the following manner to trace the paths of light. Lay a sheet of plain paper on the corkboard. Place the specimen (mirror or lens) on the paper, and trace its outline. Do not move it thereafter. Put your eye on a level with the specimen and look at the light emerging from it. Place a pin somewhere so that light from the pin goes to the specimen and then to your eye. You will see the pin's image. We'll call this PIN 1, and consider it the object. All other pins will be tools for tracing rays from PIN 1. Place two other pins between your eye and the specimen so they line up with that image exactly. Place another pin between the first pin and the specimen, so it looks in line with the others. This last pin, along with PIN 1, establish the direction of the incoming light ray. Now remove all pins except PIN 1, and use a straightedge to trace the path if the incident and emergent ray, using the pinholes as guides. It is not enough to draw one ray. To locate the positions of objects and image requires at least two different rays. So now move your eye to a new position and trace a different path from the original PIN 1. Do at least three or four rays in each case, spread out over the range of possible incident angles to the specimen. Rays from PIN 1 all diverge from PIN 1. These rays may emerge from the specimen convergent, or divergent, depending on the nature of the refracting or reflecting surfaces of the specimen. If the emergent rays converge to a common point, we call that point a If the emergent rays emerge divergent, they can be extended backwards (use a different color, or dotted lines) to locate their common point of divergence. In this case we call that apparent divergence point a Try to simulate common cases described in your textbook. (1) Flat mirror. Emergent rays divergent. Image is virtual. Where is it? (2) Glass with two parallel surfaces. Object pin on one surface. Image is within the glass. Where is it? This simulates the situation of looking at a penny on the bottom of a beaker of water. The penny seems elevated above the table. Do this and find out how great the elevation is. Use this information to determine the index of refraction of water. (3) Converging mirror. (4) Diverging mirror. (5) Converging lens. (6) Diverging lens. (7) Right angle prism. (8) Equilateral prism. (9) Spherical lens, simulated by round disk of glass. You can simulate the passage of light through a drop of water. Look for the spreading (dispersion) of white light into colors, which is the reason we see rainbows. (10) Use the geometric principles described in the theory section to determine the index of refraction of each of the transparent specimens.
Summarize your observations and analysis in an organized manner and with neat diagrams, as a short essay on Text, photo and line drawings, © 1996, 2004 by Donald E. Simanek. 1. In math, "normal" means a "perpendicular line." 2. The symbols are Greek letters: d (delta), b (beta) and g (gamma). |