CHAPTER 26 THE REFRACTION OF LIGHT
The Index of Refraction
Light travels fastest through a vacuum. When light enters another medium such as air, water, or glass, interactions with the electric and magnetic fields of matter causes the speed of light to be reduced. The ratio of the speed of light in a vacuum to the speed of light in a medium is called the index of refraction of the medium. n = c/v
The larger the index of refraction, the greater the reduction in the speed of light in the medium.
Example
The index of refraction for diamond is 2.419. Find the speed of light in a diamond.
Snell's Law
When light strikes the interface between two transparent materials such as air and water, part of the light is reflected obeying the law of reflection and part is transmitted. If the angle of incidence is not zero or 90 degrees, and the index of refraction of the two materials are different, the transmitted light will change directions. This is called refraction.

When light travels from a medium with a lower index of refraction into a medium with a higher index of refraction, it bends towards the normal as in the top diagram.
When light travels from a medium with a higher index of refraction into a medium with a lower index of refraction, it bends away from the normal as in the bottom diagram.
Refraction and Total Internal Reflection
Snell's Law of Refraction relates the angle of incidence, angle of refraction, and the index of refraction of both media. n
1 sinθ
1
= n
2 sinθ
2
Example

A light ray strikes an air to water surface at an angle of 46° with respect to the normal. The index of refraction for water is
1.33. Find the angle of refraction when the ray goes from (a) air to water, (b) water to air.
Refraction causes an apparent change in depth of an underwater object viewed from above the surface of the water.
The object appears nearer the surface than it actually is. The equation used to calculate apparent depth when the object is viewed from directly above is: d' = d(n
2
/n
1
) n
2
is the index of refraction of the medium above the surface and n
1
is the index of refraction of the medium below the surface.
Example

A coin on the bottom of a swimming pool 3.0 meters deep is seen by a swimmer out of the water directly above it. How deep does the coin appear to be?(n = 1.33)
When light passes through a transparent material with parallel sides, it is refracted twice so that a ray will continue in the same direction it originally followed. The ray will be displaced, however, so its path will be parallel to its original path.
The magnitude of the displacement depends on the index of refraction and thickness of the material as well as the angle of incidence.
Example
A ray of light strikes a glass pane(n = 1.52) with an angle of incidence of 30°. If the pane is 6.00 mm thick, find the displacement of the ray in the direction perpendicular to the incident ray.
The reason that light refracts can lead to a derivation of Snell's law. The following diagram shows why light refracts.

The part of the wave front that enters the new medium first slows down before the rest of the wavefront. This causes the change in direction called refraction. If the wavefronts are leaving the medium where they move more slowly, the reverse effect happens. Note that refraction does not occur when the angle of incidenc is 0 or 90 degrees.
Total Internal Reflection
When light travels from a medium where it moves more slowly to one where it moves faster, the angle of refraction is larger than the angle of incidence and may be found using Snell's law.
If the angle of incidence is large enough, the angle of refraction equals or exceeds 90 degrees and total internal reflection occurs.
The angle of incidence that yields an angle of refraction of 90 degrees is called the critical angle. sinθ c
= n
2
(sin 90°)/ n
1 sinθ c
= n
2
/ n
1

where n
1
> n
2
.
In diagram (a), θ
1
< θ c
, in diagram (b), θ
1
= θ c
, and in diagram
(c), θ
1
> θ c
.
Example
In the drawing, a laser beam is fired into a crown glass slab so that the angle of incidence at point A is 60°. The beam, if reflected will strike points B and C. If the glass is surrounded by air, find the first point(A,B, or C) where part of the beam will exit the glass.
The sparkle of a diamond results from total internal reflection.
If the diamond is cut so that light entering the top and striking the bottom exceeds the critical angle, it will all be reflected

back up through the top, increasing the diamond's brilliance.
Fire is the result of dispersion of white light producing colors.
Internal reflection also plays an impotant role in the health field in the use of various scopes that can sometimes be used in the place of exploratory surgery. Endoscopes, colonscopes, and bronchoscopes are some.

This is a picture from a colonscope showing a polyp(red area on left).
The Polarization of Light
Earlier we discussed what the polarization of light is and how reflected light can be plane polarized. The diagram below illustrates how this happens.
Light reflected from a horizontal, non-metallic surface is partially plane-polarized. There is an angle of incidence, called

the Brewster angle, at which the reflected ray is completely polarized in the horizontal direction and the refracted ray is partly polarized in the vertical direction. The Brewster angle can be determined by:
Tan θ
B
= n
2
/n
1
Note that θ
B
is always greater than 45° since n
2
is always greater than n
1
.
The Dispersion of Light
The index of refraction is slightly different for different frequencies of light. It is smallest on the red end of the spectrum and largest on the violet end. In the case of crown glass it varies from 1.520 for red light to 1.538 for violet light. This small difference is enough to refract different colors in different directions and is called dispersion.

Red light does not refract as much as violet because it does not slow down or speed up as much when it enters a new medium.
The Dispersion of Light
Rainbows are formed when white light from the Sun is dispersed in water droplets. We see different colors from different sets of water droplets.

Lenses
A transparent material that refracts light can be formed into a lens by curving either the front surface, the back surface, or both. A lense can be either converging or diverging depending on the curvature of its surfaces.

Notice that the focal length for a converging lens is positive and the focal length for a diverging lens is negative due to its location.
There are six types of lenses with which we should be familiar.
In all six cases, light from the object enters the lens from the left. Converging lenses are thicker at the center and diverging lenses are thinner at the center.
The Formation of Images by Lenses
Ray diagrams can be used to determine image formation by a thin lens in a fashion similar to that used with mirrors.

The difference is that two focal points are drawn, one on each side of the lens. A ray passing through the center of the lens is not refracted.
If the object is placed closer to the lens than the focal length, an upright, virtual, enlarged image is formed.
The Thin Lens Equation and the Magnification Equation
The lens equation and the magnification equation for lenses are the same as those for mirrors. There are some changes in the distance definitions.
1/f = 1/d
0
+ 1/d i

m = h i
/h
0
= -d i
/d
0
In these equations, f is focal length, d
0
is object distance, d i
is image distance, h
0
is object height, h i
is image height, and m is magnification. Note that magnification is negative for real images due to the real image being inverted.
Remember that f is positive for a converging lens and negative for a diverging lens.
Example
The distance between an object and its image formed by a diverging lens is 49.0 cm. The focal length of the lens is -233.0 cm. Find the image distance and the object distance.
Since we have a diverging lens, the image distance is negative so d
0
- d i
= 49.0 cm becomes d
0
+ d i
= 49.0 cm.
Diverging Lenses
Lenses in Combination

A microscope uses two lenses to form an image. The first image is a real, inverted, enlarged image formed by the objective. To obtain this image, the object is placed just outside the focal length of the objective.
The first image becomes the object for the eyepiece. This object is placed just inside the focal length of the eyepiece and produces an enlarged, virtual, final image that the observer sees.
The image is upside down since it was inverted by the formation of the first real image and not reinverted when the vurtual image was formed by the eyepiece.

The equation used to calculate the magnification of a microscope comes from the product of the magnification of the objective and that of the eyepiece.
The magnification of the objective is the distance from the objective to the focal point of the eyepiece divided by the focal length of the objective.
The magnification of the eyepiece is the distance to the near point(most comfortable viewing distance for most people) divided by the focal length of the eyepiece.
These combine to form the equation:
M = (L - f e
)N/f o f e
The distance L - f e
is the distance between the objective and the eyepiece minus the focal length of the eyepiece.
N is 25 cm for most people.
Example
A compound microscope has a barrel whose length is
16.0 cm and an eyepiece whose focal length is 1.4 cm.
The viewer's near point is at 25 cm from his eyes. What focal length must the objective have to produce a -320x magnification?
The Human Eye

For clear vision, the lens in the eye must form an in focus real image on the retina. In order to do that the lens must be able to change shape, therefore changing its focal length to accommodate different object distances.
When a person is nearsighted(myopic), the eyeball is too long for the maximum focal length the lens can attain.
The focused image of a distant object falls in front of the retina and a blurred image is seen. This condition is corrected by wearing a diverging lens as eyeglasses or contacts.
When a person is farsighted(hyperopic), the eyeball is too short for the minimum focal length the lens can attain. The focused image of a near object falls behind the retina and a blurred image is seen. This condition is corrected by wearing a converging lens.
As we age we all fall into this second category due to loss of flexibility of the lens and must wear reading glasses or bifocal glasses to read.

Other causes of blurred images formed by a lens are spherical aberration and chromatic aberration.
Spherical aberration occurs when the lens is large in diameter relative to its focal length. Rays too far from the principal axis are not refracted through the focal point and cause the image produced to be blurred. This is reduced by reducing the diameter of the area of the lens refracting light.
Chromatic aberration causes blurred images because the index of refraction is not the same for different frequencies. This effect produces multiple images of different colors. Chromatic aberration is greatly reduced by combining a converging lens and a diverging lens made of different kinds of glass(differenr n). This is called an achromatic lens and is found in high quality cameras.

P 811 Questions 2, 3, 4, 5, 6, 9, 12, 13, 15, 16, 19, 20, 22,
24, 26. 28, 32
P 812 Problems 3, 9, 13, 17, 23, 27, 31, 35, 39, 41, 44, 45,
47, 48, 49, 50, 52, 54, 55, 57, 59, 85