The graphical technique for locating the image of a convex mirror is shown in Figure . For convex mirrors, the image on the opposite side of the mirror is virtual, and the images on the same side of the mirror are real. Figure shows a virtual, upright, and smaller image. In comparison to the virtual image of the concave mirror, the virtual image of the convex mirror is still upright, but it is diminished (smaller) instead of enlarged and on the opposite side of the mirror instead of the same side. Again, the virtual image is formed by extending back the reflected diverging rays.
Refraction is the bending of light when the beam passes from one transparent medium into another. A transparent object allows the transmission of light, in contrast to an opaque object, which does not. Some of the light will also be reflected. The incident ray, reflected ray, normal, and refracted ray are shown in Figure 12.
The law of refraction.
When Willebrod Snell (1580–1626) observed light traveling from air into another transparent material, he found a constant ratio of the sines of the angles measured from the normal to the light ray in the material:
The constant ( n) is called the index of refraction and depends only upon the optical properties of the material. The index of refraction gives a measure of the amount of bending occurring when light travels from air into the material. It is a dimensionless number and can be located in tables of properties of materials. For example, the index of refraction of water is 1.33, and the index of refraction of crown glass varies from 1.50 to 1.62, depending upon the composition of the glass.
For the more general case of light traveling from medium 1 to medium 2, Snell's law can be written n 1 sin θ 1 = n 2 sin θ 2, where the subscripts 1 and 2 refer respectively to the angles and indices of the refraction for material 1 and material 2 respectively. A light ray traveling along the normal, with an incident angle of zero, will not be bent.
The index of refraction is also the ratio of the speed of light in a vacuum ( c) and the speed of light in that medium ( v); thus,
Consider the following problem involving both reflection and refraction. Imagine light entering an aquarium and reflecting off a mirror at the bottom. First, what will be the angle of refraction in the water if the angle in air is 30 degrees? Second, at what angle will the beam leave the water? See the setup in Figure 13.
A problem combining refraction and reflection.
Angle θ 2 is determined from θ 1, using Snell's law of refraction. Angle θ 2 = θ 2 by geometry, θ 3 = θ 4 by law of reflection, and θ 4 = θ 5 by geometry. θ 6 is related to θ 5 by Snell's law of refraction, in the same ration as θ 1 to θ 2. Therefore, θ 6—the angle of the ray leaving the water—must be 30 degrees. The problem is symmetrical.
A light ray passing through a rectangular block of transparent material will simply be displaced from its original path. For example, in passing from air to glass, the ray will bend toward the normal. Upon leaving the glass block, the ray will bend away from the normal so that the measured angles in the air on each side of the block are the same (Figure 14).
A light ray is displaced after passing through a refracting medium.
Light reflected from the surface of a material is partially polarized. A ray incident on a transparent surface at a certain angle will be partly refracted and partly reflected in a plane polarized ray. This angle of maximum plane polarization is called Brewster's angle, named for Sir David Brewster (1781–1868). The equation is tan θ = n, where n is the index of refraction of the reflecting surface.
When light travels from a material with a higher n to one with a lower n, at certain angles all of the light is reflected. This effect is called total internal reflection.
Example 1: Figure 15 illustrates ray 1 along the normal (no bending), rays 2 and 3 are refracted, and rays 5 and 6 are reflected. Ray 4 is intermediate between reflection and refraction with an angle of refraction of 90 degrees. The incident angle for this case is called the critical angle (θ). If the angle of incidence is less than θ, the light will refract, and if it is greater, the light will reflect.