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# Snell's Law Help

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McGraw-Hill Professional
Updated on Sep 12, 2011

## Light Rays At A Boundary

A qualitative example of refraction is shown in Fig. 19-2, where the refractive index of the first (lower) medium is higher than that of the second (upper) medium. A ray striking the boundary at a right angle (that is, angle of incidence equal to 0°) passes through without changing direction. However, a ray that hits at some other angle is bent; the greater the angle of incidence, the sharper is the turn the beam takes. When the angle of incidence reaches a certain critical angle , then the light ray is not refracted at the boundary but instead is reflected back into the first medium. This is known as total internal reflection.

Fig. 19-2 . Rays of light are bent more or less as they cross a boundary between media having different properties.

A ray originating in the second (upper) medium and striking the boundary at a grazing angle is bent downward. This causes distortion of landscape images when viewed from underwater. You have seen this effect if you are a scuba diver. The sky, trees, hills, buildings, people, and everything else can be seen within a circle of light that distorts the scene like a wide-angle lens.

If the refracting boundary is not flat, the principle shown by Fig. 19-2 still applies for each ray of light striking the boundary at any specific point. The refraction is considered with respect to a flat plane passing through the point and tangent to the boundary at that point. When many parallel rays of light strike a curved or irregular refractive boundary at many different points, each ray obeys the same principle individually.

## Snell’s Law

When a ray of light encounters a boundary between two substances having different indices (or indexes) of refraction, the extent to which the ray is bent can be determined according to an equation called Snell’s law .

Look at Fig. 19-3. Suppose that B is a flat boundary between two media M r and M s , whose indices of refraction are r and s , respectively. Imagine a ray of light crossing the boundary as shown. The ray is bent at the boundary whenever the ray does not strike at a right angle, assuming that the indices of refraction r and s are different.

Fig. 19-3 . A ray passing from a medium with a relatively lower refractive index to a medium with a relatively higher refractive index.

Suppose that r < s ; that is, the light passes from a medium having a relatively lower refractive index to a medium having a relatively higher refractive index. Let N be a line passing through some point P on B such that N is normal to B at P . Suppose that R is a ray of light traveling through M r that strikes B at P . Let x be the angle that R subtends relative to N at P . Let S be the ray of light that emerges from P into M s . Let y be the angle that S subtends relative to N at P . Then line N , ray R , and ray S all lie in the same plane, and y < x . The two angles x and y are equal if, but only if, ray R strikes the boundary at an angle of incidence of 0°. The following equation holds for angles x and y in this situation:

sin y /sin x = r / s

This equation also can be expressed like this:

s sin y = r sin x

Now look at Fig. 19-4. Again, let B be a flat boundary between two media M r and M s whose absolute indices of refraction are r and s , respectively. In this case, imagine that r > s ; that is, the ray passes from a medium having a relatively higher refractive index to a medium having a relatively lower refractive index. Let N, B, P, R, S, x , and y be defined as in the preceding example. As before, x = y if, but only if, ray R strikes B at an angle of incidence of 0°. Then line N , ray R , and ray S all lie in the same plane, and x < y . Snell’s law holds in this case, just as in the situation described previously:

Fig. 19-4 . A ray passing from a medium with a relatively higher refractive index to a medium with a relatively lower refractive index.

sin y /sin x = r/s and s sin y = r sin x

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