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# Trigonometry Snell’s Law Help

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By McGraw-Hill Professional
Updated on Oct 3, 2011

## From Low To High

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. 10-4. Suppose 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 at point P, as shown. The ray is bent at the boundary whenever the ray does not lie along a normal line, assuming the indices of refraction, r and s, are different.

Suppose 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 point P on B, such that N is normal to B at P. Suppose R is a ray of light traveling through M r that strikes B at P. Let θ 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 ø 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 ø ≤ θ . (The two angles θ and ø are equal if and only if ray R strikes the boundary at an angle of incidence of 0°, that is, along line N normal to the boundary at point P. ) The following equation holds for angles θ and ø in this situation:

Fig. 10–4. Snell’s law governs the behavior of a ray of light as it strikes a boundary where the index of refraction increases.

sin ø /sin = ø = r/s

The equation can also be expressed like this:

s sin ø = sin r sin ø

## From High To Low

Refer to Fig. 10-5. 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, θ , and ø be defined as in the previous example. Then line N, ray R, and ray S all lie in the same plane, and θ ≤ ø . (The angles θ and ø are equal if and only if R is normal to B. ) Snell’s law holds in this case, just as in the situation described previously:

sin ø /sin ø = r/s

Fig. 10–5. Snell’s law for a light ray that strikes a boundary where the index of refraction decreases.

and

s sin ø = r sin θ

## Determining The Critical Angle

In the situation shown by Fig. 10-5, the light ray passes from a medium having a relatively higher index of refraction, r, into a medium having a relatively lower index, s. Therefore, s < r. As angle θ increases, angle ø approaches 90°, and ray S gets closer to the boundary plane B. When θ, the angle of incidence, gets large enough (somewhere between 0 and 90°), angle ø reaches 90°, and ray S lies exactly in plane B. If angle θ increases even more, ray R undergoes total internal reflection at the boundary plane B. Then the boundary acts like a mirror.

Fig. 10–5. Snell’s law for a light ray that strikes a boundary where the index of refraction decreases.

The critical angle is the largest angle of incidence that ray R can subtend, relative to the normal N, without being reflected internally. Let’s call this angle θ c . The measure of the critical angle is the arcsine of the ratio of the indices of refraction:

θ c = arcsin ( s/r )

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