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# Snell's Law for the Refraction of Light

by Ron Kurtus (revised 8 September 2005)

When visible light enters a transparent material such as glass at an angle, the direction of the light is refracted or bent at a different angle.

The angle is determined by the initial angle and the index of refraction of the two materials. Snell's Law is an equation that determines the angle at which a ray or beam of light is refracted.

When the light passes from a material of high index of refraction to low index of refraction, there is an angle at which the light starts to be reflected at the interface of the materials. This is called the critical angle for refraction.

Questions you may have include:

• What is Snell's Law?
• How do you calculate the refracted angle?
• What is the critical angle for refraction?

This lesson will answer those questions. Useful tool: Units Conversion

## Snell's Law

Snell's Law determines the angle at which a beam of light bends, according to the initial angle and the indexes of refraction of the two materials.

### Light going from one material to another

If A is the index of refraction of the first material and a is the angle of the incoming ray or beam of light with respect to the perpendicular or normal to the surface, then b will be the angle to the normal of the ray in the second material where B is its index of refraction. Light is bent going from first material to second

The Index of Refection for each material equals the speed of light in a vacuum (c) divided by the speed of light in the material. Thus

A = c/cA

and

B = c/cB

where

• A is the index of refraction of material A
• B is the index of refraction of material A

Since index B is greater than index A, the speed of light in material B is less than the speed in material A. Thus, according to Snell's Law, the angle b is less than the angle a.

### Relationship

Snell's Law is written as:

A*sin(a) = B*sin(b)

where:

• sin(a) is the sine of angle a
• sin(b) is the sine of angle b

## Calculating values

Typically, you want to find angle b or how much the light will be bend in the second material. Using some Algebra, Snell's Law can be rewritten as:

sin(b) = A*sin(a)/B

or

b = arcsin[A*sin(a)/B]

where arcsin[A*sin(a)/B] is the arcsine or angle whose sine is A*sin(a)/B.

Thus if the first material is air (index approximately = 1), the incoming angle is 30o and the second material is glass with an index = 1.5, you can calculate the angle of the light in glass.

sin(b) = A*sin(a)/B

sin(b) = 1*sin(30o)/ 1.5

sin(b) = 0.5/ 1.5 = 0.33

b = arcsin(0.33) = 19.5o

Thus, if light hits glass at 30o, its angle will change to 19.5o within the glass. Of course, if it is a glass plate, you will then use the opposite to obtain the exit angle of 30o back into air.

## Critical angle for refraction

An interesting thing happens when light is going from a material with higher index of refraction to a lower index, such as going from water to air. There is an angle at which the light will not pass into the other material and will start to be reflected at the surface. This is call the critical angle of refraction. Light going from high index to low

In the figure above, you can see that angle b is larger than angle a when index A is larger than index B. At some angle a, angle b will equal 90 degrees. Light at critical angle for refraction

### Calculating critical angle

Using Snell's Law, we can calculate this critical angle. Let the first material be water with index of refraction = 1.33, and let the second material be air with index = 1.

We want to find angle a:

sin(a) = B*sin(b)/A

sin(a) = 1*sin(90o)/1.33

since sin(90o) = 1,

sin(a) = 1/1.33 = 0.75

a = arcsin(0.75) = 48.6o

This is the critical angle. When angle a is greater than that angle, the light will be reflected.

### Internal reflection

At angles greater than the critical angle for refraction, the light demonstrates an internal reflection. Internal reflection of light

When a is greater than the critical angle for refraction, the light is reflected off the interface of the two materials just like a mirror. (a > means "a greater than".)

The Law of Reflection holds where the incident angle equals the reflected angle. Thus b = 180o - a.

You can see this effect by filling a glass with water and observing it from below the water line. Internal reflection in glass of water
(neglecting refraction from the glass)

## Summary

Snell's Law states that when visible light enters a transparent material at an angle, the direction of the light is refracted or bent at a different angle. Snell's Law is an equation that determines the angle at which a ray or beam of light is refracted. When the light passes from a material of high index of refraction to low index of refraction, there is an angle at which the light starts to be reflected at the interface of the materials. This is called the critical angle for refraction.

## Resources and references

Ron Kurtus' Credentials

### Websites

Refraction of Light - From the University of Missouri

Electromagnetic Waves Resources

Physics Resources

### Books

Schaum's Outline of Optics by Eugene Hecht; McGraw-Hill (1974) \$16.95

Introduction to Modern Optics by Grant R. Fowles; Dover Publications (1989) \$16.95

Optics by Eugene Hecht; Addison Wesley (2001) \$108.00 - Textbook covers wave motion, electromagnetic theory, propagation of light, geometrical optics, superimposition of waves, polarization, interference, diffraction, fourier optics and lasers

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

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