Refraction

Refraction is the bending of a wave when it undergoes a change in medium. This happens due to a change in velocity of the wave in the medium, and the greater the change in velocity, the more refraction will occur. Fill a glass of water half full (or half empty if you are having a bad day) and place a pencil in it. The apparent bending of the pencil at the surface of the water is due to refraction of light.

Recall that when a wave passes from one medium to another, the frequency of the wave remains constant. Since the speed of the wave is defined as the wavelength times the frequency, or v = lf, when the velocity changes, the wavelength of the wave must also change. A term associated with refraction is the index of refraction. The index of refraction, n, is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in the substance (v). We write this as

The index of refraction will always be greater than one since light will have a slower speed in a medium other than a vacuum. To a good approximation, we consider the index of refraction in air to also be 1.00. Other indices of refraction for various substances include:

Material	Index of refraction (for light of wavelength 589 nm)
Air (at STP)	1.0003
Water	1.33
Crown Glass	1.52
Diamond	2.42

One thing to note is that wavelength affects the index of refraction, and different wavelengths of light have different indices of refraction. This is why light bends in a prism into the spectrum we see when it emerges.

We quantify the amount of bending using Snell's Law. Named for Willebrord Snell, a Dutch mathematician, he performed experiments in 1621 to find relationships between different media when light passed through it. He found that the index of refraction times the sine of the angle of the light to a perpendicular to the surface was a constant as it passed through a boundary. We write this as

where n_i is the index of refraction for the first medium, q_i is the incident angle as measured to the perpendicular, n_r is the index of refraction of the second medium, and q_r is the angle or refraction, the angle between the perpendicular and the refracted ray.

A special case occurs in a phenomenon called total internal reflection. This is what causes diamonds to sparkle and allows fiber optic cables to efficiently pass data information. In total internal reflection, the angle of refraction is 90^o and the light just skims the surface rather than passing through the boundary. It is reflected back. For this to happen, the light must be passing from a medium with a higher index of refraction to a medium with a lower index of refraction. In our example below, n₂ is our medium with the higher index of refraction, and n₁ is the medium with the lower index of refraction. Our incident angle will be in n₁. The minimum angle of incidence (q_i )
that allows this to happen is called the critical angle, q_c and can be computed as

Why?

In ray A, the angle of incidence is less than the critical angle. This means that n₂/n₁ sinq_r is less than 1, so q_i will
be less than 90^o. (Remember, the sine of an angle can never be greater than 1). Light will be refracted through the boundary. In case B, the refracted angle is 90^o, and sin q_i equals n₂/n₁. In case C, the angle of incidence is greater than the critical angle. For angles of incidence greater than q_c. Snell’s equation would make it so that sinq_r is greater than 1, which can’t happen. Therefore Snell’s law of refraction doesn’t occur in this case, and all the light is internally reflected.

The NTNU Virtual Physics Laboratory provides several excellent applets that demonstrate principles of Physics. Click Here for an applet you can run from the NTNU Virtual Physics Laboratory that will help with your understanding of refraction of a wave of light.

Another applet shows refraction in a little more detail. See the world as viewed by a fish.

Additionally, check out http://www.physicsclassroom.com/Class/refrn/ from The Physics Classroom for more information on refraction and the ray model of light. These pages rock!

For Practice Problems, Try:

Snell's Law from DCPhysics

Critical Angle from DCPhysics