How to Calculate Refraction

Written by arlene larabee
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How to Calculate Refraction
Rainbows are caused by light refracting through raindrops. (Arcobaleno image by Eagle from

Light travels in a straight line in a vacuum. Refraction is the bending of the light path that occurs at the interface (boundary) between two mediums, as the speed of the light changes. (The Resources list includes a link to a standard refraction diagram.) The incident and refractive angles of the light path are measured with respect to the normal, a line drawn perpendicular to the boundary. If light slows down (or speeds up) at the boundary, the path bends toward (or away) from the normal.

Snell's law states that n1 sine(theta1) = n2 sine(theta2). Theta1 and theta 2 are the incident and refractive angles of the light ray with respect to the normal, while n1 is the refractive index for the incident medium and n2 is the index of the refractive medium. The index of refraction, n, is the ratio of the speed of light in a vacuum to the speed of light in a given medium. The slower the light travels in a medium, the larger its value of n.

You solve for the refractive angle by rearranging the equation into the form sine(theta2) = (n1/n2) sine(theta1).

Skill level:

Things you need

  • Index of refraction table
  • Scientific calculator

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  1. 1

    Look up and record the index of refraction (n1) for the incident medium and (n2) for the refractive medium. Record the angle (theta1) of the incoming light ray with respect to the normal.

    As an example, a light ray travelling in water hits a water/glass interface. The angle of incidence of the light ray on the glass boundary is 20 degrees. You need to find the angle of refraction of the ray as it enters the glass. Record the following information: n1 = n(water) = 1.33, n2 = n(glass)= 1.55 and theta 1 = -6.67 degrees C.

  2. 2

    Calculate sine(theta2) = (n1/n2) sine(theta1)

    Using the example, the equation would be sine(theta2) = (1.33/1.55) sine 20 = (1.33/1.55)(0.3420) = 0.2935. Calculate the value of sine 20 using the sin function key on a scientific calculator.

  3. 3

    Calculate theta 2 = arcsin(theta2) = inverse of sine(theta2) if sine(theta2) is less than or equal to 1. The arcsin (or inverse sine) function computes the angle theta 2 that corresponds to the calculated value of sine(theta2).

    For the example, theta 2 = arcsin[sin(theta2)] = arcsin(0.2935) = 17.1 degrees. The light ray bent toward the normal because light slowed down as it entered the glass.

  4. 4

    Note and remember that theta 2 does not exist if sine(theta2) is larger than 1. The light is unable to refract and simply reflects off the boundary.

    This can only occur when the light speeds up at the boundary. For this situation, if the angle of incidence is greater than the critical angle C (sin C = n2/n1, n1 >n2), the light is trapped in the medium and can only reflect at the boundary.

Tips and warnings

  • Note that the index of refraction for air is 1.00029. Use the value 1.00 for air unless higher precision is needed.
  • Rainbows occur because white light refracts twice as it passes through a raindrop, once as it enters the drop and once as it leaves. The different colours making up white light have slightly different angles of refraction. When the light leaves the raindrops, the different colours are on slightly different paths and spread out into the rainbow pattern.
  • Check that your calculator is using degree units for the angles. If your calculator uses radian units, you can convert from degrees to radians by multiplying the angle in degrees by 0.0174533. If your calculator gives sin 90 =1, it is set to degrees.
  • Your scientific calculator should have an arcsin or inverse sin function. There are several ways in which scientific calculators execute or label the inverse sin function. Check your user manual for instructions if necessary.

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