Refraction and Snell's law

​​In a nutshell

Refraction is when a wave changes direction at a boundary due to a change in speed. The refractive index of a material is a measure of how much it slows down light travelling through it. Snell's law relates the angle of incidence and refraction with the refractive indexes of the two materials.

Equations

Constants

Variable definitions

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Refractive index

A wave changes speed when it enters a different medium. This causes a change in the direction that the wave travels in.

The frequency of a wave does not change during refraction. To keep the wave equation balanced as the wave changes speed, the wavelength has to also change.

The refractive index of a material measures how much the material slows down light travelling through it. It is calculated by finding the ratio between the speed of light in a vacuum, $$c$$, and the speed of light in the material, $$v$$. 

​$$n = \dfrac{c}{v}$$​​

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Example

The refractive index of air is approximately 1. 

The refractive index relates to how optically dense a medium is. A medium with a higher optical density has a higher refractive index. Light travels slower through an optically more dense material.

Snell's law

Snell's law states that the ratio of the sines of the angle of incidence and refraction is constant for light passing through a given pair of media.

$$n\sin{\theta} = constant$$​​

This isn't a very helpful equation on its own, however if light passes through two media of refractive indices $$n_1$$ and $$n_2$$ with an angle of incidence $$\theta_1$$ and an angle of refraction $$\theta_2$$, the equations can be equated to form a much more useful one:​

​$$n_{1}\sin{\theta_{1}} = n_{2}\sin{\theta_{2}}$$​​

Note: This equation is more useful for solving problems involving refraction at a boundary and will be used in the rest of the summaries and exercises. 

​$$\theta_{1}$$​​ Angle of incidence ​$$\theta_{2}$$​​ Angle of refraction ​$$n_1$$​​ Refractive index of first material ​$$n_2$$​​ Refractive index of second material

Snell's law can be used to determine that when light enters an optically more dense material, the wave slows down and it bends towards the normal. When light enters an optically less dense material, the wave speeds up and it bends away from the normal. 

Tip: Most of the time the angles will need to be determined, but if you need to remember how the ray bends use the mnemonic FAST. Faster Away, Slower Towards.

Example

A light ray goes from air to water. The refractive index of water is $$1.33$$. The light enters the water at an angle of $$35 \degree$$with respect to the normal. What is the angle of refraction?

First, write out the quantities given and check they are in the correct form:

$$n_1 = 1\newline n_2 = 1.33\newline \theta_1 = 35\degree$$​​

Then, write down the equation you need to use:

$$\sin{\theta_2} = \dfrac{n_1}{n_2}\sin{\theta_1}$$​​

Next, substitute the values into the equation:

$$\sin{\theta_2} = \dfrac{1}{1.33}\sin{35} = 0.4312...\newline \ \newline\theta_2 = \sin^{-1} ({0.4312...}) = 25.54...$$​​

Make sure to include units and round the the lowest number of significant figures of the values given in the question:

$$angle\space of \space refraction = 26\degree$$​​

The angle of refraction is $$\underline{26\degree}$$.