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# Refraction

## Refraction of light

When light travels from one material into another it changes direction. This bending is called refraction.

A pencil that is partly immersed in water appears broken at the water surface. The light reflected by the submerged part of the pencil changes direction as it leaves the water and enters the air.

The pencil appears broken because of refraction.

Light does not change direction if it hits the surface at a right angle (along the normal).

In reflection light stays inside the same material. In refraction light enters a different material.

## Refractive index

The refractive index $n$ of a material is the ratio of the speed of light in vacuum $\Tblue{c} = \Tblue{3\times 10^8\umps}$ and the speed of light $\Tred{v}$ in the material.

\begin{align*} \text{Refractive index}=&\frac{\Tblue{\text{Speed of light in vacuum}}}{\Tred{\text{Speed of light in material}}} \\ n =& \frac{\Tblue{c}}{\Tred{v}} \end{align*}

The refractive index is also called the index of refraction. It has no unit.

The speed of light in air is almost exactly the same as the speed of light in vacuum. The refractive index of air is $1.0003$.

The speed of light in water is $\Tred{2.26\times 10^8\umps}$. The refractive index is $\displaystyle\frac{\Tblue{3 \times 10^8 \umps}}{\Tred{2.26\times 10^8 \umps}}= 1.33$

Material Refractive index Speed of light (in m/s)
Vacuum $1.00$ $3.00\times 10^8$
Water (liquid) $1.33$ $2.26\times 10^8$
Glass (typical) $1.50$ $2.00\times 10^8$
Diamond $2.42$ $1.24\times 10^8$
Gemstones with a higher index sparkle more. Diamond has a refractive index of 2.4.

## Incident and refracted rays

• The incident ray is a ray of light hitting a surface between two materials. If the light travels to the new material, the resulting ray is the refracted ray.
• The normal is perpendicular to the surface.
• The angle of incidence is the angle between the normal and the incident ray.
• The angle of refraction is the angle between the normal and the refracted ray.
A ray of light refracts as it travels into a different material. $i$ is the angle of incidence and $r$ is the angle of reflection. The dotted line is the normal.

The angle of refraction does not equal the angle of incidence. If light travels to a material with a higher refractive index, the angle of refraction is smaller than the angle of incidence.

Light travelling from air into glass bends towards the normal, as glass has a higher refractive index than air.

But light coming from water into air bends away from the normal.

## Snell's law

Snell's law tells us how much a ray of light bends as it enters a given material from vacuum.

$$\frac{\Tblue{\sin i}}{\Tred{\sin r}} = n$$

$n$ is the refractive index of the material, $\Tblue{i}$ is the angle of incidence and $\Tred{r}$ is the angle of refraction.

Snell's law can be used to compute the refractive index of a material.

Light travels from air into diamond. The angle of incidence is $\Tblue{55^{\circ}}$. The angle of refraction is $\Tred{20^{\circ}}$.

The refractive index can be calculated from the angles of incidence and refraction.

$$n \, = \, \frac{\Tblue{\sin(55^\circ)}} {\Tred{\sin(20^\circ)}} \, = \, 2.4$$

The larger the value of $n$, the more the ray will bend.

When the ray of light goes into air from the material, Snell's relation is inverted: $\Tred{\sin r} = n\,\Tblue{\sin i}$.

## Total internal reflection and the critical angle

In total internal reflection, a ray of light hitting the surface of an object from the inside is completely reflected back into the object.

This can happen when light travels from a material with a higher refractive index $n$ towards a material with a lower refractive index.

The critical angle $\theta_c$ is the angle of incidence which gives an angle of refraction of $90^{\circ}.$ It can be calculated from Snell's Law. $$\Tblue{\sin \theta_c } = \frac {\Tred{1}}{n}$$

Total internal reflection happens when the angle of incidence is greater than the critical angle.

Left: the angle of incidence is smaller than the critical angle. Centre: the angle of incidence is equal to the critical angle. Right: the angle of incidence is larger than the critical angle (total internal reflection).

## Optical fibres

Optical fibre cables are based on the principle of total internal reflection.

Uses of optical fibres include:

• Telecommunications to carry large amount of data (e.g. internet) over long distances.
• Endoscopes to allow doctors to look inside a patient's body without major surgery.

Optical fibres have a core of glass with a high refractive index. They have an outer cladding of glass with a lower refractive index.

When light hits the boundary between the two layers of glass from the inside, the angle of incidence is greater than the critical value. The light undergoes total internal reflection.

Light travels along an optical fibre by total internal reflection.

Many rays can travel through optical fibre separately at the same time, which is why a lot of information can be passed on.