# 1.4 Total internal reflection

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By the end of this section, you will be able to:
• Explain the phenomenon of total internal reflection
• Describe the workings and uses of optical fibers
• Analyze the reason for the sparkle of diamonds

A good-quality mirror may reflect more than $90\text{%}$ of the light that falls on it, absorbing the rest. But it would be useful to have a mirror that reflects all of the light that falls on it. Interestingly, we can produce total reflection using an aspect of refraction.

Consider what happens when a ray of light strikes the surface between two materials, as shown in [link] (a). Part of the light crosses the boundary and is refracted; the rest is reflected. If, as shown in the figure, the index of refraction for the second medium is less than for the first, the ray bends away from the perpendicular. (Since ${n}_{1}>{n}_{2},$ the angle of refraction is greater than the angle of incidence—that is, ${\theta }_{1}>{\theta }_{2}.\right)$ Now imagine what happens as the incident angle increases. This causes ${\theta }_{2}$ to increase also. The largest the angle of refraction ${\theta }_{2}$ can be is $90\text{°}$ , as shown in part (b). The critical angle     ${\theta }_{\text{c}}$ for a combination of materials is defined to be the incident angle ${\theta }_{1}$ that produces an angle of refraction of $90\text{°}$ . That is, ${\theta }_{\text{c}}$ is the incident angle for which ${\theta }_{2}=90\text{°}$ . If the incident angle ${\theta }_{1}$ is greater than the critical angle, as shown in [link] (c), then all of the light is reflected back into medium 1, a condition called total internal reflection    . (As the figure shows, the reflected rays obey the law of reflection so that the angle of reflection is equal to the angle of incidence in all three cases.)

Snell’s law states the relationship between angles and indices of refraction. It is given by

${n}_{1}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta }_{1}={n}_{2}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta }_{2}.$

When the incident angle equals the critical angle $\left({\theta }_{1}={\theta }_{c}\right)$ , the angle of refraction is $90\text{°}$ $\left({\theta }_{2}=90\text{°}\right)$ . Noting that $\text{sin}\phantom{\rule{0.2em}{0ex}}90\text{°}=1,$ Snell’s law in this case becomes

${n}_{1}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta }_{1}={n}_{2}.$

The critical angle ${\theta }_{\text{c}}$ for a given combination of materials is thus

${\theta }_{\text{c}}={\text{sin}}^{-1}\left(\frac{{n}_{2}}{{n}_{1}}\right)\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}{n}_{1}>{n}_{2}.$

Total internal reflection occurs for any incident angle greater than the critical angle ${\theta }_{\text{c}}$ , and it can only occur when the second medium has an index of refraction less than the first. Note that this equation is written for a light ray that travels in medium 1 and reflects from medium 2, as shown in [link] .

## Determining a critical angle

What is the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air? The index of refraction for polystyrene is 1.49.

## Strategy

The index of refraction of air can be taken to be 1.00, as before. Thus, the condition that the second medium (air) has an index of refraction less than the first (plastic) is satisfied, and we can use the equation

${\theta }_{\text{c}}={\text{sin}}^{-1}\left(\frac{{n}_{2}}{{n}_{1}}\right)$

to find the critical angle ${\theta }_{\text{c}},$ where ${n}_{2}=1.00$ and ${n}_{1}=1.49.$

## Solution

Substituting the identified values gives

${\theta }_{\text{c}}={\text{sin}}^{-1}\left(\frac{1.00}{1.49}\right)={\text{sin}}^{-1}\left(0.671\right)=42.2\text{°}.$

## Significance

This result means that any ray of light inside the plastic that strikes the surface at an angle greater than $42.2\text{°}$ is totally reflected. This makes the inside surface of the clear plastic a perfect mirror for such rays, without any need for the silvering used on common mirrors. Different combinations of materials have different critical angles, but any combination with ${n}_{1}>{n}_{2}$ can produce total internal reflection. The same calculation as made here shows that the critical angle for a ray going from water to air is $48.6\text{°}$ , whereas that from diamond to air is $24.4\text{°}$ , and that from flint glass to crown glass is $66.3\text{°}$ .

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