Question

The prism shown here is used to totally reflect light at a$\mathbf{90}\mathbf{°}$angle. No surface of this prism is silvered. Use Snell's law to explain why total reflection occurs. What is the minimum refractive index of the prism for total reflection?

Short answer

The minimum refractive index of the prism for total reflection is ${\mathrm{n}}_{\mathrm{prism}}>1.1142$.

Step-by-step solution

Definition of refractive index

• The refractive index, commonly known as the index of refraction, is a measurement of how much a ray of light bends when it passes through one medium and into another.

Determine the minimum refractive index of the prism for total reflection

In this scenario, the angle of reflection is equal to the angle of incidence $45°$.

In this example, Snell's law is:

${\mathrm{n}}_{\mathrm{prism}}\times \sin {\left(45\right)}^{‐}={\mathrm{n}}_{\mathrm{air}}\times \mathrm{\theta }$

Where the$\theta$ is angle of refraction.

There is no light refracted when there is a total reflection. This happens when the following conditions are met:

role="math" localid="1663654585707" $\mathrm{sin\theta }>1$

As a result, the n of prism is:

$$\begin{gathered} {\mathrm{n}}_{\mathrm{prism}}>\frac{{\mathrm{n}}_{\mathrm{air}}}{\sin \left(45°\right)} \\ {\mathrm{n}}_{\mathrm{prism}}>\frac{1}{0.70711} \\ {\mathrm{n}}_{\mathrm{prism}}>1.4142\mathrm{a} \end{gathered}$$

As a result, if ${\mathrm{n}}_{\mathrm{prism}}>1.4142$, all light is reflected.