Question

A ray of $\mathbf{610}\mathbf{}\mathbf{nm}$light goes from air into fused quartz at an incident angle of $\mathbf{55}\mathbf{.}{\mathbf{0}}^{\mathbf{°}}$. At what incident angle must $\mathbf{470}\mathbf{}\mathbf{nm}$ light enter flint glass to have the same angle of refraction?

Short answer

The incident angle is obtained as:$71.4^{°}$

Step-by-step solution

Define Geometric Optics

Geometrical optics, often known as ray optics, is an optics model that describes light propagation using rays. In geometric optics, a ray is an abstraction that can approximate the routes along which light propagates under particular conditions.

Evaluating the angle of orange light

The light with the wave length $610\mathrm{nm}$is said to be the orange light, and it has the wavelength $470\mathrm{nm}$of blue light. We will evaluate the angle of refraction ${\mathrm{\theta }}_{\mathrm{f}}$for the orange color using Snell's law:

role="math" localid="1654057362925" $n_{\mathrm{i}}sin{\theta }_{\mathrm{i}}=n_{\mathrm{f}}sin{\theta }_{\mathrm{f}}$

We obtain the required refraction angle with which the blue light needs to be refracted with (since it is asking for the incidence angle at which this blue light would refract at the same angle). We can evaluate the required incidence angle by applying Snell's law once more.

Then, substituting in Snell's law for the orange light to find the angle by which it would refract.

So, we obtain:

$$\begin{gathered} {\theta }_f=\sin \left(\frac{n_i\sin {\theta }_i}{n_f}\right) \\ =\sin \left(\frac{1.00\times \sin \left({55}^{°}\right)}{1.456}\right) \\ =34.{24}^{°} \end{gathered}$$

Evaluating the incident angle

We already know the angle with which the orange color needs to be refracted. Then we apply Snell's law once again to find the required incidence angle.

So, we obtain:

$$\begin{gathered} {\theta }_i=\sin \left(\frac{n_f\sin \left({\theta }_f\right)}{n_i}\right) \\ =\sin \left(\frac{1.684\times \sin \left(34.{24}^{°}\right)}{1.000}\right) \\ =71.{36}^{°} \end{gathered}$$

Therefore, the blue light should be the incident on the surface of the flint angle by the incidence angle of the number obtained $71.4^{°}$in order for the blue light to be refracted at the same angle by which a blue light refracts. It is when incident on fused quartz from the air at an incidence angle$55.0^{°}$.