Question 34.5: Measuring n Using a Prism Although we do not prove it here, ...
Measuring n Using a Prism
Although we do not prove it here, the minimum angle of deviation \delta_{\min } for a prism occurs when the angle of incidence \theta_1 is such that the refracted ray inside the prism makes the same angle with the normal to the two prism faces¹ as shown in Figure 34.16. Obtain an expression for the index of refraction of the prism material in terms of the minimum angle of deviation and the apex angle Φ.
¹ The details of this proof are available in texts on optics.
Conceptualize Study Figure 34.16 carefully and be sure you understand why the light ray comes out of the prism traveling in a different direction.
Categorize In this example, light enters a material through one surface and leaves the material at another surface. Let’s apply the wave under refraction model to the light passing through the prism.
Analyze Consider the geometry in Figure 34.16, where we have used symmetry to label several angles. The reproduction of the angle Φ/2 at the location of the incoming light ray shows that \theta_2=\Phi / 2. The theorem that an exterior angle of any triangle equals the sum of the two opposite interior angles shows that \delta_{\min }=2 \alpha. The geometry also shows that \theta_1=\theta_2+\alpha.
Combine these three geometric results:
\theta_1=\theta_2+\alpha=\frac{\Phi}{2}+\frac{\delta_{\min }}{2}=\frac{\Phi+\delta_{\min }}{2}Apply the wave under refraction model at the left surface and solve for n:
(1.00) \sin \theta_1=n \sin \theta_2 \rightarrow n=\frac{\sin \theta_1}{\sin \theta_2}Substitute for the incident and refracted angles:
n=\frac{\sin \left(\frac{\Phi+\delta_{\min }}{2}\right)}{\sin (\Phi / 2)} (34.8)
Finalize Knowing the apex angle Φ of the prism and measuring \delta_{\min }, you can calculate the index of refraction of the prism material. Furthermore, a hollow prism can be used to determine the values of n for various liquids filling the prism.