Answer:
i ≅ [tex]28^{0}[/tex]
Explanation:
Let the angle of incidence be represented by i, angle of emergence be represented by e, and angle of refraction by r.
Snell's law states that;
n = [tex]\frac{sin i}{sin r}[/tex] ................ 1
where n is the refractive index of the prism.
Given that emergence = [tex]90^{0}[/tex]
But from a ray diagram for the given question, we have;
[tex]60^{0}[/tex] + ([tex]90^{0}[/tex] - r) + ([tex]90^{0}[/tex] - [tex]r^{I}[/tex]) = [tex]180^{0}[/tex] (sum of angles in a triangle) .................. 2
([tex]90^{0}[/tex] - r ) + ([tex]90^{0}[/tex] - [tex]r^{I}[/tex] ) = [tex]180^{0}[/tex] - [tex]60^{0}[/tex]
180° - (r + [tex]r^{I}[/tex]) = [tex]180^{0}[/tex] - [tex]60^{0}[/tex]
r + [tex]r^{I}[/tex] = [tex]60^{0}[/tex]
⇒ r = [tex]60^{0}[/tex] - [tex]r^{I}[/tex] ........................ 3
The refractive index of the equilateral prism = 1.5.
Applying Snell's law to the refracting surface,
[tex]\frac{sinr^{I} }{sin e}[/tex] = [tex]\frac{1}{n}[/tex]
[tex]\frac{sinr^{I} }{sin 90^{0} }[/tex] = [tex]\frac{1}{1.5}[/tex]
⇒ [tex]r^{I}[/tex] = [tex]41.81^{0}[/tex]
From equation 3,
r = [tex]60^{0}[/tex] - [tex]r^{I}[/tex]
r = [tex]60^{0}[/tex] - [tex]41.81^{0}[/tex]
r = [tex]18.19^{0}[/tex]
So that ;
n = [tex]\frac{sin i}{sin r}[/tex]
1.5 = [tex]\frac{sin i}{sin18.19^{0} }[/tex]
sin i = 0.4683
i = [tex]27.92^{0}[/tex] ≅ [tex]28^{0}[/tex]
