The angle of incidence is [tex]82.8^{\circ}[/tex]
Explanation:
Refraction occurs when a ray of light moves from one medium into another medium with different optical density. The ray of light changes direction and speed, according to Snell's law:
[tex]n_i sin \theta_i = n_r sin \theta_r[/tex]
where
[tex]n_i[/tex] is the index of refraction of the 1st medium
[tex]\theta_i[/tex] is the angle of incidence
[tex]n_r[/tex] is the index of refraction of the 2nd medium
[tex]\theta_r[/tex] is the angle of refraction
In this problem, we have the following data:
[tex]n_i = 1[/tex] (index of refraction of air)
[tex]n_r = 1.5[/tex] (index of refraction of glass)
[tex][tex]\theta_f = \frac{\theta_i}{2}[/tex][/tex] (the angle of refraction is half the angle of incidence)
Substituting into the equation and solving, we can find the angle of incidence:
[tex]sin (\theta_i) = 1.5 sin (\theta_i/2)[/tex]
By rewriting [tex]sin(\theta_i)[/tex] as
[tex]sin(\theta_i) = 2 sin (\frac{\theta_i}{2}) cos (\frac{\theta_i}{2})[/tex]
we get
[tex]2 sin (\theta_i/2) cos (\theta_i/2)=1.5 sin (\theta_i/2) \\cos(\theta_i/2)=\frac{1.5}{2}\\\frac{\theta_i}{2} = cos^{-1}(\frac{1.5}{2})=41.4^{\circ}[/tex]
So, the angle of incidence is
[tex]\theta_i = 2(41.4)=82.8^{\circ}[/tex]
Learn more about refraction here:
brainly.com/question/3183125
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