The light ray strikes the hypotenuse of the prism at an angle of incidence greater than the critical angle. Find the angle phi of the ray as it leaves the base of the prism. The incident ray is perpendicular to the left side of the prism.
(The index of refraction of the prism is 1.5. Take θ to be 43.4°.)
To find the angle phi of the ray as it leaves the base of the prism, we can use Snell's law, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the indices of refraction of the two media involved.
Let's start by assigning the variables:
Angle of incidence (θ) = 43.4°
Index of refraction of the prism (n1) = 1.5
To find the angle of refraction (phi), we need to know the index of refraction of the medium outside the prism (n2). In this case, since the incident ray is perpendicular to the left side of the prism, it means it is traveling in air. The index of refraction of air is approximately 1.0.
Now, we can use Snell's law:
n1 * sin(θ) = n2 * sin(phi)
Plugging in the values we know:
1.5 * sin(43.4°) = 1.0 * sin(phi)
Now, rearrange the equation to solve for phi:
sin(phi) = (1.5 * sin(43.4°)) / 1.0
Using a scientific calculator, calculate:
sin(phi) = 0.954
To find the angle phi, take the inverse sine (sin^-1) of 0.954:
phi = sin^-1(0.954)
Using a scientific calculator, find:
phi ≈ 70.49°
Therefore, the angle phi of the ray as it leaves the base of the prism is approximately 70.49°.