A ray of light is incident on an equilateral triangular glass prism of refractive index 3/2, Calculate the angle through which the ray is minimally deviated in the prism
A. 30.0∘
B. 37.2∘
C. 42.0∘
D. 48.6∘
To find the angle through which the ray is minimally deviated in the prism, we can use Snell's Law.
According to Snell's Law, the angle of incidence (i) and the angle of refraction (r) are related by the equation:
n1 × sin(i) = n2 × sin(r),
where n1 is the refractive index of the medium from which the light is coming (in this case, air), and n2 is the refractive index of the medium into which the light is entering (in this case, the prism).
Since the refractive index of air is approximately 1, we can rewrite the equation as:
sin(i) = (n2 / n1) × sin(r).
For an equilateral triangular prism, the angle of minimum deviation occurs when the angle of incidence is equal to the angle of emergence (e) and the emergent ray is parallel to the base of the prism. In this case, the angle of incidence is 90 degrees - (e/2).
Using the given refractive index of 3/2, we can substitute the values into the equation:
sin(90 - e/2) = (3/2) × sin(e).
To solve for e, we can use a scientific calculator or a trigonometric identity. By rearranging the equation, we get:
sin(e) = (2/3) × sin(90 - e/2).
We can then calculate the approximate value of e using a scientific calculator. By inputting the equation t0o calculate for the angle in degrees:
e ≈ 42.0°.
Therefore, the correct answer is C. 42.0°.
To calculate the angle through which the ray is minimally deviated in the prism, we can use Snell's law and the principle of minimum deviation.
The angle of incidence at the first face of the prism is given by:
sin(i) = n * sin(r)
Where i is the angle of incidence, n is the refractive index of the prism, and r is the angle of refraction inside the prism.
In an equilateral triangular prism, the angle of incidence and the angle of refraction are equal. Therefore, we can write:
sin(i) = n * sin(i)
Rearranging the equation, we have:
sin(i)/sin(i) = n
1 = n
Since n is given as 3/2, we have:
1 = 3/2
To solve for i, take the arcsine of both sides of the equation:
i = arcsin(1) = 90∘
The angle of incidence at the first face of the prism is 90∘.
Now, we can use the principle of minimum deviation to find the angle through which the ray is minimally deviated. The minimal deviation occurs when the angle of incidence is equal to the angle of emergence.
Since the prism is equilateral, the angles of incidence at the first and second faces of the prism are equal. Therefore, the angle of incidence at the second face of the prism is also 90∘.
Using Snell's law again, we have:
sin(i) = n * sin(r)
sin(90∘) = (3/2) * sin(r)
1 = (3/2) * sin(r)
To solve for r, divide both sides of the equation by 3/2:
(3/2) * sin(r) = 1
sin(r) = 2/3
To find r, take the arcsine of both sides of the equation:
r = arcsin(2/3) ≈ 37.2∘
Therefore, the angle through which the ray is minimally deviated in the prism is approximately 37.2∘.
The correct answer is option B.