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 solve this problem, we need to 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 refractive indices of the two mediums:
n1 * sin(theta1) = n2 * sin(theta2)
In this case, the incident angle (theta1) is the angle at which the ray enters the prism, and the refracted angle (theta2) is the angle at which the ray exits the prism.
Since the prism is equilateral, the angle of incidence will be equal to the angle of refraction.
Let's assume the angle of incidence is theta1, and the angle of refraction is theta2. Thus, the equation becomes:
(3/2) * sin(theta1) = sin(theta2)
Since the prism is equilateral, the angles of incidence and refraction are also the angles between the prism face and the incoming and exiting rays, respectively.
The angle through which the ray is minimally deviated is the angle of deviation (D), which is the difference between the angle of incidence and the angle of refraction.
D = theta1 - theta2
Substituting the value of sin(theta2) from the equation above, we get:
D = theta1 - (3/2) * sin(theta1)
In order to find the angle through which the ray is minimally deviated, we need to minimize the angle of deviation. We can do this by taking the derivative of D with respect to theta1, and setting it equal to zero:
dD/d(theta1) = 1 - (3/2) * cos(theta1) = 0
Simplifying, we find:
cos(theta1) = 2/3
Therefore, theta1 = cos^(-1)(2/3) ≈ 48.6°
Substituting this value of theta1 into the equation for D, we get:
D = 48.6° - (3/2) * sin(48.6°) ≈ 37.2°
Therefore, the angle through which the ray is minimally deviated in the prism is approximately 37.2°.
So, the correct option is B. 37.2∘.
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.
Step 1: Determine the angle of incidence (θ1).
Since the prism is equilateral, each angle of the prism is 60 degrees (as an equilateral triangle has three equal angles of 60 degrees). Therefore, the angle of incidence (θ1) is half of the vertex angle of the prism, which is 30 degrees (60 degrees divided by 2).
Step 2: Calculate the angle of refraction (θ2) inside the prism using Snell's law.
Snell's law states that n1 * sin(θ1) = n2 * sin(θ2), where n1 and n2 are the refractive indices of the initial medium and the prism, respectively.
Given:
Refractive index of the prism (n2) = 3/2
Angle of incidence (θ1) = 30 degrees
Using Snell's law:
sin(θ2) = (n1 / n2) * sin(θ1)
sin(θ2) = (1 / (3/2)) * sin(30)
sin(θ2) = (2/3) * (1/2)
sin(θ2) = 1/3
Taking the inverse sine:
θ2 = sin^(-1)(1/3)
θ2 = 19.47 degrees (rounded to two decimal places)
Step 3: Calculate the angle of deviation (δ) using the principle of minimum deviation.
The principle of minimum deviation states that the angle of deviation of a ray passing through a prism is the smallest when the angle of incidence is equal to the angle of emergence (when the ray is symmetrically incident on and emerging from the prism).
Since the prism is equilateral, the angle of emergence is also 30 degrees (same as the angle of incidence, θ1).
Therefore, the angle of deviation (δ) will be:
δ = (θ1 + θ2) - 60
δ = (30 + 19.47) - 60
δ = 49.47 - 60
δ = -10.53 degrees
Step 4: Calculate the minimal deviation angle.
The minimal deviation angle (Δ) is the absolute value of the angle of deviation (δ).
Δ = |δ|
Δ = |-10.53|
Δ = 10.53 degrees
Therefore, the angle through which the ray is minimally deviated in the prism is approximately 10.53 degrees.
The correct option among the given choices is not provided.