The least angle of deviation for a glass prism is equal to its refracting angle. The refractive index of glass is 1.5. Then the angle of the prism is

Options

A. `2 cos^(-1) ((3)/(4))`
B. `sin^(-1) ((3)/(4))`
C. `sin^(-1) ((3)/(2))`
D. `cos^(-1)((3)/(2))`

Answer

To solve the problem step by step, we will use the information given about the prism, the refractive index, and the relationship between the angles involved.

### Step 1: Understand the Relationship
We know that the least angle of deviation (δ_min) for a prism is equal to its refracting angle (A). Thus, we can write:
\[ δ_{min} = A \]

### Step 2: Use the Formula for Minimum Deviation
The formula for the least angle of deviation in terms of the angle of incidence (i) and the angle of the prism (A) is given by:
\[ δ_{min} = 2i - A \]
Since we established that \( δ_{min} = A \), we can substitute this into the equation:
\[ A = 2i - A \]

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ 2i = 2A \]
Dividing both sides by 2, we find:
\[ i = A \]
This means that at minimum deviation, the angle of incidence is equal to the angle of the prism.

### Step 4: Relate the Angles of Refraction
For a prism, the sum of the angles of refraction (r1 + r2) is equal to the angle of the prism (A). At minimum deviation, the angles of refraction are equal:
\[ r_1 + r_2 = A \]
Let \( r_1 = r_2 = r \). Therefore:
\[ r + r = A \]
This simplifies to:
\[ A = 2r \]
Thus, we can express the angle of refraction as:
\[ r = \frac{A}{2} \]

### Step 5: Use the Refractive Index Formula
The refractive index (μ) is given by:
\[ μ = \frac{\sin i}{\sin r} \]
Substituting \( i = A \) and \( r = \frac{A}{2} \):
\[ μ = \frac{\sin A}{\sin \left(\frac{A}{2}\right)} \]
Given that the refractive index of glass is 1.5, we can write:
\[ 1.5 = \frac{\sin A}{\sin \left(\frac{A}{2}\right)} \]

### Step 6: Cross-Multiply to Solve for Sin A
Cross-multiplying gives:
\[ \sin A = 1.5 \sin \left(\frac{A}{2}\right) \]

### Step 7: Use the Double Angle Identity
Using the identity \( \sin A = 2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right) \):
\[ 2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right) = 1.5 \sin \left(\frac{A}{2}\right) \]
Assuming \( \sin \left(\frac{A}{2}\right) \neq 0 \), we can divide both sides by \( \sin \left(\frac{A}{2}\right) \):
\[ 2 \cos \left(\frac{A}{2}\right) = 1.5 \]

### Step 8: Solve for Cosine
Dividing both sides by 2 gives:
\[ \cos \left(\frac{A}{2}\right) = \frac{1.5}{2} = \frac{3}{4} \]

### Step 9: Find the Angle A
To find the angle of the prism, we take the inverse cosine:
\[ \frac{A}{2} = \cos^{-1} \left(\frac{3}{4}\right) \]
Thus, multiplying by 2 gives:
\[ A = 2 \cos^{-1} \left(\frac{3}{4}\right) \]

### Final Answer
The angle of the prism is:
\[ A = 2 \cos^{-1} \left(\frac{3}{4}\right) \]

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