A glass prism with a refracting angle of `60^(@)` has a refractive index 1.52 for red and 1.6 for violet light. A parallel beam of white is incident on one face at an angle of incident which gives minimum deviation for red light. Find :
[Use : `sin(50^(@))=0.760 , sin(31.6^(@))=0.520 , sin (28.4^(@))=0.475 , sin (56^(@))= 0.832 , = 22//7`]
The angle of incidence at the prism is :

Options

A. `30^(@)`
B. `40^(@)`
C. `50^(@)`
D. `60^(@)`

Explanation

To find the angle of incidence at the prism for red light when the minimum deviation occurs, we can follow these steps:

### Step 1: Understand the Geometry of the Prism
The refracting angle of the prism (A) is given as \(60^\circ\). When a beam of light passes through a prism, the sum of the angles at the prism must equal \(180^\circ\). Therefore, if we denote the angle of incidence as \(i\) and the angle of refraction as \(r\), we can set up the following relationship:

\[ i + r + A = 180^\circ \]

### Step 2: Calculate the Angles
From the geometry of the prism, we can express the relationship as:

\[ i + r + 60^\circ = 180^\circ \]

This simplifies to:

\[ i + r = 120^\circ \quad \text{(1)} \]

### Step 3: Use the Minimum Deviation Condition
At minimum deviation, the angle of incidence \(i\) is equal to the angle of emergence. Thus, we can denote the angle of refraction \(r\) as \(R\). Therefore, the equation (1) can be rewritten as:

\[ 2R + 60^\circ = 180^\circ \]

From this, we can solve for \(R\):

\[ 2R = 120^\circ \implies R = 60^\circ \]

### Step 4: Apply Snell's Law
Using Snell's law at the first surface of the prism for red light, we have:

\[ n_1 \sin i = n_2 \sin R \]

Where:
- \(n_1 = 1\) (refractive index of air),
- \(n_2 = 1.52\) (refractive index for red light),
- \(R = 30^\circ\) (since \(R = 60^\circ - A\)).

Substituting the values, we get:

\[ 1 \cdot \sin i = 1.52 \cdot \sin 30^\circ \]

### Step 5: Calculate \(\sin 30^\circ\)
We know that:

\[ \sin 30^\circ = 0.5 \]

Thus, substituting this into the equation gives:

\[ \sin i = 1.52 \cdot 0.5 = 0.76 \]

### Step 6: Find the Angle of Incidence
Now we need to find the angle \(i\) such that:

\[ \sin i = 0.76 \]

From the given information, we know:

\[ \sin(50^\circ) = 0.76 \]

Thus, we conclude:

\[ i = 50^\circ \]

### Final Answer
The angle of incidence at the prism is \(50^\circ\).

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