A ray of light is incident at `65^(@)` on one face of a prism of angle of `30^(@)` and the emergent ray makes `35^(@)` with the incident ray. The refractive index of the prism is:

Options

A. 1.812
B. 1.414
C. 1.5
D. 1.33

Answer

To find the refractive index of the prism, we can follow these steps:

### Step 1: Understand the given information
- The angle of incidence (I) is \(65^\circ\).
- The angle of the prism (A) is \(30^\circ\).
- The angle of deviation (D) is \(35^\circ\).
- We need to find the refractive index (μ) of the prism.

### Step 2: Use the relationship between angles
The relationship between the angles can be expressed as:
\[ D = I + E - A \]
Where:
- \(D\) is the angle of deviation,
- \(I\) is the angle of incidence,
- \(E\) is the angle of emergence,
- \(A\) is the angle of the prism.

### Step 3: Rearrange the formula to find the angle of emergence (E)
From the equation:
\[ E = D + A - I \]
Substituting the known values:
\[ E = 35^\circ + 30^\circ - 65^\circ \]
\[ E = 0^\circ \]

### Step 4: Apply Snell's Law at the first interface
At the first interface (incident ray entering the prism), we apply Snell's law:
\[ \mu_1 \sin I = \mu \sin r \]
Where:
- \(\mu_1\) is the refractive index of air (approximately 1),
- \(I\) is the angle of incidence,
- \(r\) is the angle of refraction inside the prism.

Since \(I = 65^\circ\) and \(\mu_1 = 1\):
\[ 1 \cdot \sin(65^\circ) = \mu \cdot \sin(r) \]

### Step 5: Determine the angle of refraction (r)
Using the geometry of the prism, we know that:
\[ r + r' = A \]
Where \(r'\) is the angle of refraction at the second interface. Since \(E = 0^\circ\), we have:
\[ r' = 0^\circ \]
Thus:
\[ r + 0 = 30^\circ \implies r = 30^\circ \]

### Step 6: Substitute to find the refractive index (μ)
Now substituting \(r = 30^\circ\) back into Snell's law:
\[ \sin(65^\circ) = \mu \cdot \sin(30^\circ) \]
\[ \sin(65^\circ) = \mu \cdot \frac{1}{2} \]
\[ \mu = \frac{\sin(65^\circ)}{\frac{1}{2}} = 2 \sin(65^\circ) \]

### Step 7: Calculate the value of μ
Using a calculator:
\[ \sin(65^\circ) \approx 0.9063 \]
Thus:
\[ \mu \approx 2 \cdot 0.9063 \approx 1.8126 \]

### Final Answer
The refractive index of the prism is approximately \(1.812\).

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