If the angle of minimum deviation is of `60^@` for an equilateral prism , then the refractive index of the material of the prism is

Options

A. `1.41`
B. `1.5`
C. `1.6`
D. `1.73`

Answer

To find the refractive index of an equilateral prism given the angle of minimum deviation, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The angle of minimum deviation (δ_min) = 60°
- The angle of the prism (A) for an equilateral prism = 60°

### Step 2: Use the formula for refractive index
The formula for the refractive index (μ) of a prism in terms of the angle of minimum deviation and the angle of the prism is:
\[ \mu = \frac{\sin\left(\frac{\delta_{min} + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 3: Substitute the known values into the formula
Substituting the values we have:
- δ_min = 60°
- A = 60°

Now, substituting these values into the formula:
\[ \mu = \frac{\sin\left(\frac{60° + 60°}{2}\right)}{\sin\left(\frac{60°}{2}\right)} \]

### Step 4: Simplify the angles
Calculate the angles:
\[ \frac{60° + 60°}{2} = \frac{120°}{2} = 60° \]
\[ \frac{60°}{2} = 30° \]

### Step 5: Calculate the sine values
Now, we can find the sine values:
- \(\sin(60°) = \frac{\sqrt{3}}{2}\)
- \(\sin(30°) = \frac{1}{2}\)

### Step 6: Substitute the sine values into the equation
Now substitute these sine values into the equation for μ:
\[ \mu = \frac{\sin(60°)}{\sin(30°)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{1} = \sqrt{3} \]

### Step 7: Calculate the numerical value
The numerical value of \(\sqrt{3}\) is approximately 1.732.

### Step 8: Conclusion
Thus, the refractive index of the material of the prism is:
\[ \mu \approx 1.732 \]

### Final Answer
The refractive index of the material of the prism is approximately **1.73**.

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