If the refractive index of material of a prism is `sqrt3`, then angle of minimum diviation of the prism, which is equilateral is

Options

A. `75^(@)`
B. `60^(@)`
C. `45^(@)`
D. `30^(@)`

Explanation

To solve the problem of finding the angle of minimum deviation (Δm) for an equilateral prism with a refractive index (μ) of √3, we can follow these steps:

### Step 1: Understand the properties of the prism
An equilateral prism has an angle (A) of 60 degrees. The refractive index (μ) is given as √3.

### Step 2: Use the formula for minimum deviation
The formula relating the refractive index (μ), the angle of the prism (A), and the angle of minimum deviation (Δm) is given by:

\[ \mu = \frac{\sin\left(\frac{A + \Delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 3: Substitute the known values
For an equilateral prism, A = 60 degrees. Therefore, we have:

\[ \mu = \sqrt{3} \]
\[ \frac{A}{2} = \frac{60}{2} = 30 \text{ degrees} \]

Now, substituting these values into the formula:

\[ \sqrt{3} = \frac{\sin\left(\frac{60 + \Delta_m}{2}\right)}{\sin(30)} \]

### Step 4: Calculate sin(30)
We know that:

\[ \sin(30) = \frac{1}{2} \]

### Step 5: Rewrite the equation
Substituting sin(30) into the equation gives:

\[ \sqrt{3} = \frac{\sin\left(\frac{60 + \Delta_m}{2}\right)}{\frac{1}{2}} \]

This simplifies to:

\[ \sqrt{3} = 2 \sin\left(\frac{60 + \Delta_m}{2}\right) \]

### Step 6: Solve for sin(θ)
Rearranging gives:

\[ \sin\left(\frac{60 + \Delta_m}{2}\right) = \frac{\sqrt{3}}{2} \]

### Step 7: Find the angle corresponding to sin(θ)
We know that:

\[ \sin(60) = \frac{\sqrt{3}}{2} \]

Thus, we can equate:

\[ \frac{60 + \Delta_m}{2} = 60 \text{ degrees} \]

### Step 8: Solve for Δm
Multiplying both sides by 2 gives:

\[ 60 + \Delta_m = 120 \]

Now, subtracting 60 from both sides yields:

\[ \Delta_m = 120 - 60 = 60 \text{ degrees} \]

### Final Answer
The angle of minimum deviation (Δm) of the prism is **60 degrees**.

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