The refractive index of the material of an equilateral prism is 1.6. The angle of minimum deviation due to the prism would be

Options

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

Answer

To find the angle of minimum deviation (δ_min) for an equilateral prism with a refractive index (μ) of 1.6, we can follow these steps:

### Step 1: Identify the properties of the prism
- Given that the prism is equilateral, the angle \( A \) is \( 60^\circ \).

### Step 2: Write down the formula for refractive index
- The formula relating the refractive index \( μ \), the angle of the prism \( A \), and the angle of minimum deviation \( δ_{min} \) is given by:
\[ μ = \frac{\sin\left(\frac{A + δ_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 3: Substitute the known values into the formula
- Substitute \( A = 60^\circ \) and \( μ = 1.6 \) into the formula:
\[ 1.6 = \frac{\sin\left(\frac{60^\circ + δ_{min}}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)} \]
- Calculate \( \sin\left(\frac{60^\circ}{2}\right) = \sin(30^\circ) = \frac{1}{2} \).

### Step 4: Rearrange the equation
- Rearranging the equation gives:
\[ 1.6 = \frac{\sin\left(\frac{60^\circ + δ_{min}}{2}\right)}{\frac{1}{2}} \]
- This simplifies to:
\[ \sin\left(\frac{60^\circ + δ_{min}}{2}\right) = 1.6 \times \frac{1}{2} = 0.8 \]

### Step 5: Solve for \( \frac{60^\circ + δ_{min}}{2} \)
- Now, we need to find the angle whose sine is \( 0.8 \):
\[ \frac{60^\circ + δ_{min}}{2} = \sin^{-1}(0.8) \]
- Using a calculator, we find:
\[ \sin^{-1}(0.8) \approx 53.13^\circ \]

### Step 6: Solve for \( δ_{min} \)
- Multiply both sides by 2:
\[ 60^\circ + δ_{min} = 2 \times 53.13^\circ \]
- This gives:
\[ 60^\circ + δ_{min} \approx 106.26^\circ \]
- Now, subtract \( 60^\circ \):
\[ δ_{min} \approx 106.26^\circ - 60^\circ = 46.26^\circ \]

### Final Answer
- The angle of minimum deviation \( δ_{min} \) is approximately \( 46.26^\circ \).

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