The angle of minimum deviation for an incident ligt ray on an equillateral prism is equal to its refracting angle. The refractive index of its material is

Options

A. `1//sqrt(2)`
B. `sqrt(3)`
C. `(sqrt(3))//2`
D. `3//2`

Answer

To find the refractive index of an equilateral prism where the angle of minimum deviation is equal to its refracting angle, we can follow these steps:

### Step 1: Understand the conditions of minimum deviation
In the case of minimum deviation through a prism, the following conditions hold:
- The angle of incidence (i) is equal to the angle of emergence (e).
- The angle of refraction at both faces of the prism (r1 and r2) is equal, meaning r1 = r2 = r.

### Step 2: Relate the angles
For a prism, the angle of deviation (Δ) can be expressed as:
\[ \Delta = i + e - A \]
Where A is the angle of the prism. Since at minimum deviation, \(i = e\), we can rewrite this as:
\[ \Delta = 2i - A \]
At minimum deviation, we denote this angle as \(\Delta_{min}\).

### Step 3: Express the angle of incidence
From the equation for minimum deviation:
\[ \Delta_{min} = 2i - A \implies i = \frac{\Delta_{min} + A}{2} \]

### Step 4: Relate the angles of refraction
The angle of refraction at the prism can be expressed as:
\[ A = r_1 + r_2 = 2r \implies r = \frac{A}{2} \]

### Step 5: Substitute for an equilateral prism
For an equilateral prism, the angle A = 60°. Thus:
\[ r = \frac{60°}{2} = 30° \]

### Step 6: Substitute into the refractive index formula
The refractive index (μ) is given by:
\[ \mu = \frac{\sin i}{\sin r} \]
Substituting for \(i\) and \(r\):
\[ \mu = \frac{\sin\left(\frac{\Delta_{min} + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Since we have \(\Delta_{min} = r\) and \(A = 60°\), we can substitute:
\[ \mu = \frac{\sin\left(30° + 30°\right)}{\sin(30°)} = \frac{\sin(60°)}{\sin(30°)} \]

### Step 7: Calculate the sine values
We know:
- \(\sin(60°) = \frac{\sqrt{3}}{2}\)
- \(\sin(30°) = \frac{1}{2}\)

### Step 8: Substitute and simplify
Substituting these values into the equation for μ:
\[ \mu = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]

### Step 9: Conclusion
Thus, the refractive index of the material of the prism is:
\[ \mu = \sqrt{3} \]