A ray of light from a rarer medium strikes a denser medium at angle of incidence 60°. The reflected and refracted rays are perpendicular to each other. The refractive index of denser medium and angle of deviation respectively are

Options

A. `sqrt(3),30^@`
B. `sqrt(2),45^@`
C. `sqrt(3),60^@`
D. `sqrt(2),30^@`

Explanation

To solve the problem step by step, we will determine the refractive index of the denser medium and the angle of deviation based on the given conditions.

### Step 1: Understand the given information
- A ray of light strikes a denser medium from a rarer medium at an angle of incidence \( i = 60^\circ \).
- The reflected and refracted rays are perpendicular to each other.

### Step 2: Identify the relationship between the angles
Since the reflected and refracted rays are perpendicular, we can denote the angle of refraction as \( r \). The relationship can be expressed as:
\[ i + r = 90^\circ \]
This means:
\[ r = 90^\circ - i \]

### Step 3: Calculate the angle of refraction
Substituting the value of \( i \):
\[ r = 90^\circ - 60^\circ = 30^\circ \]

### Step 4: Use Snell's Law to find the refractive index
Snell's Law states:
\[ \mu_1 \sin i = \mu_2 \sin r \]
Where:
- \( \mu_1 \) is the refractive index of the rarer medium (air, \( \mu_1 = 1 \)).
- \( \mu_2 \) is the refractive index of the denser medium (which we need to find).
- \( i = 60^\circ \)
- \( r = 30^\circ \)

Substituting the known values into Snell's Law:
\[ 1 \cdot \sin(60^\circ) = \mu_2 \cdot \sin(30^\circ) \]

### Step 5: Substitute the sine values
We know:
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(30^\circ) = \frac{1}{2} \)

So we have:
\[ \frac{\sqrt{3}}{2} = \mu_2 \cdot \frac{1}{2} \]

### Step 6: Solve for \( \mu_2 \)
Multiplying both sides by 2:
\[ \sqrt{3} = \mu_2 \]

Thus, the refractive index of the denser medium is:
\[ \mu_2 = \sqrt{3} \]

### Step 7: Calculate the angle of deviation
The angle of deviation \( D \) is given by:
\[ D = i - r \]
Substituting the values:
\[ D = 60^\circ - 30^\circ = 30^\circ \]

### Final Answers
- The refractive index of the denser medium is \( \sqrt{3} \).
- The angle of deviation is \( 30^\circ \).

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