One face of a prism with a refrective angle of `30^@` is coated with silver. A ray of light incident on another face at an angle of `45^@` is refracted and reflected from the silver coated face and retraces its path. What is the refractive index of the prism?

Options

A. `sqrt2`
B. `3//sqrt2`
C. 1.5
D. 1.33

Explanation

To find the refractive index of the prism, we will follow these steps:

### Step 1: Understand the Geometry of the Prism
The prism has a refracting angle (A) of \(30^\circ\). A ray of light is incident on one face of the prism at an angle of \(45^\circ\). The ray is refracted into the prism and then reflects off the silver-coated face, retracing its path.

### Step 2: Identify the Angles
Let:
- \(A = 30^\circ\) (the refracting angle of the prism)
- \(i = 45^\circ\) (the angle of incidence on the first face)
- \(R_1\) = angle of refraction at the first face.

### Step 3: Apply the Geometry of the Situation
Since the ray retraces its path after reflecting from the silver-coated face, we can use the geometry of the situation:
- The sum of angles in the triangle formed by the angles at the prism's vertex is \(180^\circ\):
\[ 90^\circ - R_1 + 90^\circ + A = 180^\circ \]
Substituting \(A = 30^\circ\):
\[ 90^\circ - R_1 + 90^\circ + 30^\circ = 180^\circ \]
This simplifies to:
\[ 180^\circ - R_1 + 30^\circ = 180^\circ \]
Thus,
\[ R_1 = 30^\circ \]

### Step 4: Use Snell's Law
Now we apply Snell's Law at the first interface:
\[ n_1 \sin(i) = n_2 \sin(R_1) \]
Where:
- \(n_1 = 1\) (refractive index of air)
- \(n_2 = \mu\) (refractive index of the prism)
- \(i = 45^\circ\)
- \(R_1 = 30^\circ\)

Substituting the values:
\[ 1 \cdot \sin(45^\circ) = \mu \cdot \sin(30^\circ) \]
Using known values:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}}, \quad \sin(30^\circ) = \frac{1}{2} \]
Thus:
\[ \frac{1}{\sqrt{2}} = \mu \cdot \frac{1}{2} \]

### Step 5: Solve for the Refractive Index
Rearranging the equation gives:
\[ \mu = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Final Answer
The refractive index of the prism is \(\mu = \sqrt{2}\).

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