The cross section of a glass prism `mu=2` has the form of an isosceles right-angled triangle. It is submerged in a liquid of refractive index `1.2`. A ray is incident on to one of the equal faces perpendicularly. What is the angle between the direction of incident ray and the ray that emerges from the prism ?

Answer

To solve the problem step by step, we will analyze the situation involving the glass prism submerged in a liquid and the behavior of light as it passes through the prism.

### Step 1: Understand the Geometry of the Prism
The prism has a cross-section in the form of an isosceles right triangle. The angles of this triangle are:
- One angle = 90 degrees (right angle)
- The other two angles = 45 degrees each (isosceles property)

### Step 2: Identify the Refractive Indices
- The refractive index of the glass prism (μ) = 2
- The refractive index of the liquid = 1.2

### Step 3: Analyze the Incident Ray
The ray of light is incident perpendicularly on one of the equal faces of the prism. This means:
- Angle of incidence (i) = 0 degrees

### Step 4: Refraction at the First Surface
Since the ray is incident perpendicularly, it will pass through the first surface without any deviation:
- The ray continues straight into the prism.

### Step 5: Determine the Angle of Incidence at the Second Surface
When the ray reaches the second surface (the face opposite to the incident face), we need to find the angle of incidence at this surface.
- The ray will make an angle of 45 degrees with the normal at the second surface due to the geometry of the isosceles right triangle.

### Step 6: Calculate the Critical Angle
To determine whether the ray will emerge from the prism or undergo total internal reflection, we calculate the critical angle (θc):
\[ \sin(θc) = \frac{n_2}{n_1} = \frac{1.2}{2} \]
Calculating this gives:
\[ θc = \sin^{-1}(0.6) \approx 36.87 \text{ degrees} \]

### Step 7: Compare the Angle of Incidence with the Critical Angle
The angle of incidence at the second surface is 45 degrees, which is greater than the critical angle of 36.87 degrees. Therefore, total internal reflection occurs.

### Step 8: Reflection at the Second Surface
Since total internal reflection occurs, the angle of reflection will also be 45 degrees (by the law of reflection).

### Step 9: Determine the Emergence of the Ray
The ray will reflect off the second surface and will exit the prism through the first surface (the face where it initially entered). Since the angle of incidence at the first surface is 0 degrees, it will emerge perpendicularly.

### Step 10: Calculate the Angle Between the Incident Ray and Emergent Ray
The angle between the direction of the incident ray and the ray that emerges from the prism is the sum of the angles of incidence and reflection:
\[ \text{Angle between incident ray and emergent ray} = 45 \text{ degrees} + 45 \text{ degrees} = 90 \text{ degrees} \]

### Final Answer
The angle between the direction of the incident ray and the ray that emerges from the prism is **90 degrees**.

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