A light beam, passes from medium 1 to medium 2. Show that the emerging beam is parallel to the incident beam.

Explanation

To show that the emerging beam is parallel to the incident beam when a light beam passes from medium 1 to medium 2, we will use Snell's Law. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have two media: medium 1 (with refractive index μ1) and medium 2 (with refractive index μ2). A light beam is incident on the boundary between these two media at an angle i1.

### Step 2: Apply Snell's Law at the First Interface
According to Snell's Law, when light passes from one medium to another, the relationship between the angles of incidence and refraction is given by:
\[ \mu_1 \sin(i_1) = \mu_2 \sin(r_1) \]
where:
- i1 is the angle of incidence in medium 1,
- r1 is the angle of refraction in medium 2.

### Step 3: Apply Snell's Law at the Second Interface
Now, when the light beam exits from medium 2 back into medium 1, we again apply Snell's Law:
\[ \mu_2 \sin(r_1) = \mu_1 \sin(i_2) \]
where:
- i2 is the angle of incidence at the second interface (which is equal to the angle of emergence).

### Step 4: Relate the Angles
From the two equations derived from Snell's Law, we can express r1 in terms of i1:
1. From the first equation:
\[ \sin(r_1) = \frac{\mu_1}{\mu_2} \sin(i_1) \]
2. Substitute this into the second equation:
\[ \mu_2 \left(\frac{\mu_1}{\mu_2} \sin(i_1)\right) = \mu_1 \sin(i_2) \]
This simplifies to:
\[ \sin(i_1) = \sin(i_2) \]

### Step 5: Conclude that i1 = i2
Since the sine function is equal for the same angles, we conclude that:
\[ i_1 = i_2 \]
This means that the angle of incidence at the first interface is equal to the angle of emergence at the second interface.

### Step 6: Conclusion
Since both the incident ray and the emergent ray make the same angle (i1) with the normal, the emergent beam is parallel to the incident beam.

### Final Statement
Thus, we have shown that the emerging beam is parallel to the incident beam when light passes from medium 1 to medium 2.

---