An optical system whose cross section is shown below is constructed from two different glass isosceles wedges (each with 30 degrees - 75 degrees - 75 degrees cross section). The refractive indices of two glasses are √3 and √2 respectively. A light beam is incident at an angle of 60 degree on face AB. The angle of emergence from the face CD is?

To determine the angle of emergence from face CD of the optical system you've described, we need to analyze the light beam's behavior as it passes through the two different glass wedges. Let's break this down step by step.

Understanding the Geometry

We have two isosceles wedges, each with a cross-section of 30 degrees at the apex and 75 degrees at the base. The refractive indices of the glasses are given as √3 (for the first wedge) and √2 (for the second wedge). The light beam is incident at an angle of 60 degrees on face AB.

Step 1: Refraction at Face AB

When the light beam hits the first wedge (with refractive index √3), we can use Snell's Law to find the angle of refraction. Snell's Law states:

• n₁ * sin(θ₁) = n₂ * sin(θ₂)

Here, n₁ is the refractive index of air (approximately 1), θ₁ is the angle of incidence (60 degrees), n₂ is the refractive index of the first glass (√3), and θ₂ is the angle of refraction we want to find.

Plugging in the values:

• 1 * sin(60°) = √3 * sin(θ₂)

Calculating sin(60°) gives us √3/2, so we have:

• √3/2 = √3 * sin(θ₂)

Dividing both sides by √3:

• 1/2 = sin(θ₂)

This results in:

• θ₂ = 30 degrees

Step 2: Path Through the First Wedge

Now, the light beam travels through the first wedge. Since the wedge has a 30-degree apex angle, the beam will exit at an angle of:

• 30 degrees (angle of refraction) + 15 degrees (half of the wedge angle) = 45 degrees

Step 3: Refraction at Face CD

Next, the light beam enters the second wedge with a refractive index of √2. The angle of incidence at face CD is now 45 degrees. We apply Snell's Law again:

• n₁ * sin(θ₁) = n₂ * sin(θ₂)

In this case, n₁ is √2, θ₁ is 45 degrees, and n₂ is 1 (for air). Thus:

• √2 * sin(45°) = 1 * sin(θ₂)

Since sin(45°) is √2/2, we have:

• √2 * (√2/2) = sin(θ₂)

This simplifies to:

• 1 = sin(θ₂)

Therefore, θ₂ = 90 degrees.

Final Result

The angle of emergence from face CD is 90 degrees. This means that the light exits the second wedge perpendicularly to the surface, which is quite interesting as it indicates a total internal reflection scenario at the interface of the second wedge and air.

In summary, by applying Snell's Law at each interface and considering the geometry of the wedges, we can accurately determine the angle of emergence of the light beam from the optical system.