A Ray Of Light Strikes A Flat 200 Cm Thick Block Of Glass

A Ray Of Light Strikes A Flat 200 Cm Thick Block Of Glass N150 A

A ray of light strikes a flat, 2.00-cm-thick block of glass (n=1.50) at an angle of 30 degrees with respect to the normal. (a) Find the angle of refraction at the top surface. (b) Find the angle of incidence at the bottom surface and the refracted angle. (c) Find the lateral distance by which the beam is shifted. (d) Calculate the speed of light in the glass and (e) the time required for the light to pass through the glass block. (f) Is the travel time through the block affected by the angle of incidence? Explain.

Paper For Above instruction

The behavior of light as it passes through different media is a fundamental aspect of optics, governed primarily by the laws of refraction. When a light beam encounters a boundary between two media with different refractive indices, its speed and direction change according to Snell's Law. This analysis explores the refraction of a light beam passing through a glass block with a refractive index of 1.50, focusing on the angles involved, the lateral shift, and the time taken for the light to traverse the medium.

Refraction at the Top Surface

Initially, the incident light strikes the surface of the glass at an angle of 30 degrees with respect to the normal. Using Snell's Law, which states that n₁ sin θ₁ = n₂ sin θ₂, where n₁ is the refractive index of the incident medium (air, with n₁ ≈ 1.00), and n₂ is that of the glass (n=1.50), we can find the refraction angle inside the glass:

sin θ₂ = (n₁ / n₂) × sin θ₁ = (1.00 / 1.50) × sin 30° = (2/3) × 0.5 = 1/3 ≈ 0.333

Therefore, θ₂ = arcsin(0.333) ≈ 19.47°

Angle of Incidence at the Bottom Surface and Refraction

Since the light travels through the glass at an angle of approximately 19.47°, the incident angle at the bottom surface is the same due to the parallel surfaces of the block, meaning the incident angle inside the glass remains at approximately 19.47°. When the light reaches the bottom surface, the incident angle with respect to the normal inside the glass remains at 19.47°, and upon refraction back into air, the angle can be found similarly:

sin θ₃ = n₂ / n₁ × sin θ₂ = 1.50 / 1.00 × sin 19.47° ≈ 1.50 × 0.333 ≈ 0.5

Since sin θ₃ cannot be greater than 1, and in fact, it's exactly 0.5, the refracted angle in air is:

θ₃ = arcsin(0.5) = 30°

This confirms that the light exits the glass at the original angle of 30°, consistent with the law of reversibility of light paths in parallel-sided slabs.

Lateral Shift of the Beam

The lateral displacement, often called the lateral shift (d), occurs because the beam passes through the slab at an angle, causing it to emerge shifted laterally from its original path. The formula for lateral shift in a slab of thickness t, with an incident angle θ₁ and refraction angle θ₂, is:

d = t × (sin(θ₁ - θ₂) / cos θ₂)

Given t = 2.00 cm, θ₁ = 30°, and θ₂ ≈ 19.47°, we compute:

sin(30° - 19.47°) ≈ sin(10.53°) ≈ 0.183

cos 19.47° ≈ 0.943

d = 2.00 cm × (0.183 / 0.943) ≈ 2.00 cm × 0.194 ≈ 0.388 cm

Hence, the beam shifts laterally by approximately 0.39 cm as it passes through the glass slab.

Speed of Light in Glass

The speed of light in a medium is given by:

v = c / n

where c ≈ 3 × 10^8 m/s is the speed of light in vacuum, and n is the refractive index. Therefore:

v = (3 × 10^8 m/s) / 1.50 = 2 × 10^8 m/s

Time Required for Light to Pass Through the Glass Block

Given the thickness t = 2.00 cm = 0.02 m, and the speed v = 2 × 10^8 m/s, the transit time t_time is:

t_time = t / v = 0.02 m / 2 × 10^8 m/s = 1 × 10^-10 s

Effect of Incidence Angle on Travel Time

In a uniform, non-dispersive medium, the transit time through a slab depends primarily on the optical path length and the speed of light within the medium. When light enters at an angle, the geometrical path length increases compared to the thickness measured perpendicular to the surface. Thus, the actual optical path length (L) is:

L = t / cos θ₂ ≈ 0.02 m / 0.943 ≈ 0.0212 m

The time is then:

t_actual = L / v ≈ 0.0212 m / 2 × 10^8 m/s ≈ 1.06 × 10^-10 s

This reveals that the travel time slightly increases with the incidence angle due to the increased path length. Hence, the time is affected by the angle of incidence, increasing as the angle becomes steeper.

Conclusion

The refraction of light through a glass slab involves changes in direction and speed governed by Snell's Law. The angles of incidence and refraction are interconnected, and the phenomenon results in a lateral shift of the beam and an increase in the actual path length depending on the incident angle. The transit time is affected by these factors, emphasizing the importance of considering geometrical and optical properties in practical applications such as optical fibers, lenses, and other photonics devices.

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