Question

If the critical angle for total internal reflection from a medium to vacuum is \(30^{\circ}\) then velocity of light in the medium is \(-\mathrm{ms}^{-1}\left(\right.\) take \(\left.\mathrm{c}=3.0 \times 10^{8} \mathrm{~ms}^{-1}\right)\) (A) \(2.0 \times 10^{8}\) (B) \(1.5 \times 10^{8}\) (C) \(10^{8}\) (D) \(1.5 \times 10^{-8}\)

Short answer

The velocity of light in the medium is \(1.5 \times 10^8 ms^{-1}\). The correct answer is (B).

Step-by-step solution

Find the refractive index of the medium

Given the critical angle for total internal reflection is 30 degrees, we can use Snell's law to find the refractive index of the medium: \[ n = \frac{\sin{\theta_2}}{\sin{\theta_1}} \] Here, \(\theta_1\) is the critical angle (30°) and \(\theta_2\) is the angle at which the refracted light ray will be at 90° (perpendicular to the interface). Plug the values: \[ n = \frac{\sin{(90^\circ)}}{\sin{(30^\circ)}} \] \[ n = \frac{1}{\frac{1}{2}} \] \[ n = 2 \]

Find the velocity of light in the medium

Now that we have the refractive index, we can find the velocity of light in the medium using the formula: \[ v = \frac{c}{n} \] Plug in the given speed of light in vacuum, c = \(3.0 \times 10^8 ms^{-1}\), and the refractive index, n = 2: \[ v = \frac{3.0 \times 10^8}{2} \] \[ v = 1.5 \times 10^8 ms^{-1} \] So, the velocity of light in the medium is \(1.5 \times 10^8 ms^{-1}\). The correct answer is (B).

Key concepts

Critical Angle

The critical angle is a key concept linked to the phenomenon of total internal reflection. When light travels from a denser medium to a less dense medium, like glass to air, there's a specific angle of incidence where the refracted light ray skims along the boundary without passing into the second medium. This specific angle is known as the critical angle.

• At any angle below the critical angle, light will both reflect and refract.
• At the critical angle, light refracts at 90°, traveling along the interface.
• Beyond the critical angle, all light is internally reflected, and no refraction occurs.

This optical behavior is crucial in the design of optical fibers and other technologies that rely on reflecting light.

Refractive Index

The refractive index, denoted as 'n', measures how much a ray of light bends when it enters a material from another medium. It's a dimensionless number that states how fast light travels through a substance relative to vacuum.

• The refractive index of a medium is calculated using: \[ n = \frac{\text{Speed of light in vacuum}}{\text{Speed of light in the medium}} \]
• A higher refractive index means light travels slower in that medium.
• In the given problem, the refractive index is determined using the sine of the critical angle, where \( n = \frac{1}{\sin(\theta_\text{c})} \).

Understanding this concept is vital for calculating the speed of light in various materials, which is an everyday necessity in optics.

Velocity of Light

The velocity of light refers to how fast light travels through a medium. In a vacuum, light reaches its maximum speed, which is \( 3.0 \times 10^8 \mathrm{~ms}^{-1} \). However, this speed reduces when light travels through different media, such as glass or water.

• The velocity of light in a medium can be found using: \[ v = \frac{c}{n} \]
• Here, \( c \) is the speed of light in vacuum, and \( n \) is the refractive index of the medium.
• The velocity decreases because photons are delayed by interactions with the atoms in the medium.

These principles are crucial when designing lenses or fiber optics, where precise control over light speed is essential.

Snell's Law

Snell's Law forms the basis for understanding refraction, describing how light bends as it passes between different media. Expressed mathematically, this law relates the angles of incidence and refraction to the refractive indices of the two media.

• Snell's Law is represented as: \[ n_1 \sin{\theta_1} = n_2 \sin{\theta_2} \]
• Where \( n_1 \) and \( n_2 \) are the refractive indices of the respective media, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction.
• This law is crucial for determining how light behaves at the boundary of two media, explaining why lenses focus light and how prisms disperse it.

Understanding Snell's Law is essential for fields like optics, ensuring accurate design and function of instruments reliant on light manipulation.