Problem 21 - University Physics with Modern Physics

Question

A ray of light is traveling in a glass cube that is totally immersed in water. You find that if the ray is incident on the glass-water interface at an angle to the normal larger than \(48.7^{\circ}\) , no light is refracted into the water. What is the refractive index of the glass?

Step-by-Step Solution

Answer

The refractive index of glass is approximately 1.77.

Step 1: Understand the Critical Angle Concept

When a ray of light passes from a medium with higher refractive index to a medium with a lower refractive index, total internal reflection occurs if the angle of incidence is greater than the critical angle. In this case, since no light is refracted into water beyond 48.7°, this is the critical angle ( abla_c").

Step 2: Apply Snell’s Law

Snell's Law relates the refractive indices of two media and the sine of the angle of incidence and refraction: \[ n_1 imes \sin(\theta_1) = n_2 imes \sin(\theta_2) \]For the critical angle, \( \theta_2 = 90^\circ \) because the refracted ray travels along the boundary. So \( \sin(\theta_2) = 1 \).

Step 3: Rearrange Snell’s Law for Critical Angle

Using the critical angle, Snell's Law becomes \[ n_1 \times \sin(\theta_c) = n_2 \]where \( n_1 \) is the refractive index of glass, \( \theta_c = 48.7^\circ \), and \( n_2 = 1.333 \) (the refractive index of water).

Step 4: Calculate Refractive Index of Glass

Rearrange the equation to find \( n_1 \) (the refractive index of the glass):\[ n_1 = \frac{n_2}{\sin(\theta_c)} \]Substitute \( n_2 = 1.333 \) and \( \sin(48.7^\circ) \) into the equation:\[ n_1 = \frac{1.333}{\sin(48.7^\circ)} \approx 1.77 \].

Step 5: Final Step: Verify Calculation

Calculate \( \sin(48.7^\circ) \) which is approximately 0.752, and then verify the refractive index calculation:\[ n_1 \approx \frac{1.333}{0.752} \approx 1.77 \]This confirms the refractive index of glass is approximately 1.77.

Key Concepts

Snell's Law

Snell's Law is a fundamental principle that governs the behavior of light as it travels between different media. Imagine a ray of light moving from one medium into another. Snell's Law helps us calculate how much the light ray bends, changing its direction, at the boundary between the two media.

The law is mathematically expressed as: \[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \] where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the first and second medium, respectively.
• \( \theta_1 \) is the angle of incidence, the angle between the incoming ray and the normal to the surface.
• \( \theta_2 \) is the angle of refraction, the angle between the refracted ray and the normal.

When light passes from a denser medium like glass into a less dense medium like water, it bends away from the normal if \(n_1 > n_2\). Snell's Law is essential for understanding phenomena like lenses, prisms, and fiber optics.

Always remember, when applying Snell's Law, the sine of the angles is used, and this law only applies when light passes from one medium into another, not when it reflects.

critical angle

The critical angle is a special angle of incidence that occurs when light travels from a medium with a higher refractive index to one with a lower refractive index. Above this angle, total internal reflection occurs, meaning that all the light is reflected back into the initial medium rather than passing through the interface.

Understanding this is fundamental for applications in fiber optic cables where light needs to be confined inside a core.

• The critical angle \( \theta_c \) can be found when the angle of refraction \( \theta_2 \) is \( 90^{\circ} \). At this point, the refracted ray runs along the surface of the two media.
• Using Snell's Law, when the refracted angle is \( 90^{\circ} \), we have: \[ n_1 \sin(\theta_c) = n_2 \sin(90^{\circ}) \] which simplifies to \[ n_1 \sin(\theta_c) = n_2 \]
• Thus, \( \theta_c = \arcsin\left( \frac{n_2}{n_1} \right) \).

In scenarios like the exercise, when no light is refracted beyond the critical angle, it results in total internal reflection, a concept crucial in the design of many optical devices.

total internal reflection

Total internal reflection is an optical phenomenon that happens when light traveling through a medium hits the boundary with another medium at an angle greater than the critical angle. When this condition is met, none of the light crosses the boundary; instead, all of it reflects back into the original medium.

Let's explore the key factors:

• Occurs when light moves from a more dense medium, like glass, into a less dense medium, like air or water.
• Requires the angle of incidence to be greater than the critical angle \( \theta_c \).
• Applications of total internal reflection include optical fibers, where light signals are confined within the core to transmit data efficiently over long distances without loss.

This principle is beneficial in directing light as it prevents light from escaping, allowing it to "bounce" along the walls of the medium. Understanding total internal reflection helps in designing efficient lighting systems, enhancing the performance of optical instruments, and crucially in enhancing technologies like endoscopes and telecommunications.