Problem 4 - University Physics with Modern Physics

Question

Standing by a pool filled with water, under what condition will you see a reflection of the scenery on the opposite side through total internal reflection of the light from the scenery? a) Your eyes are level with the water. b) You observe the pool at an angle of \(41.8^{\circ}\) c) Under no condition. d) You observe the pool at an angle of \(48.2^{\circ}\)

Step-by-Step Solution

Answer

Answer: (c) Under no condition

Step 1: Determine the critical angle

First, we need to calculate the critical angle. We can do this using Snell's law: $$ n_{1} \times \sin(\theta_{1}) = n_{2} \times \sin(\theta_{2}) $$ where \(n_{1}\) and \(n_{2}\) are the refractive indices of the two media, and \(\theta_{1}\) and \(\theta_{2}\) are the angles of incidence and refraction, respectively. Here, \(n_{1}\) is the refractive index of water (approximately 1.33) and \(n_{2}\) is the refractive index of air (approximately 1). For total internal reflection to happen, \(\theta_{2}\) needs to be \(90^{\circ}\). Therefore, we can rewrite Snell's law as: $$ 1.33 \times \sin(\theta_{1c}) = 1 \times \sin(90^{\circ}) $$

Step 2: Calculate the critical angle

Solve the equation in the previous step for \(\theta_{1c}\), the critical angle: $$ \sin(\theta_{1c}) = \frac{1}{1.33} $$ $$ \theta_{1c} = \sin^{-1} \left(\frac{1}{1.33}\right) $$ $$ \theta_{1c} \approx 48.6^{\circ} $$

Step 3: Compare the critical angle to the given angles

Now that we have calculated the critical angle, let's assess each option to see if total internal reflection can occur: a) Your eyes are level with the water: In this case the incident angle is \(0^{\circ}\), which is less than the critical angle. So, total internal reflection will not occur. b) You observe the pool at an angle of \(41.8^{\circ}\): This angle is also less than the critical angle, so total internal reflection will not occur. c) Under no condition: We have not yet proven that this is true, so let's check the last option. d) You observe the pool at an angle of \(48.2^{\circ}\): This angle is slightly less than the critical angle, so total internal reflection will not occur.

Step 4: Conclusion

Since none of the given angles led to total internal reflection, we can conclude that the answer is (c) Under no condition.

Key Concepts

Optics

Optics is the branch of physics that deals with the study of light. It involves understanding how light behaves as it travels through different materials or media, and how it interacts with objects. Optics defines the foundation for numerous technologies and is essential to understanding phenomena like reflection, refraction, and diffraction.

When you look at a pool of water, it's more than just simple observation. The light travels from the scenery through the water to your eyes, bending and reflecting in specific ways as described by optical principles. These principles explain why we see reflections, refraction, and even rainbow formation through water droplets. Two significant ideas in optics are refraction and reflection, both of which are key to understanding total internal reflection.

Snell's Law

Snell's Law is fundamental in understanding how light refracts when it crosses the boundary between different media, like air and water. The law is given by:

• \( n_{1} \times \sin(\theta_{1}) = n_{2} \times \sin(\theta_{2}) \)

Where \( n_1 \) and \( n_2 \) are the refractive indices of the two media, \( \theta_1 \) is the angle of incidence, and \( \theta_2 \) is the angle of refraction. It describes how the light bends, or refracts, at the interface. If \( n_1 \) is greater than \( n_2 \), the light will bend away from the normal as it passes into the less dense medium.

For the exercise, you need Snell's law to find the critical angle, the specific angle at which total internal reflection occurs. This law helps mathematicians and physicists calculate how different angles influence the path of light, vital for applications such as lenses, prisms, and even eyeglasses.

Critical Angle

The critical angle is a unique concept in the study of optics, especially when it comes to total internal reflection. It is the angle of incidence beyond which light cannot pass through the boundary and is instead reflected back entirely inside the medium. For water (refractive index of 1.33) to air (refractive index of 1), this critical angle is approximately \( 48.6^{\circ} \).

Understanding the critical angle is crucial because it marks the transition from refraction to total internal reflection. Below this angle, light exits the medium and enters the adjacent one. Beyond this angle, light reflects back, which can be useful in fiber optics, where light needs to be guided through long lengths of optical fibers without escaping.

In your exercise, calculating the critical angle allows you to determine if any of the given observation angles would cause total internal reflection.

Refractive Index

The refractive index is a key factor in determining how light propagates through a medium. It is a measure of how much the speed of light is reduced inside the medium compared to vacuum. It is represented as:

• \( n = \frac{c}{v} \)

where \( c \) is the speed of light in vacuum, and \( v \) is the speed of light in the medium.

Different materials have different refractive indices, affecting how much light bends when entering from one medium to another. For instance, the refractive index of air is approximately 1, meaning light travels almost as fast through air as through vacuum, while water has a refractive index of about 1.33, meaning light slows down substantially.

Understanding refractive indices helps predict how much light will bend or reflect when it hits a new material. This is critical for the analysis in the exercise, where knowing these indices enables the calculation of the critical angle and determining conditions for total internal reflection.