Question

You are under water in a pond and look up at the smooth surface of the water, noticing the sun in the sky. Is the sun in fact higher in the sky than it appears to you while under water, or is it lower?

Short answer

Answer: The sun appears higher in the sky than its actual position to an observer underwater due to refraction.

Step-by-step solution

Understanding refraction

Refraction is the change in direction of a wave due to a change in its speed as it passes from one medium to another. In this case, when the sun is in the sky, its light travels through the air and then enters the water. The speed of light changes at the air-water boundary, causing the light to refract. This bending of light occurs due to the difference in the index of refraction of the two mediums (air and water).

Identifying Snell's Law

Snell's Law is used to determine the angle of refraction when light passes from one medium to another. It is derived from the relationship between the angles of incidence and refraction with the indices of refraction of the incident and transmitted media. Mathematically, Snell's Law is expressed as: n_1 \times \sin{\theta}_1 = n_2 \times \sin{\theta}_2 Here, n_1 and n_2 are the indices of refraction of the two media (air and water, respectively), while \theta_1 and \theta_2 are the angles of incidence and refraction, respectively.

Comparing the indices of refraction

In this scenario, the index of refraction for air (n_1) is approximately 1, while the index of refraction for water (n_2) is approximately 1.33. Since n_2 > n_1, when light passes from air to water, it bends towards the normal (a line perpendicular to the surface at the point of incidence).

Analyzing the apparent position of the sun

As the light enters the water and bends towards the normal, the angle \theta_2 becomes smaller than the angle \theta_1. This bending of light creates the illusion that the sun is at a higher position in the sky than its actual location. Therefore, to an observer underwater, the sun appears higher in the sky than it actually is.

Conclusion

Based on the analysis of refraction and the application of Snell's Law, it can be concluded that the sun appears higher in the sky than it actually is to an observer underwater.

Key concepts

Snell's Law

When light travels from one medium to another, its path bends. The formula governing this behavior is called Snell's Law. It relates the angles of incidence and refraction to the indices of refraction of the two media involved. Snell's Law can be mathematically expressed as follows:\[ n_1 \times \sin{\theta}_1 = n_2 \times \sin{\theta}_2 \]

• \( n_1 \) and \( n_2 \) are the indices of refraction for the initial and secondary media, respectively.
• \( \theta_1 \) represents the angle of incidence, and \( \theta_2 \) represents the angle of refraction.

When light moves from air to water, it slows down due to water's higher index of refraction, causing the light path to bend. Understanding Snell's Law helps us predict how much this path will bend based on the properties of the materials.

index of refraction

The index of refraction, denoted by \( n \), tells us how much a substance can bend light. It is a crucial value for understanding refraction and comparing different materials. The higher the index, the more the light will slow down when entering the substance from air or vacuum, and the more it will bend.

• Air typically has an index of approximately 1.
• Water has a higher index, around 1.33, meaning it bends light more than air.

This difference is why light changes direction as it crosses from air into water. Calculating the bending of light at this boundary is essential for comprehending optical illusions, such as the apparent position of the sun when viewed from underwater.

angle of incidence

The angle of incidence is the angle formed between the incoming light ray and the normal. The normal is an imaginary line perpendicular to the surface at the point of contact where light enters another medium. It acts as a reference point from which both the angle of incidence and angle of refraction are measured. In the context of underwater observation, when sunlight hits the water surface, it forms an incidence angle with the normal of the surface. This angle plays a significant role in determining how the light will bend, as described by Snell's Law. The greater this angle in air (which has a lower index of refraction), the more noticeable the bending effect when light enters water.

angle of refraction

As light penetrates a new medium, it refracts at a different angle, known as the angle of refraction. This angle is measured between the refracted ray and the normal. It often differs from the angle of incidence due to the change in the light's velocity. For example, when sunlight travels from air into water, the angle of refraction is smaller than the angle of incidence. This difference arises because water has a greater index of refraction than air, causing the light to bend towards the normal. As a result, submerged observers perceive objects like the sun as higher in the sky than they truly are. This effect, explained by Snell's Law, is crucial for understanding how our perception can be distorted by changes in light's path.