Problem 17 - University Physics with Modern Physics

Question

A speck of dirt is embedded 3.50 cm below the surface of a sheet of ice \((n = 1.3092)\). What is its apparent depth when viewed at normal incidence?

Step-by-Step Solution

Answer

The apparent depth is approximately 2.67 cm.

Step 1: Understand the Problem

We are asked to find the apparent depth of a speck of dirt embedded 3.50 cm below the surface of ice with a refractive index of 1.3092. This involves understanding how light refracts when it changes medium.

Step 2: Use Snell's Law for Normal Incidence

For normal incidence, that is, when light travels perpendicularly to the surface, Snell's Law simplifies, because the angle of incidence is 0 degrees. Thus, there is no deviation in the path of light due to refraction at the surface.

Step 3: Apply the Apparent Depth Formula

The apparent depth \((d')\) can be calculated using the formula: \( d' = \frac{d}{n} \), where \((d)\) is the actual depth and \((n)\) is the refractive index of the medium (ice).

Step 4: Calculate Apparent Depth

Given \((d = 3.50 \, \text{cm})\) and \((n = 1.3092)\), plug these values into the formula to find the apparent depth: \[d' = \frac{3.50}{1.3092} \]This results in \((d' \approx 2.67 \, \text{cm})\).

Key Concepts

Snell's Law

Snell's Law is fundamental in understanding how light behaves when it passes between different media. This law relates the angles and refractive indices of two media when light is refracted. The standard form of Snell's Law is given by the equation: \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \) where \( n_1 \) and \( n_2 \) are the refractive indices, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively. In simpler terms, Snell's Law tells us that the direction light takes through different substances changes depending on their refractive index. At normal incidence: When light hits a surface straight on (perpendicular), Snell’s Law simplifies. Because the angle of incidence is zero degrees, the light does not bend and continues straight through.

Apparent Depth

Apparent depth is an optical illusion that occurs due to the refraction of light. When we view an object submerged in a transparent medium, it often appears shallower than it really is. This happens because light bends as it travels through mediums with different refractive indices.The formula to calculate apparent depth is: \( d' = \frac{d}{n} \) - \( d' \): apparent depth - \( d \): actual depth - \( n \): refractive index of the mediumApparent depth is important in fields like physics and engineering, especially when precise measurements are needed through mediums that cause refraction. In the exercise, the actual depth of the speck of dirt was 3.50 cm, but its apparent depth was calculated to be approximately 2.67 cm due to the refractive properties of the ice.

Optics

Optics is the branch of physics that studies light, its properties, and its interaction with matter. Light behaves in fascinating ways that allow it to be bent, focused, or split into colors. Understanding optics helps us comprehend and manipulate light for various technological and scientific purposes. Main optics concepts include:

• Reflection: The bouncing back of light when it hits a surface.
• Refraction: The bending of light as it passes through different media.
• Dispersion: The splitting of light into its component colors.

These concepts are used in a wide range of applications, from simple lenses in glasses to complex scientific instruments like telescopes and microscopes. By mastering optics, we are able to create technologies that enhance our ability to observe the world.

Light Refraction

Light refraction is a phenomenon that occurs when light travels from one medium to another, such as from air to water. This change in medium causes the light to bend, changing its speed and direction. Refraction is a key concept in understanding how lenses work and why objects appear distorted underwater. Factors influencing refraction include:

• Refractive Index: Different materials slow down light to varying degrees. This index quantifies that effect, impacting the angle and extent of bending.
• Wavelength: Different colors of light bend differently. For example, blue light bends more than red light.

Refraction is beautifully demonstrated in everyday experiences, such as when a straw appears bent in a glass of water or when the sun appears higher in the sky as it sets. Understanding refraction allows us to craft solutions and technologies, like corrective lenses for vision improvement and intricate optical devices.