Problem 25 - College Physics

Question

A speck of dirt is embedded 3.50 \(\mathrm{cm}\) below the surface of a sheet of ice having a refractive index of \(1.309 .\) What is the apparent depth of the speck, when viewed from directly above?

Step-by-Step Solution

Answer

The apparent depth is approximately 2.67 cm.

Step 1: Understanding Refraction and Apparent Depth

When light travels from a denser medium (ice) to a less dense medium (air), it bends away from the normal. Due to this bending, the object appears to be at a different depth than it actually is. This is known as apparent depth.

Step 2: Apply the Apparent Depth Formula

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

Step 3: Substitute Known Values

Given that the actual depth \(d\) is \(3.50\) cm and the refractive index \(n\) is \(1.309\), substitute these values into the formula:\[ d_a = \frac{3.50}{1.309} \]

Step 4: Calculate the Apparent Depth

Perform the division to find the apparent depth:\[ d_a = \frac{3.50}{1.309} \approx 2.67 \text{ cm} \]Therefore, the apparent depth of the speck when viewed from directly above is approximately \(2.67\) cm.

Key Concepts

Refractive Index

The refractive index is a crucial concept in understanding how light behaves when it travels from one medium to another. It is a dimensionless number that describes how much a medium slows down the passage of light compared to a vacuum. The higher the refractive index, the more the medium slows down the light. For example, ice has a refractive index of about 1.309, which means light travels slower in ice compared to air, which has a refractive index close to 1.
The refractive index is used to determine the bending of light, also known as refraction, when it enters a different medium. This bending is quantified using Snell's Law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equivalent to the ratio of the refractive indices of the two media.
Understanding refractive index helps us predict how much the path of light will bend as it enters mediums like water, glass, or in our example, ice.

Light Refraction

Light refraction is the bending of light as it passes from one medium into another with a different refractive index. This phenomenon is why objects appear displaced or distorted when viewed through water or glass. When light enters a denser medium at an angle, it slows down and bends towards the normal (an imaginary line perpendicular to the surface). Conversely, when light exits a denser medium into a less dense one, it speeds up and bends away from the normal.
The degree of bending depends on the refractive index of the two media. The denser the medium, the more the light will bend. This is why ice, with a refractive index of 1.309, alters the path of light differently than air.
Refraction is responsible for many optical phenomena, including the apparent depth of objects in a medium and the colorful displays seen in phenomena like rainbows.

Medium Density

Medium density refers to the compactness of particles within a given material and is directly linked to its refractive index. In general, the denser a medium is, the higher its refractive index, because more closely packed particles slow down the passage of light more.
Ice, being a solid form of water, has particles more tightly packed compared to liquid water. This is why it has a different refractive index and affects light differently.
Medium density influences not only refraction but also other optical properties like reflection and absorption, which explains why different materials have diverse optical behaviors.

Depth Calculation

Depth calculation involves determining the apparent versus actual depth of an object when viewed through a medium. This is particularly useful in scenarios where light refraction causes illusions of displacement.
Using the formula for apparent depth: \[ d_a = \frac{d}{n} \] where \(d_a\) is the apparent depth, \(d\) is the actual depth, and \(n\) is the refractive index, one can determine how objects submerged in a medium like water or ice appear shallower than they actually are.
For example, if an object is embedded 3.50 cm within a piece of ice with a refractive index of 1.309, the apparent depth can be calculated as:\[ d_a = \frac{3.50}{1.309} \approx 2.67 \text{ cm} \]This calculation helps in various practical applications, from designing optical instruments to understanding natural optical phenomena.