Problem 76 - University Physics with Modern Physics

Question

A layer of benzene \((n=1.50) 4.20 \mathrm{cm}\) deep floats on water \((n=1.33)\) that is 6.50 \(\mathrm{cm}\) deep. What is the apparent distance from the upper benzene surface to the bottom of the water layer when it is viewed at normal incidence?

Step-by-Step Solution

Answer

The apparent distance is 7.69 cm.

Step 1: Identify the Refractive Indices and Depths

The refractive index of benzene is \( n_1 = 1.50 \) and its depth is 4.20 cm. The refractive index of water is \( n_2 = 1.33 \) and its depth is 6.50 cm. Our goal is to find the apparent distance from the top of the benzene layer to the bottom of the water layer when viewed from above at normal incidence.

Step 2: Calculate Apparent Depth of Benzene

Using the formula for apparent depth, \( d' = \frac{d}{n} \), where \( d \) is the actual depth and \( n \) is the refractive index, find the apparent depth of the benzene layer: \[ d'_\text{benzene} = \frac{4.20 \, \text{cm}}{1.50} = 2.80 \, \text{cm} \].

Step 3: Calculate Apparent Depth of Water

Next, calculate the apparent depth of the water, as seen from the top surface of the benzene: \[ d'_\text{water} = \frac{6.50 \, \text{cm}}{1.33} \approx 4.89 \, \text{cm} \].

Step 4: Add Apparent Depths

The apparent distance from the benzene surface to the bottom of the water layer is the sum of the apparent depths of both layers. Calculate: \[ \text{Apparent distance} = d'_\text{benzene} + d'_\text{water} = 2.80 \, \text{cm} + 4.89 \, \text{cm} = 7.69 \, \text{cm} \].

Key Concepts

Refractive Index

The refractive index, often denoted as \( n \), is a measure of how much light bends when it enters a medium from another. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium.
For example, if the refractive index of a material is 1.50, it means that light travels 1.50 times slower in that medium than it does in a vacuum.
In the context of our exercise, the refractive indices are critical in determining how much light is bent when it passes through the benzene and water layers.

• The refractive index of benzene is 1.50.
• For water, it's 1.33.

Understanding these values helps us calculate the apparent depth, which we'll explore further in connection with normal incidence.

Normal Incidence

Normal incidence refers to a situation where light rays strike a surface directly at a perpendicular angle. This means they hit the surface at an angle of 90 degrees.
When light enters a medium at normal incidence, it doesn't change direction, but its speed changes according to the refractive index of that medium.
This concept simplifies the calculations of apparent depth because there's no refraction at oblique angles to account for.

• Light travels straight through each layer (e.g., from air into benzene, then into water).
• This direct passage helps us easily apply the apparent depth formula without needing to correct for angle.

In our exercise, this means that we focus mainly on calculating how light slows down in each medium rather than how it bends.

Optics

Optics is a field of physics that studies the behavior and properties of light. This includes its interactions with various media, such as refraction and reflection. When dealing with optics, understanding how light travels through different substances is crucial.
The calculation of apparent depth is an essential application in optics. It's based on how light, while passing through transparent substances like benzene and water, appears to come from different depths due to the change in speed, which is governed by the refractive index.

• The apparent depth of benzene and water is determined by the speed reduction of light in each medium.
• We use the formula \( d' = \frac{d}{n} \) to compute perceived distances where 'd' is the actual depth and 'n' is the refractive index of the medium.

These insights help provide a clearer understanding of how we perceive depth through layered liquids.

Benzene and Water Layers

Let's explore the specifics of the benzene and water layers in our exercise. These two media have different refractive indices, which affects how we perceive depth through them.
Benzene is the top layer with a refractive index of 1.50 and a depth of 4.20 cm. Water, the bottom layer, has a refractive index of 1.33 and is deeper at 6.50 cm.
When looking through these layers at normal incidence, the light's speed changes, affecting the perceived thickness of each layer.

• Benzene appears thinner than it is because light travels slower in benzene than in water.
• Water appears shallower due to a similar light-speed reduction in relation to air.

By understanding these dynamics, we can calculate the total apparent distance from the top of benzene to the bottom of water as 7.69 cm, summing up the apparent depths of both layers. This calculation illustrates how the layering of liquids impacts optical perception.