Problem 21 - Physics

Question

A small logo is embedded in a thick block of crown glass \((n=1.52), 3.20 \mathrm{~cm}\) beneath the top surface of the glass. The block is put under water, so there is \(1.50 \mathrm{~cm}\) of water above the top surface of the block. The logo is viewed from directly above by an observer in air. How far beneath the top surface of the water does the logo appear to be?

Step-by-Step Solution

Answer

The logo appears to be 4.30 cm beneath the water surface.

Step 1: Understand the Problem

We have a logo embedded in a crown glass block under water. Observers look from above in air. We need to find out how far beneath the water surface the logo appears to be.

Step 2: Define Refraction and Apparent Depth

Refraction causes objects to appear closer than their actual depth when viewed across boundaries between different media. The apparent depth \(d'\) of an object viewed through a medium like water or glass can be calculated using the formula: \(d' = \frac{d}{n_{rel}}\), where \(d\) is the actual depth and \(n_{rel}\) is the relative refractive index of the medium with respect to the medium above it.

Step 3: Calculate Apparent Depth in Glass

First, calculate the apparent depth due to the glass alone. Given glass refractive index \(n_1 = 1.52\) and refractive index of water \(n_2 = 1.33\), we use the formula: \(d'_g = \frac{d}{n_1/n_2}\). Here, \(d = 3.20 \text{ cm}\).

Step 4: Apparent Depth Calculation Formula for Glass

Calculate: \[d'_g = \frac{3.20}{1.52/1.33} = \frac{3.20 \times 1.33}{1.52}\]Perform the multiplication and division to find the apparent depth in the glass.

Step 5: Apparent Depth in Glass

After calculating: \[d'_g = 2.80 \text{ cm}\]This means within the glass, due to refraction, the logo appears to be at 2.80 cm from the top of the glass.

Step 6: Calculate Overall Apparent Depth

Next, consider the 1.50 cm of water added above the glass. The apparent depth due to water needs to be added to the apparent depth in the glass. Since water to air interface doesn't affect depth significantly, add the water depth.

Step 7: Overall Apparent Depth Calculation

The total apparent depth from the surface of the water is:\[d' = 2.80\text{ cm (apparent depth in glass)} + 1.50 \text{ cm (water layer)}\]Thus, \(d' = 2.80 + 1.50 = 4.30\text{ cm}\).

Step 8: Result Interpretation

The logo appears to be 4.30 cm beneath the water surface when viewed from above. This accounts for the refraction effects through both the water and glass surfaces.

Key Concepts

apparent depth

When viewing objects immersed in liquids or solids like water and glass, they often appear at different depths than they actually are. This phenomenon is known as apparent depth, which results from light bending, or refracting, as it moves through different media.

Apparent depth is an optical illusion where objects appear closer to the surface than they are. It's calculated using a special formula where the object's actual depth is divided by the relative refractive index of the media involved: - Apparent depth ( d' ) = Actual depth ( d ) / Relative refractive index ( n_{rel} ).

This formula helps in understanding how light's path alteration affects our perception of depth. You can see this in regular activities, like noticing how a straw in a glass of water seems bent at the surface.

crown glass

Crown glass is a type of glass often used in optics, like lenses and prisms, due to its clarity and stability in refractive properties. It is a common medium for studying refraction because of its well-defined optical characteristics.

The refractive index of crown glass is typically around 1.52, signifying how much it bends light compared to a vacuum (which has a refractive index of 1). This quality makes it suitable for applications involving light transmission and focusing.

For educational purposes as seen with the embedded logo scenario, crown glass is perfect for demonstrating principles of light and refraction, teaching students about how light behavior changes across materials.

refractive index

Refractive index is a measure of how much light bends, or refracts, when entering a medium from another. It's a crucial concept in optics, determining the speed at which light travels through different materials.

A medium with a refractive index greater than 1 indicates light will slow down and bend towards the normal (an imaginary line perpendicular to the surface) as it enters that medium from air. Conversely, as light exits into a medium with a lower refractive index, it speeds up and bends away from the normal.

• The refractive index of water is approximately 1.33.
• Crown glass, for instance, has a refractive index of 1.52.

Understanding refractive index helps in calculating apparent depths and is pivotal in disciplines like physics and engineering.

relative refractive index

The relative refractive index helps describe how light transitions between two different media, such as from water to glass. It is calculated as the ratio of the refractive index of one medium with respect to another.

Mathematically, n_{rel} = n_1/ n_2, where n_1 is the refractive index of the first medium (e.g., glass) and n_2 is that of the second medium (e.g., water). This factor is essential for determining apparent depth when an object is viewed through multiple layers.

Using this ratio, students can better understand light's behavior in complex systems where multiple media are involved, like the crown glass problem where the apparent depth of an object under water needs taking into account both media effects.