Problem 101 - University Physics with Modern Physics

Question

A glass plate 3.50 \(\mathrm{cm}\) thick, with an index of refraction of 1.55 and plane parallel faces, is held with its faces horizontal and its lower face 6.00 \(\mathrm{cm}\) above a printed page. Find the position of the image of the page formed by rays making a small angle with the normal to the plate.

Step-by-Step Solution

Answer

The image of the page is formed approximately 8.26 cm from the top surface of the glass plate.

Step 1: Understanding the Problem

We are given a glass plate with a thickness of 3.50 cm and a refractive index of 1.55. We need to find the position of the image of a page located 6 cm below the plate when viewed through this glass plate.

Step 2: Calculate Apparent Thickness of the Glass

When light passes through a medium with a refractive index greater than 1, objects appear closer than they are. The apparent thickness, \( t' \), can be calculated using the formula: \[ t' = \frac{t}{n} \] where \( t \) is the actual thickness (3.50 cm) and \( n \) is the refractive index (1.55). Therefore, \[ t' = \frac{3.50}{1.55} \approx 2.26 \text{ cm} \]

Step 3: Determine the Position of the Image

The image of the page will appear at a distance equal to the apparent thickness of the glass plus the distance from the bottom of the glass to the page. This gives: \[ ext{Image position} = t' + 6.00 \text{ cm} \approx 2.26 + 6.00 \text{ cm} = 8.26 \text{ cm} \] from the top surface of the glass.

Key Concepts

Refractive Index

The refractive index is a measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum. It is expressed as:

• n = \( \frac{c}{v} \)

where \( c \) is the speed of light in a vacuum and \( v \) is the speed of light in the medium. The refractive index describes how light bends when it enters a new medium.
The higher the refractive index, the more the light is slowed down in the medium.
In the given exercise, the glass plate has a refractive index of 1.55, meaning light travels slower in the glass than in air. Knowing the refractive index helps calculate the apparent thickness of the glass, making objects viewed through it appear closer.

Apparent Depth

Apparent depth is a phenomenon observed when light passes through mediums with different refractive indices. An object immersed in a medium appears to be at a different position than it actually is due to the bending of light rays.
The apparent depth, \( t' \), is calculated by adjusting the actual depth \( t \) with the refractive index \( n \):

• \( t' = \frac{t}{n} \)

Using the formula, we adjust for how objects appear shallower or deeper due to refraction.
In the exercise, the glass's actual thickness is 3.50 cm, but it appears to be about 2.26 cm thick when viewed through a medium with n = 1.55.
This concept explains why the printed page beneath the glass appears closer to the observer.

Geometrical Optics

Geometrical optics focuses on the principles governing the behavior and propagation of light rays. It uses geometry to analyze how light interacts with objects, especially in terms of reflection and refraction.
When light rays travel through a glass plate with a known refractive index, they bend upon entering and leaving the glass.
This bending causes optical illusions such as the apparent depth.
In our task, geometric principles are used to determine the position of the image of a printed page under a glass plate.

• Light entering the glass slows down, bending towards the normal.
• Upon exiting, it speeds up, bending away from the normal.

This results in the apparent shift of the page's image. Understanding geometrical optics provides insight into everyday phenomena like the way objects look when viewed through glasses or submerged in water.