Problem 43 - University Physics with Modern Physics

Question

A glass plate 2.50 mm thick, with an index of refraction of 1.40, is placed between a point source of light with wavelength 540 nm (in vacuum) and a screen. The distance from source to screen is 1.80 cm. How many wavelengths are there between the source and the screen?

Step-by-Step Solution

Answer

9,349 wavelengths are between the source and the screen.

Step 1: Convert Given Units

Convert the thickness of the glass plate from millimeters to centimeters. We know that 1 mm = 0.1 cm, so 2.50 mm = 0.25 cm.

Step 2: Calculate the Wavelength in Glass

In a medium other than vacuum, the wavelength changes. Use the formula \( \lambda_m = \frac{\lambda_0}{n} \), where \( \lambda_0 = 540 \) nm and \( n = 1.40 \). Thus, \( \lambda_m = \frac{540 \text{ nm}}{1.40} \approx 385.7 \text{ nm} \). Convert this to centimeters: \( 385.7 \text{ nm} \approx 3.857 \times 10^{-5} \text{ cm} \).

Step 3: Determine the Number of Wavelengths in Glass

Using the wavelength in the glass, calculate how many wavelengths fit in the 0.25 cm thickness of the glass. \( N_{glass} = \frac{0.25 \text{ cm}}{3.857 \times 10^{-5} \text{ cm}} \approx 6479 \text{ wavelengths} \).

Step 4: Determine the Number of Wavelengths in Air

Calculate the number of wavelengths corresponding to the remaining distance (1.80 cm - 0.25 cm = 1.55 cm) using \( \lambda_0 = 540 \) nm, which equals \( 5.40 \times 10^{-5} \) cm. Therefore, \( N_{air} = \frac{1.55 \text{ cm}}{5.40 \times 10^{-5} \text{ cm}} \approx 2870 \text{ wavelengths}. \)

Step 5: Total Number of Wavelengths

Add the wavelengths from the path through the glass and the path through the air to find the total number of wavelengths: \( N_{total} = N_{glass} + N_{air} = 6479 + 2870 = 9349 \text{ wavelengths}. \)

Step 6: Conclusion: Result Interpretation

The total number of wavelengths between the source and the screen is the sum of wavelengths in the glass and air segments. This gives us the comprehensive optical distance in terms of the original light wavelength.

Key Concepts

Index of Refraction

The index of refraction, often denoted as \( n \), is a measure of how much light slows down in a medium compared to its speed in a vacuum. This concept is crucial in wave optics because it influences how light behaves as it travels through different materials. Here's how it works:

• In a vacuum, light travels at its maximum speed, approximately \( 3 \times 10^8 \) meters per second.
• When light enters a medium, like glass or water, it slows down, resulting in a lower speed.
• The index of refraction is calculated as the ratio of the speed of light in a vacuum to the speed of light in the medium: \( n = \frac{c}{v} \).

For example, if the index of refraction of glass is 1.40, this means light travels 1.40 times slower in glass than in a vacuum. This slowing effect alters not only the speed but also the path and, crucially, the wavelength of light in the medium.

Wavelength in Medium

Wavelength refers to the distance between consecutive crests or troughs of a wave. In optics, the wavelength of light is a fundamental property. However, it is not always constant; it changes when light moves through different materials.The wavelength in a medium is given by \( \lambda_m = \frac{\lambda_0}{n} \), where \( \lambda_0 \) is the wavelength in a vacuum, and \( n \) is the index of refraction.

• If the vacuum wavelength is 540 nm and the index of refraction of the glass is 1.40, the wavelength in the glass becomes approximately 385.7 nm.
• This change is due to the fact that the speed of light is reduced in the medium, causing the wave “compression.”

This altered wavelength is essential for calculations involving phase and optical path length, allowing us to determine how the wave propagates through the medium.

Optical Path Length

Optical path length (OPL) is a concept that helps us understand how light travels in different media. It combines the physical distance the light travels with the medium's effect on light. The formula for optical path length is:\[OPL = n \, \times \text{ physical distance}.\]

• For example, if light travels through a 0.25 cm thick glass plate with an index of refraction of 1.40, the OPL becomes 0.35 cm.
• Similarly, light traveling through air over a distance of 1.55 cm maintains almost the same OPL, assuming air's refractive index is close to 1.

The optical path length is foundational for calculating the phase shift in light and determining how many wavelengths fit between two points, which is fundamental in wave optics for understanding light behavior and interference.

Light Transmission

Light transmission describes how light waves pass through a material. Whether reflecting, refracting, or being absorbed, each material affects light differently. In contexts like lenses and optical fibers, understanding light transmission becomes crucial:

• Materials with higher close-to-unity transmittance allow more light to pass through with minimal loss.
• The glass plate, with its index of refraction, will affect the light's transmission by altering its speed and wavelength, creating potential for reflection or absorption.

For applications relying on accurate light passage, adjusting for the index of refraction ensures precision. By calculating light behavior through these media, engineers and scientists can enhance device design, improving everything from eyeglasses to advanced communication systems, fostering better technological integration into daily life.