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Chapter 8: Problem 8

What is the wavelength of light if its frequency is \(3.66 \times 10^{6} \mathrm{~s}^{-1}\) ?

Short Answer

Expert verified
The wavelength is approximately 81.97 meters.

Step by step solution

01

Understand the Formula

To determine the wavelength of light, we use the formula \( c = \lambda u \) where \( c \) is the speed of light (approximately \( 3.00 \times 10^{8} \text{ m/s} \)), \( \lambda \) is the wavelength, and \( u \) is the frequency of the light.
02

Rearrange the Formula

We need to solve for the wavelength \( \lambda \). Rearrange the formula to \( \lambda = \frac{c}{u} \).
03

Substitute the Values

Substitute \( c = 3.00 \times 10^{8} \text{ m/s} \) and \( u = 3.66 \times 10^{6} \text{ s}^{-1} \) into the equation: \[ \lambda = \frac{3.00 \times 10^{8}}{3.66 \times 10^{6}}. \]
04

Calculate the Wavelength

Perform the division to find \( \lambda \): \[ \lambda = \frac{3.00 \times 10^{8}}{3.66 \times 10^{6}} \approx 81.97 \text{ meters}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed of Light
The speed of light is an essential constant in physics, symbolized by the letter "c". It represents how fast light travels in a vacuum, which is approximately \( 3.00 \times 10^{8} \text{ m/s} \). This speed is incredibly fast and forms a fundamental part of many equations and theories in science.
  • Light travels so swiftly that it can circle the Earth over 7 times in just a single second!
  • "c" is not just a value for light, but it also sets the speed limit for the entire universe, according to Einstein's theory of relativity.
Whether we're dealing with lasers, the sun's rays, or radio waves, the speed remains constant in a vacuum.
It's crucial because it helps us understand how light and all electromagnetic radiation behave in our universe.
Frequency of Light
Frequency of light, often symbolized as \( u \), tells us how often the waves of light pass a certain point each second. It is measured in Hertz (Hz). In simpler terms, if we think of light like a wave, frequency tells us how many waves fit in a second.
  • A higher frequency means there are more waves per second.
  • A lower frequency means fewer waves per second.
In our example, the frequency of light is \( 3.66 \times 10^{6} \text{ s}^{-1} \), which indicates the number of light waves per second. Frequency is inversely related to wavelength, meaning as one increases, the other decreases.
Formula for Wavelength
The formula for calculating the wavelength \( \lambda \) of light from its frequency \( u \) and the speed of light \( c \) is:\[ \lambda = \frac{c}{u} \]Wavelength refers to the distance between successive crests of a wave. Calculating it gives us insight into the properties and behavior of the light.
To use the formula effectively:
  • Ensure you know the speed of light, \( c \), usually \( 3.00 \times 10^{8} \text{ m/s} \).
  • Use the given frequency, \( u \).
  • Substitute these values into the formula.
  • Perform the division to solve for \( \lambda \).
In our exercise, substituting \( c \) and \( u \) correctly leads to a wavelength of about 81.97 meters, showing us the relationship between speed, frequency, and wavelength.

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