Problem 5 - College Physics

Question

\(\bullet\) Consider electromagnetic waves propagating in air. (a) Determine the frequency of a wave with a wavelength of (i) \(5.0 \mathrm{km},\) (ii) \(5.0 \mu \mathrm{m},\) (iii) 5.0 \(\mathrm{nm}\) . (b) What is the wavelength (in meters and nanometers) of (i) gamma rays of frequency \(6.50 \times 10^{21} \mathrm{Hz}\) , (ii) an AM station radio wave of frequency 590 \(\mathrm{kHz} ?\)

Step-by-Step Solution

Answer

(a) Frequencies: (i) \(6.0 \times 10^4\) Hz, (ii) \(6.0 \times 10^{13}\) Hz, (iii) \(6.0 \times 10^{16}\) Hz. (b) Wavelengths: (i) 4.62 x 10^-14 m (4.62 x 10^-5 nm), (ii) 508.47 m (5.0847 x 10^11 nm).

Step 1: Understanding the Relationship Between Frequency and Wavelength

The relationship between the frequency \(f\) and wavelength \(\lambda\) of electromagnetic waves is given by the formula:\[ c = \lambda \cdot f \]where \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8\) m/s.

Step 2: Calculate Frequency for Wavelength of 5.0 km

Given the wavelength \(\lambda = 5.0\) km, convert it to meters first, \(5.0 \text{ km} = 5000 \text{ m}\). Now, use the relation:\[ f = \frac{c}{\lambda} = \frac{3 \times 10^8 \text{ m/s}}{5000 \text{ m}} = 6.0 \times 10^4 \text{ Hz} \]

Step 3: Calculate Frequency for Wavelength of 5.0 micrometers

Convert the wavelength \(\lambda = 5.0 \mu m\) to meters, \(5.0 \mu \text{m} = 5.0 \times 10^{-6} \text{ m}\). Using the formula:\[ f = \frac{3 \times 10^8 \text{ m/s}}{5.0 \times 10^{-6} \text{ m}} = 6.0 \times 10^{13} \text{ Hz} \]

Step 4: Calculate Frequency for Wavelength of 5.0 nm

Convert the wavelength \(\lambda = 5.0 \text{ nm}\) to meters, \(5.0 \text{ nm} = 5.0 \times 10^{-9} \text{ m}\). Using the formula:\[ f = \frac{3 \times 10^8 \text{ m/s}}{5.0 \times 10^{-9} \text{ m}} = 6.0 \times 10^{16} \text{ Hz} \]

Step 5: Calculate Wavelength for Gamma Rays Frequency 6.50 x 10^21 Hz

Using the frequency \(f = 6.50 \times 10^{21} \text{ Hz}\), the wavelength \(\lambda\) is:\[ \lambda = \frac{c}{f} = \frac{3 \times 10^8 \text{ m/s}}{6.50 \times 10^{21} \text{ Hz}} = 4.62 \times 10^{-14} \text{ m} \]To convert to nanometers, \(\lambda = 4.62 \times 10^{-5} \text{ nm}\).

Step 6: Calculate Wavelength for AM Station Frequency 590 kHz

Convert the frequency from kHz to Hz, \(590 \text{ kHz} = 590 \times 10^3 \text{ Hz}\). The wavelength \(\lambda\) is:\[ \lambda = \frac{c}{f} = \frac{3 \times 10^8 \text{ m/s}}{590 \times 10^3 \text{ Hz}} = 508.47 \text{ m} \]Converting to nanometers, \(\lambda = 5.0847 \times 10^{11} \text{ nm}\).

Key Concepts

Wavelength and Frequency Relationship

Electromagnetic waves exhibit a fascinating reciprocal relationship between wavelength and frequency. This relationship is governed by the equation \(c = \lambda \cdot f\), where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(f\) is the frequency. Simply put, if you know any two of these variables, you can easily determine the third one.
The speed of light (denoted by \(c\)) is constant, meaning that if the wavelength increases, the frequency must decrease, and vice versa.

• A longer wavelength corresponds to a lower frequency.
• A shorter wavelength corresponds to a higher frequency.

This inverse relationship helps explain why different types of electromagnetic waves, like radio waves and gamma rays, have such varying properties. Understanding this principle is foundational when studying the behavior of electromagnetic waves.

Speed of Light Constant

The speed of light in a vacuum is an unfaltering constant: \(3 \times 10^8\) meters per second (m/s). This constant is crucial because it connects wavelength and frequency in the equation \(c = \lambda \cdot f\).
Wherever electromagnetic waves travel, this speed remains unchangeable when in a vacuum. However, it can be slightly reduced when passing through different media, like air or water. Despite these alterations in speed in various media, the relationship between wavelength and frequency stays intact because both are affected proportionally to maintain the product equal to the speed of light.
Knowing this constant allows for simplified calculations across the electromagnetic spectrum, from gamma rays to radio waves, making it an essential component of physics and engineering.

Gamma Rays

Gamma rays are a type of electromagnetic radiation with incredibly high frequencies and correspondingly short wavelengths. They fall at the extreme end of the electromagnetic spectrum. For instance, waves with a frequency as high as \(6.50 \times 10^{21}\) Hz are classified as gamma rays.

• Gamma rays can travel through most materials, making them useful in medical imaging and cancer treatment radiation therapy.
• They originate from radioactive decay and celestial phenomena.

Conversion between their high frequency and their incredibly small wavelength requires precise calculation. In most practical applications, we use the speed of light constant in \(\lambda = \frac{c}{f}\) to determine their minute wavelength measurements, as demonstrated in scientific studies and technologies that employ gamma rays.

Radio Waves

Radio waves are part of the electromagnetic spectrum that have the longest wavelength and the lowest frequency. Their frequencies typically range from thousands (kilohertz) to millions (megahertz) of cycles per second. These characteristics make them suitable for wireless communication such as AM and FM radio broadcasting, television, and mobile phones.
For example, an AM radio wave might have a frequency of \(590\) kHz, which can be converted to a wavelength by rearranging the light-speed equation to \(\lambda = \frac{c}{f}\). Understanding how to perform this conversion is integral to designing and analyzing communication systems that use radio waves.

Unit Conversion in Physics

Physics often requires converting between different units to solve problems accurately, particularly in calculating electromagnetic wave properties. One common conversion is from kilometers or various sub-units to meters, as the standard unit for wavelength in physics problems. For example:

• Convert kilometers (km) to meters (m): multiply by \(1000\).
• Convert micrometers (\(\mu m\)) to meters: multiply by \(10^{-6}\).
• Convert nanometers (nm) to meters: multiply by \(10^{-9}\).

Frequency units might also change from kilohertz (kHz) to hertz (Hz) by multiplying by \(10^3\). These conversions ensure uniformity and prevent calculation errors. Familiarity with these unit conversions is an essential skill in physics academia and industry.