Problem 11 - College Physics

Question

\(\bullet\) Medical rays. Medical xays are taken with electromagnetic waves having a wavelength around 0.10 nm. What are the frequency, period, and wave number of such waves?

Step-by-Step Solution

Answer

Frequency: \( 3 \times 10^{18} \) Hz, Period: \( 3.33 \times 10^{-19} \) s, Wave number: \( 6.28 \times 10^{10} \) m\(^{-1}\).

Step 1: Identify Wavelength

The given wavelength of medical X-rays is 0.10 nm. Convert this wavelength into meters for ease of calculation: \[ 0.10 \, \text{nm} = 0.10 \times 10^{-9} \, \text{m} = 1.0 \times 10^{-10} \, \text{m} \]

Step 2: Calculate Frequency

Use the speed of light formula \( c = \lambda f \) to find the frequency \( f \), where \( c = 3 \times 10^8 \, \text{m/s} \), and \( \lambda = 1.0 \times 10^{-10} \, \text{m} \). Rearrange the formula to solve for \( f \): \[ f = \frac{c}{\lambda} = \frac{3 \times 10^8}{1.0 \times 10^{-10}} = 3 \times 10^{18} \, \text{Hz} \]The frequency of the X-ray waves is \( 3 \times 10^{18} \) Hz.

Step 3: Calculate Period

The period \( T \) is the reciprocal of the frequency \( f \). Use the formula \( T = \frac{1}{f} \):\[ T = \frac{1}{3 \times 10^{18}} \approx 3.33 \times 10^{-19} \, \text{s} \]The period of the X-ray waves is approximately \( 3.33 \times 10^{-19} \) seconds.

Step 4: Calculate Wave Number

The wave number \( k \) is given by the formula \( k = \frac{2\pi}{\lambda} \). Using \( \lambda = 1.0 \times 10^{-10} \, \text{m} \):\[ k = \frac{2\pi}{1.0 \times 10^{-10}} = 2\pi \times 10^{10} \approx 6.28 \times 10^{10} \, \text{m}^{-1} \]The wave number of the X-ray waves is approximately \( 6.28 \times 10^{10} \, \text{m}^{-1} \).

Key Concepts

Electromagnetic Waves

Electromagnetic waves are a type of wave that can travel through the vacuum of outer space. They are composed of oscillating electric and magnetic fields, which are perpendicular to each other and to the direction the wave travels. These waves include a wide spectrum of types, such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
Each type of electromagnetic wave has its own range of wavelengths and frequencies. X-rays, for instance, are high-frequency and short-wavelength waves, making them useful in medical imaging.
A key property of electromagnetic waves is that they all travel at the speed of light in a vacuum, which is approximately \(3 \times 10^8 \, \text{m/s}\). This speed allows us to relate different properties of the waves, such as frequency and wavelength, using the formula \(c = \lambda f\), where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(f\) is the frequency.

Frequency Calculation

To calculate the frequency of an electromagnetic wave, we use the relationship between speed, frequency, and wavelength stated by the formula \(c = \lambda f\). In this equation, \(c\) represents the speed of light, \(\lambda\) is the wavelength, and \(f\) is the frequency.
By rearranging the formula to solve for frequency, \(f\), we get: \(f = \frac{c}{\lambda}\). This calculation involves dividing the speed of light by the given wavelength. For example, with X-rays having a wavelength of \(0.10 \, \text{nm}\), or \(1.0 \times 10^{-10} \, \text{m}\), the frequency is calculated as follows:

• Identify the wavelength: \(1.0 \times 10^{-10} \, \text{m}\).
• Use the formula: \(f = \frac{3 \times 10^8}{1.0 \times 10^{-10}}\).

The result is a frequency of \(3 \times 10^{18} \, \text{Hz}\). This means the wave oscillates \(3 \times 10^{18}\) times every second!

Wave Number

Wave number is an important concept in the study of waves, especially in fields like spectroscopy. It is defined as the number of wavelengths per unit distance and is often represented by the symbol \(k\).
Wave number provides a more convenient measure than wavelength for some calculations, especially in quantum mechanics. The formula used to calculate the wave number, \(k\), is \(k = \frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength.
For X-rays with a wavelength of \(1.0 \times 10^{-10} \, \text{m}\), calculating the wave number involves substituting into the formula:

• Wave number, \(k = \frac{2\pi}{1.0 \times 10^{-10}}\).
• This results in \(k \approx 6.28 \times 10^{10} \, \text{m}^{-1}\).

This high wave number reflects the very short wavelength and high energy of X-rays.

Wavelength Conversion

Wavelength conversion is often required when working with measurements given in units like nanometers (nm) or angstroms. These smaller units are commonly used because many electromagnetic waves, especially those in the visible and higher frequency spectrums, have very small wavelengths.
To convert wavelength from nanometers to meters, which is a more standard unit for calculations, you use the conversion factor: \(1 \, \text{nm} = 10^{-9} \, \text{m}\).
For instance, if an X-ray wavelength is 0.10 nm, the conversion to meters would be:

• 0.10 nm \(= 0.10 \times 10^{-9} \, \text{m}\)
• Which equals \(1.0 \times 10^{-10} \, \text{m}\)

Having the wavelength in meters makes it easier to use physics formulas involving light speed and frequency, ensuring calculations are precise and consistent.