Problem 5 - University Physics with Modern Physics

Question

Medical x rays are taken with electromagnetic waves having a wavelength of around 0.10 nm in air. What are the frequency, period, and wave number of such waves?

Step-by-Step Solution

Answer

Frequency is \( 3.0 \times 10^{18} \text{ Hz} \), period is \( 3.33 \times 10^{-19} \text{ seconds} \), and wave number is \( 6.283 \times 10^{10} \text{ m}^{-1} \).

Step 1: Given data

We are provided with the wavelength of the x-ray, which is \( \lambda = 0.10 \text{ nm} \). We first need to convert this into meters for standard SI units. Thus, \( 0.10 \text{ nm} = 0.10 \times 10^{-9} \text{ meters} = 1.0 \times 10^{-10} \text{ meters} \).

Step 2: Calculate frequency

The speed of electromagnetic waves in air (and vacuum) is approximately \( c = 3.00 \times 10^8 \text{ meters/second} \). The frequency \( f \) can be calculated using the equation \( c = \lambda f \). Rearranging gives us \( f = \frac{c}{\lambda} \). Substituting the given values gives us \[ f = \frac{3.00 \times 10^8 \text{ meters/second}}{1.0 \times 10^{-10} \text{ meters}} = 3.0 \times 10^{18} \text{ Hz} \].

Step 3: Calculate period

The period \( T \) is the reciprocal of the frequency: \( T = \frac{1}{f} \). Substituting the frequency we found: \[ T = \frac{1}{3.0 \times 10^{18} \text{ Hz}} = 3.33 \times 10^{-19} \text{ seconds} \].

Step 4: Calculate wave number

The wave number \( k \) is calculated using the formula \( k = \frac{2\pi}{\lambda} \). Substituting the wavelength into the formula, we get: \[ k = \frac{2\pi}{1.0 \times 10^{-10} \text{ meters}} = 6.283 \times 10^{10} \text{ m}^{-1} \].

Key Concepts

Electromagnetic Waves

Electromagnetic waves are an important concept in physics. They consist of oscillating electric and magnetic fields traveling through space at the speed of light. These waves do not require a medium; they can propagate through the vacuum of space. The speed of electromagnetic waves in a vacuum is a universally recognized constant, approximately \( 3.00 \times 10^8 \text{ meters/second} \). This speed remains the same across all frequencies of electromagnetic waves, which consist of a broad spectrum including radio waves, microwaves, visible light, and x-rays.

• Generated by charged particles, such as electrons, moving through electrical and magnetic fields.
• Can travel outside Earth’s atmosphere and cover vast areas in space.
• Fundamental to the functionality of devices like radios, microwaves, and x-ray machines.

X-Rays

X-rays are a high-energy type of electromagnetic wave. Known for their short wavelengths, typically in the range of 0.01 to 10 nanometers, x-rays can penetrate most materials. This property makes them especially useful in medical imaging and security scans. Discovered by Wilhelm Röntgen in 1895, x-rays have since played a pivotal role in various fields.

• Used in medical diagnostics to view the inside of the body, like bones and tissues.
• Safety regulations are important, as excessive exposure can damage human tissues.
• Also employed in industrial settings to inspect products and materials.

Understanding their nature and behavior is crucial for harnessing their benefits while minimizing potential hazards.

Wavelength to Frequency Conversion

To convert the wavelength of a wave to its frequency, one can use the formula \( c = \lambda f \), where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( f \) is the frequency. This relationship shows that wavelength and frequency are inversely proportional; as one increases, the other decreases. Calculating one from the other involves rearranging this formula to \( f = \frac{c}{\lambda} \).

• Useful in physics to determine unseen properties of electromagnetic waves.
• Helps in understanding the energy of the wave, as energy is directly related to frequency.
• Crucial in designing technologies like antennas, which need precise frequency control.

Simplifying such transformations make it easier to interpret various wave-related phenomena.

Wave Number Calculation

Wave number involves measuring the spatial frequency of a wave, representing the number of wave cycles in a given unit of space. It's a fundamental property of waves and is related to both wavelength and frequency. The wave number \( k \) is calculated using the formula \( k = \frac{2\pi}{\lambda} \).

• Gives insight into the spatial variability of a wave, often used in physics and engineering.
• Expressed in inverse meters (\( \text{m}^{-1} \)), and helps in understanding wave behavior in mediums.
• Useful in atomic and molecular spectroscopy, where it simplifies the description of spectral lines.

Understanding the wave number can help determine how waves interact with matter at microscopic levels and improve techniques in spectroscopy and imaging technologies.