Problem 41 - Physics

Question

Some of the X-rays produced in an X-ray machine have a wavelength of \(2.1 \mathrm{nm}\). What is the frequency of these electromagnetic waves?

Step-by-Step Solution

Answer

The frequency of the X-rays is approximately \(1.43 \times 10^{17}\, \mathrm{Hz}\).

Step 1: Identify the given values

We are given the wavelength of the X-rays, \(\lambda = 2.1\, \mathrm{nm}\). To use this in calculations, convert the wavelength from nanometers to meters: \(2.1\, \mathrm{nm} = 2.1 \times 10^{-9}\, \mathrm{m}\).

Step 2: Identify the relationship between wavelength and frequency

Electromagnetic waves follow the equation \(c = \lambda u\), where \(c\) is the speed of light (\(3.00 \times 10^8\, \mathrm{m/s}\)), \(\lambda\) is the wavelength in meters, and \(u\) is the frequency in Hertz (Hz).

Step 3: Rearrange the formula to solve for frequency

To find the frequency, rearrange the formula: \(u = \frac{c}{\lambda}\).

Step 4: Substitute the known values into the equation

Substitute \(\lambda = 2.1 \times 10^{-9}\, \mathrm{m}\) and \(c = 3.00 \times 10^8\, \mathrm{m/s}\) into the equation: \[ u = \frac{3.00 \times 10^8\, \mathrm{m/s}}{2.1 \times 10^{-9}\, \mathrm{m}} \]

Step 5: Calculate the frequency

Perform the division to find the frequency: \[ u \approx 1.43 \times 10^{17}\, \mathrm{Hz} \] This is the frequency of the X-rays with a wavelength of \(2.1\, \mathrm{nm}\).

Key Concepts

Electromagnetic Waves

Electromagnetic waves are a fascinating aspect of physics because they include a wide range of phenomena, from visible light to X-rays, radio waves, and beyond. These waves are produced when electric charges oscillate. This oscillation creates alternating electric and magnetic fields that move through space.

Electromagnetic waves do not need a medium to travel through. This means they can move through a vacuum, such as outer space, at the speed of light. These waves move via a specific mechanism:

• Electric field creates a magnetic field
• Magnetic field, in turn, generates an electric field

This cycle allows the wave to propagate indefinitely.

Each type of electromagnetic wave has a different wavelength and frequency, leading to diverse applications ranging from medical imaging (in the case of X-rays) to communication technology (in the case of radio waves). Understanding this helps us grasp the full spectrum of electromagnetic waves and their impact on modern technology.

Wavelength and Frequency

Wavelength and frequency are two crucial characteristics that describe electromagnetic waves. They are inversely related, which means as one increases, the other decreases.

**Wavelength**

• It is the distance between successive peaks of a wave.
• Measured in meters, but often seen in smaller units such as nanometers for X-rays.

**Frequency**

• The number of waves that pass through a given point per second.
• Measured in Hertz (Hz), where 1 Hz equals one cycle per second.

The relationship between wavelength and frequency can be expressed with the formula:\[ c = \lambda u \]where:

• \(c\) is the speed of light, \(3.00 \times 10^8\) m/s
• \(\lambda\) is the wavelength in meters
• \(u\) is the frequency in Hertz

Using this relationship, you can calculate one if you know the other and the speed of light. This fundamental equation helps us understand how these waves behave and how we can utilize them.

Speed of Light

The speed of light is one of the most fundamental constants in physics. It represents the highest speed at which all massless particles and associated fields and influences of matter can travel through the vacuum of space. The exact speed of light is \(3.00 \times 10^8\) meters per second (m/s).

Light - and all other electromagnetic waves - travel at this constant speed in a vacuum. However, when these waves pass through different media, their speed can change due to interactions with the matter in the medium. Despite this, in many calculations involving electromagnetic waves, such as with X-ray frequency, the speed of light in a vacuum is used as a constant.

• Because this speed is a constant, it provides a reliable way to link wavelength and frequency through the formula \(c = \lambda u\)
• It underscores the universality and reliability of physical laws concerning wave propagation

Knowing the speed of light allows us to solve for other properties of electromagnetic waves, like calculating the frequency of X-rays if the wavelength is known. It emphasizes the deep connection between various physical constants and their role in understanding the universe.