Question

One type of electromagnetic radiation has a frequency of 107.1 \(\mathrm{MHz}\) , another type has a wavelength of \(2.12 \times 10^{-10} \mathrm{m},\) and another type of electromagnetic radiation has photons with energy equal to \(3.97 \times 10^{-19} \mathrm{J} / \mathrm{photon}\) . Identify each type of electromagnetic radiation and place them in order of increasing photon energy and increasing frequency.

Short answer

The electromagnetic radiations can be ordered as follows: Order of increasing photon energy: 1. Type 1 (\( E_1 \approx 4.76 \times 10^{-24} \mathrm{J} \)) 2. Type 3 (\( E_3 = 3.97 \times 10^{-19} \mathrm{J/photon} \)) 3. Type 2 (\( E_2 \approx 9.94 \times 10^{-16} \mathrm{J} \)) Order of increasing frequency: 1. Type 1 (\( f_1 = 107.1 \times 10^6 \mathrm{Hz} \)) 2. Type 3 (\( f_3 \approx 5.99 \times 10^{14} \mathrm{Hz} \)) 3. Type 2 (\( f_2 \approx 1.42 \times 10^{18} \mathrm{Hz} \))

Step-by-step solution

Type 1: Frequency Given

We are given the frequency of type 1 electromagnetic radiation: \( f_1 = 107.1 \mathrm{MHz} \), we need to convert this into Hz by multiplying by \( 10^6 \). So, \( f_1 = 107.1 \times 10^6 \mathrm{Hz} \). Now we can find the wavelength using the equation: \( f \times \lambda = c \). \( \lambda_1 = \frac{c}{f_1} = \frac{3 \times 10^8 \mathrm{m/s} }{107.1 \times 10^6 \mathrm{Hz}} \) And finally, we can find the photon energy using the equation: \( E = h \times f \). \( E_1 = (6.626 \times 10^{-34} \mathrm{Js} ) \times (107.1 \times 10^6 \mathrm{Hz}) \)

Type 2: Wavelength Given

We are given the wavelength of type 2 electromagnetic radiation: \( \lambda_2 = 2.12 \times 10^{-10} \mathrm{m} \). Now we can find the frequency using the equation: \( f \times \lambda = c \). \( f_2 = \frac{c}{\lambda_2} = \frac{3 \times 10^8 \mathrm{m/s}}{2.12 \times 10^{-10} \mathrm{m}} \) And finally, we can find the photon energy using the equation: \( E = h \times f \). \( E_2 = (6.626 \times 10^{-34} \mathrm{Js}) \times (\frac{3 \times 10^8 \mathrm{m/s}}{2.12 \times 10^{-10} \mathrm{m}}) \)

Type 3: Photon Energy Given

We are given the photon energy of type 3 electromagnetic radiation: \( E_3 = 3.97 \times 10^{-19} \mathrm{J/photon} \). Now we can find the frequency using the equation: \( E = h \times f \). \( f_3 = \frac{E_3}{h} = \frac{3.97 \times 10^{-19} \mathrm{J}}{6.626 \times 10^{-34} \mathrm{Js}} \) And finally, we can find the wavelength using the equation: \( f \times \lambda = c \). \( \lambda_3 = \frac{c}{f_3} = \frac{3 \times 10^8 \mathrm{m/s}}{(\frac{3.97 \times 10^{-19} \mathrm{J}}{6.626 \times 10^{-34} \mathrm{Js}})} \) Now that we have all the information, let's compare the energies and frequencies.

Comparing Energies and Frequencies

We need to calculate and compare the energies \( E_1, E_2, E_3 \) and frequencies \( f_1, f_2, f_3 \) to find the order. Using a calculator, we find: - \( E_1 \approx 4.76 \times 10^{-24} \mathrm{J} \) - \( E_2 \approx 9.94 \times 10^{-16} \mathrm{J} \) - \( f_1 = 107.1 \times 10^6 \mathrm{Hz} \) - \( f_2 \approx 1.42 \times 10^{18} \mathrm{Hz} \) Order of increasing photon energy: 1. Type 1 (\( E_1 \)) 2. Type 3 (\( E_3 \)) 3. Type 2 (\( E_2 \)) Order of increasing frequency: 1. Type 1 (\( f_1 \)) 2. Type 3 (\( f_3 \)) 3. Type 2 (\( f_2 \))

Key concepts

Photon Energy

Photon energy is a fundamental concept in the study of electromagnetic radiation. It refers to the energy carried by a single photon, the smallest unit of light. The energy of a photon is directly related to its frequency: that means higher frequency photons have more energy. This relationship is described by the equation:\[ E = h \times f \]Where:

• \( E \) is the photon energy (in joules),
• \( h \) is Planck's constant (approximately \( 6.626 \times 10^{-34} \mathrm{Js} \)),
• \( f \) is the frequency of the photon (in hertz).

This equation implies that if you know the frequency, you can calculate the energy, and vice versa.
Notably, different types of electromagnetic radiation, like radio waves and X-rays, have varying photon energies, affecting how they interact with matter.

Frequency

Frequency is a crucial parameter in the study of waves, including electromagnetic waves. It is defined as the number of wave cycles that pass a point per second. Measured in hertz (Hz), frequency gives us a way to understand the energy and behavior of different types of electromagnetic radiation.
The relationship between frequency \( f \) and wavelength \( \lambda \) is given by the wave equation:\[ f \times \lambda = c \]Where \( c \) is the speed of light, approximately \( 3 \times 10^8 \mathrm{m/s} \). This equation shows that if the frequency is high, the wavelength must be short to maintain the constant speed of light.
Different frequencies correspond to different types of electromagnetic radiation. For example:

• Radio waves have low frequencies.
• Visible light has much higher frequencies.
• Gamma rays have the highest frequencies.

Wavelength

Wavelength is the distance between successive crests (or troughs) of a wave. In electromagnetic radiation, it characterizes the type of radiation. Typically measured in meters, wavelengths range from very long, as in radio waves, to extremely short, as in gamma rays.
Electromagnetic waves all travel at the speed of light \( c \) through a vacuum, which ties wavelength and frequency together through the formula:\[ \lambda = \frac{c}{f} \]Understanding wavelength is crucial as it enables us to categorize electromagnetic radiation into bands on the electromagnetic spectrum. Different wavelengths have practical applications:

• Radio waves, with long wavelengths, are used in communication.
• Microwaves aid in cooking and radar technology.
• Visible light allows us to see.
• Ultraviolet light aids in disinfection and studying stars.

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all types of electromagnetic radiation. Radiation varies in energy, frequency, and wavelength, covering a vast spectrum. Starting from long-wavelength radio waves to short-wavelength gamma rays, each type fits on the spectrum based on its particular wavelength and frequency.
The electromagnetic spectrum includes:

• Radio Waves: Long wavelengths and low frequencies.
• Microwaves: Slightly shorter wavelengths used in heating and communication.
• Infrared: Felt as heat, used in thermal imaging.
• Visible Light: The tiny portion we can see, ranging from red to violet.
• Ultraviolet: Beyond violet, useful yet potentially harmful.
• X-Rays: Penetrating radiation used in medical imaging.
• Gamma Rays: Very high energy, used in cancer treatment.

Understanding where each type of radiation fits on the spectrum helps us use it effectively in technology, medicine, and research. Each portion has specific characteristics and interactions with matter, making the spectrum a foundational concept in physics.