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 \(3.97 \times 10^{-19} \mathrm{J} /\) photon. Identify each type of electromagnetic radiation and place them in order of increasing photon energy and increasing frequency.

Short answer

The types of electromagnetic radiation are identified as Radio waves (Radiation #1), Gamma rays (Radiation #2), and Infrared (Radiation #3). The order of increasing photon energy is Infrared, Gamma rays, and Radio waves. The order of increasing frequency is Radio waves, Infrared, and Gamma rays.

Step-by-step solution

Find the frequency and energy of each type of radiation

Given information: 1. Frequency: \(f_1 = 107.1 \: MHz = 107.1 \times 10^6 Hz\) 2. Wavelength: \(\lambda_2 = 2.12 \times 10^{-10} m\) 3. Photon energy: \(E_3 = 3.97 \times 10^{-19} J\) Constants: 1. Speed of light, \(c = 3.00 \times 10^8 m/s\) 2. Planck's constant, \(h = 6.626 \times 10^{-34} J \cdot s\) Now, we will find other parameters for each type of radiation. For radiation #1: Frequency is given, so we need to find the energy. \(E_1 = h \cdot f_1 = (6.626 \times 10^{-34}) \cdot (107.1 \times 10^6)\) \(E_1 = 7.08 \times 10^{-24} J\) For radiation #2: Wavelength is given, so we need to find the frequency and energy. Frequency: \(f_2 = \frac{c}{\lambda_2} = \frac{3.00 \times 10^8}{2.12 \times 10^{-10}}\) \(f_2 = 1.42 \times 10^{18} Hz\) Energy: \(E_2 = h \cdot f_2 = (6.626 \times 10^{-34}) \cdot (1.42 \times 10^{18})\) \(E_2 = 9.42 \times 10^{-16} J\) For radiation #3: Energy is given, so we need to find the frequency. Frequency: \(f_3 = \frac{E_3}{h} = \frac{3.97 \times 10^{-19}}{6.626 \times 10^{-34}}\) \(f_3 = 5.99 \times 10^{14} Hz\)

Identify each type of radiation

We will identify each type of radiation based on their energy: 1. Radiation #1: \(E_1 = 7.08 \times 10^{-24} J\) (Radio waves) 2. Radiation #2: \(E_2 = 9.42 \times 10^{-16} J\) (Gamma rays) 3. Radiation #3: \(E_3 = 3.97 \times 10^{-19} J\) (Infrared) Now, we can order them based on increasing photon energy and increasing frequency.

Order radiation types based on energy and frequency

Order of increasing photon energy: 1. Infrared (Radiation #3) 2. Gamma rays (Radiation #2) 3. Radio waves (Radiation #1) Order of increasing frequency: 1. Radio waves (Radiation #1) 2. Infrared (Radiation #3) 3. Gamma rays (Radiation #2)

Key concepts

Frequency of Radiation

The frequency of electromagnetic radiation, denoted by the symbol \( f \), is the number of times the wave oscillates, or cycles, in one second. It is measured in hertz (Hz), where one hertz equals one cycle per second. Higher-frequency radiation has more energy, explaining why gamma rays, with frequencies in the order of \( 10^{18} Hz \), are much more energetic than radio waves, with frequencies around \( 10^6 Hz \).

Understanding the frequency of radiation is crucial for characterizing different types of electromagnetic waves, ranging from radio waves at the low end of the spectrum to gamma rays at the high end. The type of radiation used in the exercise, with a frequency of 107.1 MHz, falls in the category of radio waves, often used in FM radio broadcasting.

Wavelength of Radiation

The wavelength of radiation, represented as \( \lambda \), is the physical distance between successive peaks of the wave. It is inversely related to frequency: as the frequency increases, the wavelength decreases, and vice versa. This relationship is governed by the equation \( c = \lambda \cdot f \), where \( c \) is the speed of light. The wavelength says a lot about the energy and type of the radiation; for example, the radiation with a wavelength of \( 2.12 \times 10^{-10} m \) in the given exercise is indicative of gamma rays—high-energy radiation often emitted by radioactive materials or cosmic phenomena.

Wavelengths can range from very long (kilometers for radio waves) to extremely short (fractions of a nanometer for gamma rays), and identifying the type of radiation through its wavelength is a key aspect in various scientific fields such as astronomy, medicine, and communications.

Photon Energy

Photon energy is the energy carried by a single photon, the fundamental particle of light. The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. Expressed mathematically, the energy \( E \) of a photon can be calculated using Planck's equation, \( E = h \cdot f \), where \( h \) is Planck's constant. The higher the frequency of the light, the higher the energy of each photon. In the exercise's context, photons with energy equal to \( 3.97 \times 10^{-19} J \) correspond to infrared radiation, which is less energetic than visible light and is commonly experienced as heat.

Photon energy is a critical concept in understanding electromagnetic interactions and plays a significant role in areas like spectroscopy, quantum mechanics, and the functioning of various electronic devices.

Speed of Light

The speed of light, denoted \( c \), is a fundamental physical constant representing the speed at which light travels in a vacuum. It is approximately \( 3.00 \times 10^8 m/s \). This constant is vital in linking wavelength and frequency of electromagnetic waves through the equation \( c = \lambda \cdot f \). Since the speed of light is constant, if we know either the frequency or wavelength of a type of radiation, we can easily find the other quantity.

The concept of the speed of light extends beyond the scope of electromagnetic waves—it is also central to Einstein's theory of relativity and has implications in understanding the structure of the Universe and the nature of time and space.

Planck's Constant

Planck's constant, symbolized by \( h \), is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. Its value is approximately \( 6.626 \times 10^{-34} J \cdot s \). This constant is a cornerstone in the equation \( E = h \cdot f \), which provides the energy of a photon given its frequency. Planck's constant captures the inherent granularity of quantum phenomena, where energy exchange occurs in discrete packets, or 'quanta', rather than in a continuous flow.

The introduction of Planck's constant revolutionized physics by laying the groundwork for the quantum theory, which explains the behavior of particles at the atomic and subatomic levels, opening doors to new technologies such as semiconductors, lasers, and quantum computers.