Glossary

13.4 Energy and momentum of electromagnetic waves

3 min read•august 7, 2024

Electromagnetic waves pack a punch, carrying both energy and momentum as they zip through space. They're like invisible couriers, delivering power and exerting pressure on everything they touch. Understanding their behavior is key to grasping how light and radio waves work.

This section dives into the nitty-gritty of electromagnetic wave energy and momentum. We'll explore the , , and . These concepts help explain phenomena from solar sails to comet tails, showing how waves shape our universe.

Energy and Intensity of Electromagnetic Waves

Poynting Vector and Energy Density

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  • The Poynting vector S⃗\vec{S} represents the directional (the rate of energy transfer per unit area) of an electromagnetic wave
    • Mathematically defined as the cross product of the electric field E⃗\vec{E} and the magnetic field H⃗\vec{H}: S⃗=E⃗×H⃗\vec{S} = \vec{E} \times \vec{H}
    • Points in the direction of wave propagation and is perpendicular to both the electric and magnetic fields
  • Energy density uu is the amount of energy stored in the electromagnetic fields per unit volume
    • Consists of two components: electric field energy density uE=12ϵ0E2u_E = \frac{1}{2}\epsilon_0E^2 and magnetic field energy density uB=12μ0B2u_B = \frac{1}{2\mu_0}B^2
    • The total energy density is the sum of these two components: u=uE+uB=12ϵ0E2+12μ0B2u = u_E + u_B = \frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2

Energy Flux and Intensity

  • Energy flux, also known as the Poynting vector magnitude ∣S⃗∣|\vec{S}|, quantifies the rate at which energy flows through a unit area perpendicular to the direction of propagation
    • For a plane electromagnetic wave, the energy flux is given by ∣S⃗∣=1μ0EB|\vec{S}| = \frac{1}{\mu_0}EB, where EE and BB are the amplitudes of the electric and magnetic fields, respectively
  • II is the time-averaged energy flux of an electromagnetic wave
    • Defined as the average of the Poynting vector magnitude over one period of oscillation: I=⟨∣S⃗∣⟩=12ϵ0μ0E02I = \langle |\vec{S}| \rangle = \frac{1}{2}\sqrt{\frac{\epsilon_0}{\mu_0}}E_0^2, where E0E_0 is the peak amplitude of the electric field
    • Intensity decreases with the square of the distance from the source (inverse-square law) in the far-field region

Power Transmitted by Electromagnetic Waves

  • The power PP transmitted by an electromagnetic wave through a surface is the integral of the Poynting vector over that surface: P=∫S⃗⋅dA⃗P = \int \vec{S} \cdot d\vec{A}
    • For a uniform plane wave propagating through a surface of area AA perpendicular to the direction of propagation, the power is simply the product of intensity and area: P=IAP = IA
  • The power radiated by an isotropic point source (radiating equally in all directions) can be found using the intensity at a distance rr: P=4πr2IP = 4\pi r^2 I
    • This relation is useful for determining the power output of electromagnetic sources like antennas or light bulbs

Momentum and Pressure of Electromagnetic Waves

Momentum Density and Radiation Pressure

  • Electromagnetic waves carry momentum, with the g⃗\vec{g} given by the Poynting vector divided by the square of the speed of light: g⃗=S⃗c2\vec{g} = \frac{\vec{S}}{c^2}
    • The direction of the momentum density is the same as the direction of wave propagation
  • Radiation pressure pp is the force per unit area exerted by an electromagnetic wave on a surface that absorbs or reflects the wave
    • For a plane wave normally incident on a perfectly absorbing surface, the radiation pressure is equal to the energy density of the wave: p=u=12ϵ0E2+12μ0B2p = u = \frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2
    • For a perfectly reflecting surface, the radiation pressure is twice the energy density: p=2up = 2u
  • Examples of radiation pressure include:
    • Solar sails, which use the radiation pressure from sunlight to propel spacecraft
    • Comet tails, which are formed by the radiation pressure from the Sun pushing dust particles away from the comet nucleus

Key Terms to Review (21)

Compton scattering: Compton scattering is the phenomenon in which a photon collides with a charged particle, typically an electron, resulting in a transfer of energy and momentum between them. This process illustrates the particle-like behavior of electromagnetic waves, as the photon loses energy and changes direction, while the electron gains kinetic energy and recoils. The effects of Compton scattering provide crucial insights into the dual nature of light and help explain energy conservation and momentum conservation in electromagnetic interactions.
E=hf: The equation e=hf describes the relationship between the energy (e) of a photon and its frequency (f), where h is Planck's constant. This equation highlights the quantum nature of electromagnetic radiation, showing that energy is quantized and directly proportional to frequency. It connects to the wave-particle duality of light, illustrating how photons carry energy that can be calculated based on their oscillation rate.
Electromagnetic radiation: Electromagnetic radiation is a form of energy that travels through space at the speed of light, consisting of oscillating electric and magnetic fields. This energy manifests as waves that can vary in wavelength and frequency, spanning a wide spectrum from radio waves to gamma rays. Understanding electromagnetic radiation is essential for exploring how these waves are generated, how they carry energy and momentum, and their role in various physical phenomena.
Energy Density: Energy density refers to the amount of energy stored in a given system or region of space per unit volume. It plays a crucial role in understanding how energy is distributed in electromagnetic fields, impacting both the behavior of electromagnetic waves and the functioning of capacitors. This concept helps quantify the energy that can be harnessed from these systems, providing insight into their efficiency and performance.
Energy Flux: Energy flux refers to the rate at which energy is transferred or radiated through a given area, often measured in watts per square meter (W/m²). This concept is crucial in understanding how electromagnetic waves carry energy through space, allowing us to comprehend their impact on various physical processes and interactions with matter.
Frequency: Frequency is the number of occurrences of a repeating event per unit time, commonly measured in Hertz (Hz). This concept is essential in understanding how electric circuits operate, particularly when analyzing power and energy, as well as in the behavior of alternating current and voltage. It also plays a crucial role in characterizing electromagnetic waves, where frequency determines properties such as wavelength and energy.
Frequency-wavelength relationship: The frequency-wavelength relationship describes how the frequency of a wave is inversely proportional to its wavelength. This means that as the frequency of an electromagnetic wave increases, its wavelength decreases, and vice versa. This relationship is fundamental to understanding the behavior and properties of electromagnetic waves, including their energy and momentum.
Intensity: Intensity refers to the amount of energy that electromagnetic waves carry per unit area in a direction perpendicular to that area. It is a crucial concept as it quantifies how much power is transmitted by these waves, which has implications for understanding their energy and momentum, as well as their polarization characteristics. The intensity of electromagnetic waves can influence various phenomena, such as how they interact with materials and how they can be measured in different environments.
Maxwell's equations: Maxwell's equations are a set of four fundamental equations in electromagnetism that describe how electric and magnetic fields interact and propagate. They unify the concepts of electricity and magnetism into a coherent framework, highlighting the relationships between electric charges, currents, and electromagnetic fields. These equations are essential for understanding phenomena like electromagnetic waves, which play a crucial role in technologies such as radio, microwaves, and light.
Momentum density: Momentum density is defined as the amount of momentum per unit volume in a given field, particularly in the context of electromagnetic waves. This concept relates to how electromagnetic fields carry momentum as they propagate through space, demonstrating that light and other electromagnetic waves have both energy and momentum associated with them. Understanding momentum density helps explain the interactions between electromagnetic waves and matter, including phenomena like radiation pressure.
Momentum of photons: The momentum of photons refers to the quantity of motion carried by a photon, which is a massless particle of light. Unlike classical objects, photons possess momentum even though they have no rest mass, and their momentum is directly related to their energy and frequency. This concept is crucial in understanding the behavior of electromagnetic waves and their interaction with matter, particularly in processes like radiation pressure and the photoelectric effect.
P=e/c: The equation $p=\frac{e}{c}$ relates the momentum (p) of electromagnetic waves to their energy (e) and the speed of light (c). This equation highlights the intrinsic connection between energy and momentum in the context of electromagnetic radiation, illustrating how they are interrelated in the behavior of waves in space. Understanding this relationship is key to grasping how electromagnetic waves carry energy and momentum through various media.
Photoelectric effect: The photoelectric effect is the phenomenon where electrons are emitted from a material, typically a metal, when it is exposed to electromagnetic radiation, such as light. This effect demonstrates the particle nature of light, showing that photons can transfer energy to electrons, allowing them to overcome the work function of the material. The photoelectric effect is significant in understanding how electromagnetic waves carry energy and momentum.
Planck's constant (h): Planck's constant (h) is a fundamental physical constant that relates the energy of a photon to the frequency of its associated electromagnetic wave. It plays a critical role in quantum mechanics, bridging the gap between classical physics and quantum phenomena by establishing that energy is quantized and occurs in discrete packets called quanta. This constant not only helps define the energy of electromagnetic waves but also has significant implications for the behavior of particles at microscopic scales.
Planck's Relation: Planck's Relation describes the relationship between the energy of a photon and its frequency, given by the equation $$E = h u$$, where $$E$$ is the energy, $$h$$ is Planck's constant, and $$ u$$ is the frequency. This fundamental principle highlights how electromagnetic waves, including light, carry energy quantized in discrete amounts proportional to their frequency, influencing various aspects of quantum mechanics and electromagnetic wave behavior.
Power Transmitted: Power transmitted refers to the rate at which energy is transferred through electromagnetic waves. This concept is crucial in understanding how electromagnetic waves carry energy away from a source and how that energy can be harnessed or utilized, particularly in applications like communication, heating, and light generation. The power transmitted also relates to the intensity of these waves and how they interact with materials and fields.
Poynting Vector: The Poynting vector is a vector that represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. It is mathematically defined as the cross product of the electric field vector and the magnetic field vector, scaled by the permeability of free space. This vector is crucial for understanding how electromagnetic energy propagates through space and interacts with matter.
Radiation pressure: Radiation pressure is the pressure exerted by electromagnetic radiation on a surface, resulting from the momentum carried by photons. This phenomenon arises because light, although massless, has momentum that can transfer to objects when it is absorbed or reflected. The amount of radiation pressure depends on the intensity of the light and the area over which it acts, influencing various physical processes in astrophysics and engineering.
Speed of light (c): The speed of light (c) is the maximum speed at which information and matter can travel in the universe, valued at approximately 299,792,458 meters per second in a vacuum. This constant plays a crucial role in the relationship between energy, momentum, and electromagnetic waves, highlighting how light behaves as both a particle and a wave. The speed of light also serves as a foundation for various physical theories, including Einstein's theory of relativity, emphasizing its significance in understanding the fundamental nature of the universe.
Wave-particle duality: Wave-particle duality is the concept in quantum mechanics that posits that every particle or quantum entity exhibits both wave and particle properties. This duality is crucial in understanding how electromagnetic waves, such as light, can behave like particles (photons) in certain situations while also demonstrating wave-like behavior through interference and diffraction.
Wavelength: Wavelength is the distance between consecutive peaks or troughs in a wave, commonly measured in meters. It plays a crucial role in determining the behavior of electromagnetic waves, affecting their energy, frequency, and how they interact with matter. Understanding wavelength helps explain wave phenomena such as interference, diffraction, and the propagation of light across different media.
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