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16.3 Energy Carried by Electromagnetic Waves

3 min read•june 24, 2024

carry energy through space, combining electric and magnetic fields. The of these waves depends on field amplitudes, while the describes energy flow direction and magnitude.

Wave energy is linked to both amplitude and frequency. Higher amplitudes mean more photons, while higher frequencies result in more energetic photons. This interplay shapes the diverse properties of electromagnetic radiation across the spectrum.

Energy in Electromagnetic Waves

Energy density of electromagnetic waves

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  • Quantifies the amount of energy stored per unit volume in an electromagnetic wave
  • Proportional to the square of the E02E_0^2 and the square of the B02B_0^2
    • equation: u=12ϵ0E02+12μ0B02u = \frac{1}{2}\epsilon_0E_0^2 + \frac{1}{2\mu_0}B_0^2
      • uu: energy density
      • ϵ0\epsilon_0: (vacuum)
      • μ0\mu_0: (vacuum)
  • In an electromagnetic wave, the electric and magnetic field amplitudes are related by E0=cB0E_0 = cB_0
    • cc: in vacuum (3×1083 \times 10^8 m/s)
    • Substituting this relation into the energy density equation gives u=ϵ0E02=1μ0B02u = \epsilon_0E_0^2 = \frac{1}{\mu_0}B_0^2
    • Shows that the energy density can be expressed using either the electric or magnetic field amplitude

Poynting vector and energy intensity

  • Poynting vector S⃗\vec{S} represents the direction and magnitude of energy flow in an electromagnetic wave
    • Defined as S⃗=1μ0E⃗×B⃗\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}
      • E⃗\vec{E}: electric field vector
      • B⃗\vec{B}: magnetic field vector
      • Cross product of E⃗\vec{E} and B⃗\vec{B} gives the direction of energy flow (perpendicular to both fields)
  • Magnitude of the Poynting vector is called the II or
    • Energy equation: I=∣S⃗∣=1μ0EBI = |\vec{S}| = \frac{1}{\mu_0}EB
      • For a , E=E0sin⁡(kx−ωt)E = E_0\sin(kx - \omega t) and B=B0sin⁡(kx−ωt)B = B_0\sin(kx - \omega t)
      • Average energy intensity over one period: Iavg=12ϵ0cE02=12μ0cB02I_{avg} = \frac{1}{2}\epsilon_0cE_0^2 = \frac{1}{2\mu_0}cB_0^2
    • Measures the rate of energy flow per unit area (W/m²)
  • Energy intensity is related to the energy density by I=ucI = uc
    • Energy flows at the cc in the direction of the Poynting vector
  • exerted by electromagnetic waves on surfaces they interact with

Wave amplitude vs photon frequency

  • Energy of an electromagnetic wave depends on both its amplitude and frequency
  • Energy density is proportional to the square of the wave amplitude (E02E_0^2 or B02B_0^2)
    • Doubling the amplitude quadruples the energy density (4 times more energy per unit volume)
    • Amplitude determines the number of photons in the wave (more photons = higher amplitude)
  • Photon energy is directly proportional to the frequency ff of the electromagnetic wave
    • Photon energy equation: E=hfE = hf
      • hh: (6.626×10−346.626 \times 10^{-34} J⋅s)
    • Higher frequency photons have more energy than lower frequency photons
      • () have higher photon energies than (radio waves)
  • Total energy of an electromagnetic wave depends on both:
    1. Number of photons (related to amplitude)
    2. Energy of each photon (related to frequency)
  • () use high-amplitude, low-frequency waves to heat food efficiently

Electromagnetic Wave Properties

  • describes how electromagnetic waves travel through space and various media
  • encompasses all types of electromagnetic radiation, from radio waves to gamma rays
  • Polarization refers to the orientation of the electric field oscillations in an electromagnetic wave

Key Terms to Review (24)

Electric Field Amplitude: The electric field amplitude is the maximum value of the electric field in an electromagnetic wave. It represents the strength or intensity of the electric field and is a crucial parameter in understanding the energy carried by electromagnetic waves.
Electromagnetic Spectrum: The electromagnetic spectrum is the entire range of electromagnetic radiation, from the longest wavelengths of radio waves to the shortest wavelengths of gamma rays. This spectrum encompasses all the different forms of light, including visible light, that make up the world around us and enable various scientific and technological applications.
Electromagnetic Waves: Electromagnetic waves are a type of energy that travels through space or a medium in the form of oscillating electric and magnetic fields. These waves are responsible for various phenomena, including heat transfer, magnetism, and the propagation of electric fields, and are fundamental to our understanding of Maxwell's equations and the electromagnetic spectrum.
Energy density: Energy density is the amount of energy stored in a given system or region of space per unit volume. It is often used to describe the energy stored in capacitors and electromagnetic fields.
Energy Density: Energy density is a measure of the amount of energy stored or carried per unit volume or mass of a material or system. It is a crucial concept in understanding the storage and transmission of energy in various physical contexts.
Energy flux: Energy flux is the rate of energy transfer per unit area perpendicular to the direction of propagation of an electromagnetic wave. It is measured in watts per square meter (W/m²).
Energy Intensity: Energy intensity refers to the amount of energy consumed per unit of output or activity, often measured in terms of energy used per square meter, per product, or per dollar of economic output. This concept helps assess how efficiently energy is used in various processes, especially in the context of energy carried by electromagnetic waves, as it illustrates how much energy is transmitted versus the useful work done. Understanding energy intensity allows for a deeper analysis of energy consumption patterns and their environmental impacts.
Gamma Rays: Gamma rays are a type of high-energy electromagnetic radiation with the shortest wavelength and highest frequency in the electromagnetic spectrum. They are produced by the radioactive decay of atomic nuclei and have the ability to penetrate deep into matter, making them useful in various applications.
Intensity: Intensity is the power transferred per unit area in the direction of wave propagation. It is measured in watts per square meter ($$W/m^2$$).
Irradiance: Irradiance is the measure of the power of electromagnetic radiation per unit area incident on a surface. It is the radiant flux received by a surface per unit area, and is commonly used to describe the intensity of sunlight, or other sources of electromagnetic radiation, incident on a surface at a given location.
Magnetic Field Amplitude: Magnetic field amplitude refers to the maximum value or strength of the magnetic field in an electromagnetic wave. It represents the peak intensity of the magnetic component of the electromagnetic radiation, which oscillates in a sinusoidal pattern over time and space.
Microwave Ovens: Microwave ovens are a type of kitchen appliance that use electromagnetic radiation in the microwave frequency range to heat and cook food. They work by exciting the water molecules within the food, causing them to vibrate and generate heat through friction.
Permeability of free space: Permeability of free space, denoted as $\mu_0$, is a physical constant that describes the extent to which a magnetic field can penetrate and affect a vacuum. Its value is $4\pi \times 10^{-7}$ Tm/A.
Permeability of Free Space: The permeability of free space, denoted as $\mu_0$, is a fundamental physical constant that describes the magnetic properties of a vacuum or free space. It is a measure of the ability of free space to support the formation of a magnetic field in response to an electric current or changing electric field.
Permittivity of Free Space: Permittivity of free space is a fundamental physical constant that measures the ability of a vacuum to permit electric field lines. It plays a crucial role in electrostatics, affecting the strength of electric fields and the behavior of charge distributions in free space.
Photon frequency: Photon frequency refers to the number of oscillations or cycles of a photon that occur in one second. This frequency is directly related to the energy carried by the photon, with higher frequencies corresponding to higher energy levels. Understanding photon frequency is crucial in grasping how electromagnetic waves transmit energy across space and how different frequencies correspond to different types of electromagnetic radiation, such as visible light, radio waves, and X-rays.
Planck's constant: Planck's constant is a fundamental physical constant that represents the smallest possible change in energy or action. It is a crucial quantity in quantum mechanics, relating the energy of a photon to its frequency and the angular momentum of a particle to its spin.
Plane Wave: A plane wave is a type of electromagnetic wave in which the electric and magnetic fields oscillate in planes perpendicular to the direction of propagation. It is a fundamental concept in the study of the energy carried by electromagnetic waves.
Poynting vector: The Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. It is given by the cross product of the electric field and the magnetic field vectors, denoted as $\mathbf{S} = \mathbf{E} \times \mathbf{H}$.
Radiation pressure: Radiation pressure is the pressure exerted by electromagnetic radiation on any surface due to the transfer of momentum from the photons to that surface. It is a key concept in understanding how light and other forms of electromagnetic waves can exert force.
Radiation Pressure: Radiation pressure is the pressure exerted on a surface by the momentum of electromagnetic radiation, such as light or other forms of radiant energy. It is a consequence of the transfer of momentum from photons to the surface they strike.
Speed of light: The speed of light in a vacuum is the constant speed at which all electromagnetic waves propagate, approximately $3 \times 10^8$ meters per second. It is a fundamental constant in physics denoted by the symbol $c$.
Speed of Light: The speed of light is the maximum velocity at which all electromagnetic radiation, including visible light, can travel through a vacuum. It is a fundamental constant in physics that has profound implications across various topics in electromagnetism and relativity.
Wave Propagation: Wave propagation refers to the movement and transmission of waves through a medium or space. It describes how waves, such as electromagnetic waves, transport energy and information without the physical movement of the medium itself.
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16.4 Momentum and Radiation Pressure