3.3 Energy and Momentum of Electromagnetic Waves

Electromagnetic waves carry energy and momentum, key concepts in understanding their behavior and interactions. This section explores how to quantify these properties, from energy density to the Poynting vector, and their conservation in various phenomena.

We'll dive into the relationships between intensity, electric, and magnetic fields, and how they change in different media. These ideas are crucial for applications like solar sails, antenna design, and understanding cosmic phenomena.

Energy density and Poynting vector

Energy Density in Electromagnetic Waves

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• Energy density represents the amount of energy stored per unit volume in electric and magnetic fields
• Total energy density sums electric field energy density and magnetic field energy density
• Calculate electric field energy density using $u_e = \frac{1}{2}\epsilon_0E^2$
• Determine magnetic field energy density with $u_m = \frac{1}{2}\mu_0H^2$
• In vacuum, electric and magnetic energy densities are equal
  • Results in total energy density of $u_{total} = u_e + u_m = \epsilon_0E^2$
• Energy density varies with the square of field amplitudes
  • Doubling field strength quadruples energy density

Poynting Vector and Energy Flow

• Poynting vector measures energy flux (power per unit area) of electromagnetic waves
• Represents direction and magnitude of energy flow
• Define Poynting vector as cross product of electric and magnetic field vectors $\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$
• Time-averaged Poynting vector describes average energy flow over one wave cycle
• Calculate time-averaged Poynting vector magnitude using $S_{avg} = \frac{1}{2}\epsilon_0cE_0^2$
• Poynting vector direction perpendicular to both electric and magnetic field vectors
  • Follows right-hand rule (thumb points in Poynting vector direction when fingers curl from E to B)
• Examples of Poynting vector applications
  • Solar radiation (energy flow from sun to Earth)
  • Antenna radiation patterns (energy distribution in radio waves)

Electromagnetic wave energy and momentum

Energy of Electromagnetic Waves

• Calculate energy carried by electromagnetic wave using [E = hf](https://www.fiveableKeyTerm:e_=_hf)
  • h represents Planck's constant (6.626 x 10^-34 J·s)
  • f denotes wave frequency
• Energy directly proportional to frequency
  • Higher frequency waves (gamma rays) carry more energy than lower frequency waves (radio waves)
• Quantize electromagnetic energy in discrete packets called photons
• Photon energy determines its interactions with matter (photoelectric effect, Compton scattering)
• Calculate total energy in a wave by integrating Poynting vector over area and time
  • $E_{total} = \int_A \int_t \vec{S} \cdot d\vec{A} dt$

Momentum of Electromagnetic Waves

• Relate electromagnetic wave momentum to energy using $p = \frac{E}{c}$
  • c represents speed of light in vacuum (3 x 10^8 m/s)
• Express photon momentum as $p = \frac{h}{\lambda}$
  • λ denotes wavelength
• Momentum of electromagnetic waves causes radiation pressure
  • Calculate pressure for perfect absorption using $P = \frac{I}{c}$
  • I represents wave intensity
• Angular momentum of circularly polarized waves relates to photon spin
  • Each photon carries ±ℏ angular momentum (+ for right circular, - for left circular polarization)
• Examples of electromagnetic wave momentum
  • Solar sail propulsion in spacecraft
  • Comet tail formation due to solar radiation pressure

Intensity, Electric, and Magnetic Fields

Intensity and Field Amplitude Relationships

• Electromagnetic wave intensity proportional to square of electric field amplitude $I \propto E_0^2$
• Intensity also proportional to square of magnetic field amplitude $I \propto B_0^2$
• Relate electric and magnetic field amplitudes in vacuum $E_0 = cB_0$
• Express time-averaged intensity using electric field amplitude $I = \frac{1}{2}\epsilon_0cE_0^2$
• Calculate time-averaged intensity using magnetic field amplitude $I = \frac{1}{2}\frac{c}{\mu_0}B_0^2$
• Define root-mean-square (RMS) values of fields
  • Electric field RMS: $E_{RMS} = \frac{E_0}{\sqrt{2}}$
  • Magnetic field RMS: $B_{RMS} = \frac{B_0}{\sqrt{2}}$

Intensity Variations and Media Effects

• Intensity decreases with square of distance from point source (inverse square law)
  • $I \propto \frac{1}{r^2}$, where r represents distance from source
• Modify intensity-field relationships in media other than vacuum
  • Account for material's permittivity (ε) and permeability (μ)
  • Replace ε₀ with ε and μ₀ with μ in intensity formulas
• Examples of intensity applications
  • Calculating safe distances from radioactive sources
  • Determining power requirements for communication satellites

Energy and Momentum Conservation for Waves

Conservation of Energy in Electromagnetic Systems

• Total energy remains constant in isolated systems with electromagnetic waves
• Account for energy transformations between different forms
  • Electromagnetic to kinetic (photoelectric effect)
  • Electromagnetic to thermal (microwave heating)
• Energy conservation in absorption and emission processes
  • Change in matter's energy equals energy of absorbed or emitted radiation
• Apply conservation of energy to phenomena like fluorescence and phosphorescence
  • Absorbed high-energy photons re-emitted as lower-energy photons

Conservation of Momentum in Wave-Matter Interactions

• Conserve momentum of electromagnetic waves in interactions with matter
• Radiation pressure results from momentum transfer to surfaces
  • Calculate radiation pressure on perfectly reflecting surface $P = \frac{2I}{c}$
• Photon recoil occurs when atoms emit or absorb photons
  • Basis for laser cooling techniques in atomic physics
• Demonstrate energy and momentum conservation in Compton scattering
  • Change in photon wavelength: $\Delta \lambda = \frac{h}{m_ec}(1-\cos\theta)$
  • θ represents scattering angle, m_e denotes electron mass
• Apply conservation principles to pair production and annihilation
  • Photon energy converts to particle-antiparticle pair masses and kinetic energies
  • Particle-antiparticle annihilation produces photons with conserved total energy and momentum