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 = 1 2 ϵ 0 E 2 u_e = \frac{1}{2}\epsilon_0E^2 u e = 2 1 ϵ 0 E 2
Determine magnetic field energy density with u m = 1 2 μ 0 H 2 u_m = \frac{1}{2}\mu_0H^2 u m = 2 1 μ 0 H 2
In vacuum, electric and magnetic energy densities are equal
Results in total energy density of u t o t a l = u e + u m = ϵ 0 E 2 u_{total} = u_e + u_m = \epsilon_0E^2 u t o t a l = u e + u m = ϵ 0 E 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 S ⃗ = 1 μ 0 E ⃗ × B ⃗ \vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B} S = μ 0 1 E × B
Time-averaged Poynting vector describes average energy flow over one wave cycle
Calculate time-averaged Poynting vector magnitude using S a v g = 1 2 ϵ 0 c E 0 2 S_{avg} = \frac{1}{2}\epsilon_0cE_0^2 S a vg = 2 1 ϵ 0 c E 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 t o t a l = ∫ A ∫ t S ⃗ ⋅ d A ⃗ d t E_{total} = \int_A \int_t \vec{S} \cdot d\vec{A} dt E t o t a l = ∫ A ∫ t S ⋅ d A d t
Momentum of Electromagnetic Waves
Relate electromagnetic wave momentum to energy using p = E c p = \frac{E}{c} p = c E
c represents speed of light in vacuum (3 x 10^8 m/s)
Express photon momentum as p = h λ p = \frac{h}{\lambda} p = λ h
Momentum of electromagnetic waves causes radiation pressure
Calculate pressure for perfect absorption using P = I c P = \frac{I}{c} P = c I
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 ∝ E 0 2 I \propto E_0^2 I ∝ E 0 2
Intensity also proportional to square of magnetic field amplitude I ∝ B 0 2 I \propto B_0^2 I ∝ B 0 2
Relate electric and magnetic field amplitudes in vacuum E 0 = c B 0 E_0 = cB_0 E 0 = c B 0
Express time-averaged intensity using electric field amplitude I = 1 2 ϵ 0 c E 0 2 I = \frac{1}{2}\epsilon_0cE_0^2 I = 2 1 ϵ 0 c E 0 2
Calculate time-averaged intensity using magnetic field amplitude I = 1 2 c μ 0 B 0 2 I = \frac{1}{2}\frac{c}{\mu_0}B_0^2 I = 2 1 μ 0 c B 0 2
Define root-mean-square (RMS) values of fields
Electric field RMS: E R M S = E 0 2 E_{RMS} = \frac{E_0}{\sqrt{2}} E RMS = 2 E 0
Magnetic field RMS: B R M S = B 0 2 B_{RMS} = \frac{B_0}{\sqrt{2}} B RMS = 2 B 0
Intensity decreases with square of distance from point source (inverse square law)
I ∝ 1 r 2 I \propto \frac{1}{r^2} I ∝ r 2 1 , 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 = 2 I c P = \frac{2I}{c} P = c 2 I
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: Δ λ = h m e c ( 1 − cos θ ) \Delta \lambda = \frac{h}{m_ec}(1-\cos\theta) Δ λ = m e c h ( 1 − cos θ )
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