Chapter 8: Electromagnetic Waves

DISPLACEMENT CURRENT

Maxwell's Modification to Ampere's Law

  • Problem: Original Ampere's law inconsistent with charge conservation
  • Solution: Addition of displacement current term
  • Modified Ampere's Law: ∮B·dl = μ₀(I + Id)
  • Displacement Current: Id = ε₀(dΦE/dt)
  • Physical Significance: Changing electric field produces magnetic field

Displacement Current in Capacitor

  • During Charging/Discharging: Id = ε₀A(dE/dt) = I (conduction current)
  • Ensures Continuity: Conduction current in wires equals displacement current between plates

ELECTROMAGNETIC WAVES

Maxwell's Equations

  1. Gauss's Law for Electricity: ∮E·dA = q/ε₀
  2. Gauss's Law for Magnetism: ∮B·dA = 0
  3. Faraday's Law: ∮E·dl = -dΦB/dt
  4. Ampere-Maxwell Law: ∮B·dl = μ₀(I + ε₀(dΦE/dt))

Wave Equation from Maxwell's Equations

\[ \nabla^2\vec{E} = \mu_0\varepsilon_0\frac{\partial^2\vec{E}}{\partial t^2} \]
\[ \nabla^2\vec{B} = \mu_0\varepsilon_0\frac{\partial^2\vec{B}}{\partial t^2} \]
\[ c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3 \times 10^8 \text{ m/s} \]

Properties of EM Waves

  • Transverse Waves: E and B oscillate perpendicular to direction of propagation
  • Perpendicular Fields: E ⊥ B
  • Speed: c = 3 × 10⁸ m/s in vacuum
  • No Medium Required: Can propagate through vacuum
  • Energy Transport: Carry energy and momentum

Plane Electromagnetic Waves

\[ \vec{E} = E_0\sin(kx - \omega t)\hat{j} \]
\[ \vec{B} = B_0\sin(kx - \omega t)\hat{k} \]
  • Relationship: E₀/B₀ = c
  • Wave Vector: k = 2π/λ
  • Angular Frequency: ω = 2πf

ENERGY IN ELECTROMAGNETIC WAVES

Energy Density

  • Electric Field Contribution: uₑ = (1/2)ε₀E²
  • Magnetic Field Contribution: uₘ = (1/2)B²/μ₀
  • Total Energy Density: u = uₑ + uₘ = ε₀E²

Poynting Vector

\[ \vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B}) \]
  • Physical Meaning: Energy flux (energy per unit area per unit time)
  • Direction: Direction of wave propagation
  • Magnitude: S = EB/μ₀ = E²/(μ₀c)
  • Average Value: Sav = (1/2)E₀B₀/μ₀ = (1/2)cε₀E₀²

Radiation Pressure

  • Definition: Pressure exerted by EM waves on surface
  • Perfect Absorber: p = u (energy density)
  • Perfect Reflector: p = 2u
  • Relation to Intensity: p = I/c (absorber), p = 2I/c (reflector)

ELECTROMAGNETIC SPECTRUM

Radio Waves

  • Frequency: 10⁴ - 10¹² Hz
  • Wavelength: 10⁻⁴ - 10⁴ m
  • Production: Accelerating charges in antenna
  • Applications: Broadcasting, communications, radar

Microwaves

  • Frequency: 10⁹ - 10¹² Hz
  • Wavelength: 10⁻³ - 10⁻¹ m
  • Production: Klystron, magnetron
  • Applications: Radar, satellite communication, microwave ovens

Infrared Waves

  • Frequency: 10¹² - 10¹⁴ Hz
  • Wavelength: 10⁻⁵ - 10⁻³ m
  • Production: Hot bodies, molecular vibrations
  • Applications: Thermal imaging, heating, night vision

Visible Light

  • Frequency: 4 × 10¹⁴ - 7.5 × 10¹⁴ Hz
  • Wavelength: 400 - 750 nm
  • Production: Electronic transitions in atoms
  • Applications: Vision, photography, illumination

Ultraviolet Waves

  • Frequency: 10¹⁵ - 10¹⁷ Hz
  • Wavelength: 10⁻⁸ - 10⁻⁷ m
  • Production: Electronic transitions in atoms
  • Applications: Sterilization, detecting forged documents

X-rays

  • Frequency: 10¹⁷ - 10²⁰ Hz
  • Wavelength: 10⁻¹¹ - 10⁻⁸ m
  • Production: Inner electronic transitions, bremsstrahlung
  • Applications: Medical imaging, crystallography

Gamma Rays

  • Frequency: > 10²⁰ Hz
  • Wavelength: < 10⁻¹¹ m
  • Production: Nuclear transitions, radioactive decay
  • Applications: Cancer treatment, nuclear medicine

PROPAGATION OF ELECTROMAGNETIC WAVES

In Vacuum

  • Speed: c = 3 × 10⁸ m/s
  • No Dispersion: All frequencies travel at same speed

In Matter

  • Speed: v = c/n
  • Refractive Index: n = √(εᵣμᵣ)
  • Dispersion: Different frequencies travel at different speeds

Reflection and Refraction

  • Reflection: Obeys law of reflection (θᵢ = θᵣ)
  • Refraction: Obeys Snell's law (n₁sinθ₁ = n₂sinθ₂)
  • Total Internal Reflection: Occurs when angle of incidence exceeds critical angle

COMMUNICATION SYSTEMS

Basic Elements

  1. Transmitter: Converts information to EM waves
  2. Transmission Medium: Space or transmission lines
  3. Receiver: Detects and decodes EM waves

Modulation

  • Need: To transmit low-frequency signals efficiently
  • Types:
    • Amplitude Modulation (AM): Varying amplitude of carrier wave
    • Frequency Modulation (FM): Varying frequency of carrier wave
    • Phase Modulation (PM): Varying phase of carrier wave

Bandwidth

  • Definition: Range of frequencies in a signal
  • Importance: Determines information-carrying capacity

KEY FORMULAS

  • Displacement Current: Id = ε₀(dΦE/dt)
  • Speed of EM Waves: c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s
  • Relation Between E and B: E₀/B₀ = c
  • Energy Density: u = ε₀E² = B²/μ₀
  • Poynting Vector: S = (1/μ₀)(E × B)
  • Average Intensity: Iav = (1/2)cε₀E₀²
  • Radiation Pressure: p = I/c (absorber), p = 2I/c (reflector)