Electromagnetic Waves

🎯 Key Points

• c=1/√(μ₀ε₀)=3×10⁸ m/s; E, B, and propagation direction are mutually perpendicular; E₀/B₀=c
• Displacement current I_d=ε₀(dΦ_E/dt) - Maxwell's fix that makes Ampere's law work even where there's no actual charge flow (e.g. between capacitor plates)
• EM spectrum (increasing frequency/energy): Radio < Microwave < Infrared < Visible < UV < X-ray < Gamma ray
• Radiation pressure = I/c; intensity I=½ε₀cE₀²
• Energy is shared EQUALLY between electric and magnetic fields on time-average in an EM wave

Maxwell's Equations (Concept)

• Gauss's law for E: electric flux from charge
• Gauss's law for B: no magnetic monopoles (magnetic flux always zero)
• Faraday's law: changing B creates E
• Ampere-Maxwell law: changing E creates B (displacement current)
• Displacement current: I_d = ε₀ × dΦ_E/dt (fills gap between capacitor plates)

Properties of EM Waves

• Speed in vacuum: c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s
• E, B, and propagation direction are mutually perpendicular
• E₀/B₀ = c (ratio of field amplitudes)
• Transverse waves; can travel in vacuum
• Carry energy and momentum: pressure = I/c (intensity/speed)

Electromagnetic Spectrum

• Radio waves (> 0.1 m): radio, TV, MRI
• Microwaves (1 mm - 0.1 m): microwave ovens, radar, satellite
• Infrared (700 nm - 1 mm): thermal imaging, remote controls, heating
• Visible (400-700 nm): VIBGYOR (violet to red)
• Ultraviolet (10-400 nm): sterilization, vitamin D synthesis, skin cancer
• X-rays (0.01-10 nm): medical imaging, crystallography
• Gamma rays (< 0.01 nm): cancer treatment, nuclear reactions

The electromagnetic spectrum: all bands travel at c, with wavelength falling and frequency (and photon energy) rising from radio to gamma rays. Image: Inductiveload, NASA, CC BY-SA 3.0, via Wikimedia Commons.

Applications

• Radio: AM (amplitude modulation), FM (frequency modulation)
• Microwaves: heating water molecules (dielectric heating)
• X-rays: discovered by Roentgen; Bragg diffraction for crystal structure

Displacement Current in Detail

• Displacement current arises from a changing electric field (not actual charge flow) and was introduced by Maxwell to make Ampere's law consistent with charge conservation in situations like a charging capacitor
• Total current (conduction + displacement) is continuous across any circuit, including the gap between capacitor plates
• Modified Ampere-Maxwell law: ∮B·dl = μ₀(I_c + I_d)

Energy in EM Waves

• EM waves carry energy equally shared between electric and magnetic fields on time-average
• Average energy density: u_avg = ½ε₀E₀² (electric) + B₀²/2μ₀ (magnetic), each contributing equally
• Intensity (average power per unit area): I = ½ε₀cE₀²

Sources of EM Waves by Band

• Radio and microwaves: produced by oscillating currents in electronic circuits and antennas
• Infrared: produced by hot bodies and molecular vibrations
• Visible and ultraviolet: produced by electronic transitions in atoms
• X-rays: produced by sudden deceleration of high-energy electrons striking a target (bremsstrahlung) or inner-shell electron transitions
• Gamma rays: produced by nuclear transitions and radioactive decay

Nature and Mathematical Description of EM Waves

• For a wave travelling along x, the fields vary as E = E₀ sin(kx − ωt) along one transverse axis (say y) and B = B₀ sin(kx − ωt) along the perpendicular axis (z) - E and B oscillate in phase, reaching their maxima and zeros together
• The wave number k = 2π/λ and angular frequency ω = 2πf are linked by the wave speed ω/k = c = fλ
• EM waves are purely transverse (E and B both perpendicular to the propagation direction) and, unlike mechanical waves, require no material medium - they travel through vacuum
• An accelerating (or oscillating) electric charge is the fundamental source of an electromagnetic wave, radiating at the frequency of its oscillation

Speed of EM Waves in a Material Medium

• In a medium of permeability μ and permittivity ε, the wave speed is v = 1/√(με), which is always less than the vacuum speed c = 1/√(μ₀ε₀)
• The refractive index is n = c/v = √(μ_r ε_r); for most transparent non-magnetic media μ_r ≈ 1, so n ≈ √ε_r
• When an EM wave enters a medium its frequency stays fixed (set by the source) while its speed and wavelength both decrease by the factor n

🚀 JEE Advanced Edge

Radiation pressure - absorption vs reflection: A surface that fully ABSORBS incident EM radiation experiences pressure P=I/c; a surface that fully REFLECTS it experiences DOUBLE the pressure, P=2I/c, since the wave's momentum is reversed rather than just absorbed (analogous to elastic vs inelastic collision momentum transfer).

Calculating fields from intensity: Since I=½ε₀cE₀², given a known intensity (e.g. solar constant ≈1400 W/m² at Earth) you can solve for E₀=√(2I/ε₀c), and then B₀=E₀/c - a common way numericals connect EM wave energy to field amplitudes.

Worked problem: Sunlight has an intensity of 1400 W/m² at Earth's surface. Find the radiation pressure on a perfectly absorbing surface, and compare it to a perfectly reflecting one. Approach: P_absorb=I/c=1400/(3×10⁸)≈4.67×10⁻⁶ Pa. P_reflect=2I/c≈9.33×10⁻⁶ Pa - twice as much.

Transverse Nature and Field Relations in EM Waves

• In an EM wave, the electric field E, magnetic field B, and direction of propagation are mutually perpendicular, so the wave is transverse.
• E and B oscillate in phase: both reach maxima and zeros at the same instant and position.
• The magnitudes are related by c = E naught divided by B naught, where c is the speed of light.
• The direction of propagation is along the vector E cross B.
• EM waves need no material medium and travel through vacuum at c = 3 times 10 to the power 8 metre per second.

Speed of EM Waves from Fundamental Constants

• Maxwell showed that the speed of an EM wave in vacuum is c = 1 divided by square root of (mu naught times epsilon naught).
• Substituting the measured values of mu naught and epsilon naught gives c approximately 3 times 10 to the power 8 metre per second, matching the known speed of light.
• This agreement led Maxwell to conclude that light itself is an electromagnetic wave.
• In a medium the speed is v = 1 divided by square root of (mu times epsilon), always less than c.
• The refractive index is n = c divided by v = square root of (mu r times epsilon r) for a non-magnetic medium.

Radiation Pressure and Momentum of EM Waves

• EM waves carry momentum as well as energy; when they strike a surface they exert a radiation pressure.
• For total absorption, the momentum delivered is p = U divided by c, where U is the energy absorbed.
• For a perfectly reflecting surface the momentum transfer, and hence the pressure, is doubled.
• This effect explains why a comet's tail points away from the Sun and underpins the concept of solar sails.
• The pressure is tiny for ordinary light but measurable with sensitive apparatus.