Electromagnetic Waves

Electromagnetic waves carry momentum $p = \frac{U}{c}$ where $U$ is the energy of the wave.

Momentum transfer calculations:

1. Complete absorption: Matter receives momentum $p = \frac{U}{c}$
2. Complete reflection: Matter receives momentum $p = \frac{2U}{c}$ (twice as much due to momentum reversal)
3. Partial absorption/reflection: Momentum transferred is between these values

Physical consequences:

• EM waves exert radiation pressure on objects
• Force exerted equals rate of momentum transfer: $F = \frac{dp}{dt} = \frac{1}{c}\frac{dU}{dt} = \frac{P}{c}$ (for absorption)
• For reflection, force is doubled: $F = \frac{2P}{c}$
• This explains phenomena like comet tails pointing away from the sun

The momentum of EM waves demonstrates the particle-like properties of radiation, complementing the wave-like properties.

Derivation from Maxwell's equations:

1. Start with Maxwell's equations in vacuum:

   • $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ (Faraday's law)
   • $\nabla \times \vec{B} = \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$ (Ampere's law with displacement current)

2. Take the curl of Faraday's law: $\nabla \times (\nabla \times \vec{E}) = -\nabla \times \frac{\partial \vec{B}}{\partial t} = -\frac{\partial}{\partial t}(\nabla \times \vec{B})$

3. Substitute Ampere's law: $\nabla \times (\nabla \times \vec{E}) = -\mu_0\epsilon_0\frac{\partial^2 \vec{E}}{\partial t^2}$

4. Using vector identity $\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2\vec{E}$ and $\nabla \cdot \vec{E} = 0$ in vacuum: $-\nabla^2\vec{E} = -\mu_0\epsilon_0\frac{\partial^2 \vec{E}}{\partial t^2}$

5. Final wave equation: $\nabla^2\vec{E} = \mu_0\epsilon_0\frac{\partial^2 \vec{E}}{\partial t^2}$

This tells us:

• Light is a wave phenomenon that satisfies the wave equation
• Wave speed is $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$
• Light requires no medium for propagation (unlike mechanical waves)
• Electric and magnetic components are interdependent and create each other

Comparing electromagnetic spectrum regions:

Radio waves (λ: km to m):

• Penetrate atmosphere easily
• Diffract around obstacles
• Interact with conducting materials and electronics

Microwaves (λ: cm to mm):

• Absorbed by water molecules (cooking principle)
• Used in radar and communications
• Partially absorbed by atmosphere

Infrared (λ: mm to 700 nm):

• Strongly absorbed by atmosphere (especially water vapor)
• Associated with thermal radiation
• Emitted by molecules during vibrations and rotations

Visible light (λ: 700-400 nm):

• Passes through atmosphere with minimal absorption ("atmospheric window")
• Interacts with electron energy levels in atoms
• Primary range for photosynthesis

Ultraviolet (λ: 400-10 nm):

• High-energy UV strongly absorbed by ozone layer
• Damages biological molecules (DNA)
• Ionizing at shorter wavelengths

X-rays and Gamma rays (λ: <10 nm):

• Highly penetrating through matter
• Strongly ionizing
• Generally blocked by atmosphere

Atmospheric interaction depends on:

• Molecular resonances (specific wavelengths interact with atmospheric molecules)
• Ozone layer specifically absorbs UV radiation
• Water vapor absorbs in infrared regions
• Atmospheric density (ionosphere reflects certain radio frequencies)

The "atmospheric windows" allow astronomy in visible light and radio frequencies.

Maxwell's displacement current directly led to the prediction of electromagnetic waves through these key steps:

1. The introduction of displacement current completed Maxwell's equations by adding the term $\epsilon_0\frac{d\Phi_E}{dt}$ to Ampere's law

2. This created a fundamental symmetry in electromagnetic theory:

   • Changing magnetic fields produce electric fields (Faraday's law)
   • Changing electric fields produce magnetic fields (Ampere-Maxwell law)

3. Mathematical analysis of the complete equations revealed wave solutions with:

   • Speed of propagation = $\frac{1}{\sqrt{\mu_0\epsilon_0}} = c$
   • Mutually perpendicular, oscillating electric and magnetic fields
   • Propagation direction perpendicular to both fields

4. When Maxwell calculated the speed using known values of $\mu_0$ and $\epsilon_0$, he found it matched the measured speed of light (3×10⁸ m/s)

This profound insight revealed that light is an electromagnetic wave, unifying optics with electromagnetism and demonstrating that visible light is just one part of a much broader electromagnetic spectrum.

Maxwell's displacement current maintains continuity by:

• Accounting for the changing electric field when conduction current isn't conserved
• Providing a "substitute current" where no physical charge carriers exist
• Mathematically defined as: i_d = ε₀(dΦ_E/dt), where Φ_E is electric flux
• Acting as the missing link that ensures i₁ = i₂ + i_d (incoming equals outgoing plus displacement)
• Completing Ampere's law to make it consistent with conservation of charge

Example: In a charging capacitor, conduction current flows in the wires but not between the plates. The displacement current between the plates equals the conduction current in the wires, maintaining continuity throughout the circuit.

The relationship between charge accumulation and displacement current is:

• The rate of charge accumulation in a volume equals the difference between incoming and outgoing conduction currents: d(q_inside)/dt = i₁ - i₂

• This rate of charge accumulation exactly equals the displacement current: i_d = d(q_inside)/dt

• From Gauss's law: q_inside = ε₀Φ_E, where Φ_E is the electric flux through the surface

• Therefore: i_d = ε₀(dΦ_E/dt) = d(q_inside)/dt

This relationship ensures that the total current (conduction + displacement) is always continuous across any boundary, preserving the fundamental conservation of charge.

To verify Faraday's Law for electromagnetic waves:

1. Select a rectangular path perpendicular to the propagation direction
2. Calculate the electric field circulation around this path:
   • Sum $\oint \vec{E} \cdot d\vec{l}$ for all segments
   • For a wave with $E = E_0 \sin(\omega t - kx)$, this gives a value proportional to the difference in field values at different positions
3. Calculate the rate of change of magnetic flux through the area:
   • $\Phi_B = \int \vec{B} \cdot d\vec{S}$ through the enclosed area
   • Find $-\frac{d\Phi_B}{dt}$
4. Verify that $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

For a valid electromagnetic wave, these calculations must yield equal values, confirming that a changing magnetic field produces an electric field according to Faraday's Law.

To demonstrate that $E_0 = cB_0$ for electromagnetic waves:

1. Set up an experiment using a rectangular path perpendicular to wave propagation:

   • Create a rectangular loop with width $(x_2-x_1)$ and height $l$
   • Orient it so the electric field runs parallel to height $l$
   • Place it perpendicular to the wave's propagation direction

2. Apply Faraday's Law: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

   • Left side: $E_0l[\sin(\omega t-kx_2) - \sin(\omega t-kx_1)]$
   • Right side: $-\frac{d}{dt}\int_{x_1}^{x_2} B_0l\sin(\omega t-kx)dx$

3. After integration and derivation:

   • $-\frac{d\Phi_B}{dt} = -cB_0l[\sin(\omega t-kx_2) - \sin(\omega t-kx_1)]$

4. For Faraday's Law to hold true, we must have: $E_0l[\sin(\omega t-kx_2) - \sin(\omega t-kx_1)] = -cB_0l[\sin(\omega t-kx_2) - \sin(\omega t-kx_1)]$

This can only be satisfied when $E_0 = cB_0$, confirming this fundamental relationship between electric and magnetic field amplitudes in electromagnetic waves.

The wavelength and frequency of electromagnetic waves are inversely proportional:

• As wavelength increases, frequency decreases
• As wavelength decreases, frequency increases
• This relationship is described by the equation: c = λf where c is the speed of light (3×10⁸ m/s), λ is wavelength, and f is frequency

Example: Radio waves have wavelengths of meters to kilometers and low frequencies (10⁴-10⁸ Hz), while gamma rays have extremely short wavelengths (10⁻¹² m and shorter) and very high frequencies (10²⁰ Hz and higher).

Electromagnetic radiation generation mechanisms:

1. Radio waves & Microwaves:

   • Generated by accelerating charges in electrical circuits
   • AC circuits with inductors and capacitors produce oscillating currents
   • Wavelength determined by circuit components and oscillation frequency

2. Infrared:

   • Emitted by atoms and molecules in hot bodies
   • Thermal vibration of molecules produces radiation
   • Temperature determines peak wavelength (higher temp = shorter wavelength)

3. Visible light & Ultraviolet:

   • Produced by electron transitions between energy levels in atoms
   • Higher energy transitions produce shorter wavelengths
   • Energy difference determines exact wavelength

4. X-rays:

   • Generated when fast-moving electrons decelerate inside a metal target
   • Also produced by high-energy electron transitions in heavy atoms
   • Electron energy determines X-ray wavelength

5. Gamma rays:

   • Emitted by unstable atomic nuclei during radioactive decay
   • Result from nuclear transitions, not electron transitions
   • Nuclear binding energy differences determine wavelength

The underlying principle: Higher energy processes produce shorter wavelength radiation, following E = hf = hc/λ (where h is Planck's constant).