Electromagnetic Waves

From Maxwell's equations in vacuum:

• Faraday's law: $\oint \vec{E}\cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
• Ampere's law with displacement current: $\oint \vec{B}\cdot d\vec{l} = \mu_0\epsilon_0\frac{d\Phi_E}{dt}$

For a plane wave propagating in x-direction:

• Faraday's law requires $E_0 = cB_0$
• Wave equations: $E = E_0\sin(\omega t - kx)$ and $B = B_0\sin(\omega t - kx)$
• Wave speed: $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$

The relationship between field maximums: $E_0 = cB_0$, meaning the amplitudes of electric and magnetic fields are proportional, with proportionality constant equal to the speed of light.

For an electromagnetic wave:

1. Instantaneous energy density: $u = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0}B^2$
2. Substituting $E = E_0\sin(\omega t - kx)$ and $B = B_0\sin(\omega t - kx)$ where $B_0 = \frac{E_0}{c}$
3. Over time, $\sin^2$ terms average to $\frac{1}{2}$
4. Electric energy density equals magnetic energy density: $\frac{1}{4}\epsilon_0 E_0^2 = \frac{1}{4\mu_0}B_0^2$
5. Average total energy density: $u_{av} = \frac{1}{2}\epsilon_0 E_0^2 = \frac{1}{2\mu_0}B_0^2$

Example: For $E_0 = 50\text{ N/C}$, $u_{av} = \frac{1}{2}(8.85×10^{-12})(50)^2 = 1.1×10^{-8} \text{ J/m}^3$

Maxwell's four equations:

1. Gauss's law for electricity: $\oint \vec{E}\cdot d\vec{S} = \frac{q}{\epsilon_0}$

   • Electric fields originate from electric charges
   • Field lines begin on positive charges and end on negative charges
   • The electric flux through a closed surface is proportional to enclosed charge

2. Gauss's law for magnetism: $\oint \vec{B}\cdot d\vec{S} = 0$

   • No magnetic monopoles exist
   • Magnetic field lines always form closed loops
   • The magnetic flux through any closed surface equals zero

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

   • Changing magnetic fields produce electric fields
   • Base principle for electrical generators and transformers
   • Electric field lines circulate around changing magnetic flux

4. Ampere's law with Maxwell's addition: $\oint \vec{B}\cdot d\vec{l} = \mu_0 i + \mu_0\epsilon_0\frac{d\Phi_E}{dt}$

   • Electric currents produce magnetic fields
   • Changing electric fields also produce magnetic fields
   • Base principle for electromagnets and electromagnetic waves

Collectively, they:

• Form a complete, symmetrical description of electromagnetism
• Reveal the interdependence of electric and magnetic fields
• Predict electromagnetic waves propagating at speed $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$
• Unify electricity, magnetism, and optics into a single theory

The relationship $E_0 = cB_0$ demonstrates:

Physical significance:

• Electric and magnetic fields in EM waves are fundamentally linked, not independent
• Neither field exists alone in a propagating wave; they create each other
• The ratio is precisely the speed of light, connecting electromagnetism to optics
• Field strengths are perfectly balanced to maintain propagation

Theoretical implications:

• Reveals the symmetry in Maxwell's equations
• Shows that electric and magnetic fields are different aspects of a single electromagnetic field
• Demonstrates that the speed of light emerges naturally from electromagnetic theory
• Provided crucial evidence for Einstein to develop special relativity

Historical significance:

• Helped Maxwell predict electromagnetic waves before experimental confirmation
• Led to understanding light as an electromagnetic phenomenon
• Unified previously separate domains of electricity, magnetism, and optics
• Showed that the EM wave speed matched the measured speed of light, confirming light's electromagnetic nature

This relationship represents one of the most profound unifications in physics, showing how seemingly distinct phenomena are manifestations of the same fundamental interaction.

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).

The intensity of an electromagnetic wave is defined as:

$I = u_{av} \cdot c$

Where:

• $I$ = intensity (W/m²)
• $u_{av}$ = average energy density (J/m³)
• $c$ = speed of light (m/s)

This represents the energy crossing per unit area per unit time perpendicular to the direction of propagation.

In terms of the maximum electric field, intensity can also be expressed as: $I = \frac{1}{2}\epsilon_0 E_0^2 c$

Example: For an electromagnetic wave with electric field $E_0 = 50$ N/C, the intensity would be approximately 3.3 W/m².

Displacement Current Derivation

1. For a parallel-plate capacitor:

   • Electric field between plates: $E = \frac{Q}{\epsilon_0 A}$
   • Electric flux: $\Phi_E = EA = \frac{Q}{\epsilon_0}$

2. Displacement current definition: $i_d = \epsilon_0\frac{d\Phi_E}{dt} = \epsilon_0\frac{d}{dt}\left(\frac{Q}{\epsilon_0}\right) = \frac{dQ}{dt}$

3. For capacitor charged by constant current $i$:

   • Conduction current in wires: $i_c = \frac{dQ}{dt} = i$
   • Therefore: $i_d = i_c = i$

Conceptual Significance

• Current Continuity: Displacement current maintains continuity of current through the complete circuit, including regions where no physical charges flow

• Maxwell's Modification: Added displacement current term to Ampère's Law, creating the generalized Ampère-Maxwell Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0\epsilon_0\frac{d\Phi_E}{dt}$

• Resolving Conservation Issues: Without displacement current, Ampère's Law would violate charge conservation in circuits with time-varying fields

• Electromagnetic Waves: Displacement current is essential for the existence of electromagnetic waves, as it couples time-varying electric and magnetic fields