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🔬Modern Optics

🔬modern optics review

1.2 Maxwell's equations and electromagnetic waves

5 min readLast Updated on July 22, 2024

Maxwell's equations are the foundation of electromagnetic theory. They describe how electric and magnetic fields interact and propagate through space. These equations explain the behavior of light and other electromagnetic waves.

Electromagnetic waves have fascinating properties. They can travel through a vacuum at the speed of light, carry energy and momentum, and exhibit polarization. Understanding these properties is crucial for many modern technologies, from telecommunications to medical imaging.

Electromagnetic Theory

Maxwell's equations: forms and significance

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  • Maxwell's equations in differential form encapsulate the fundamental laws of electromagnetism
    • Gauss's law for electric fields E=ρε0\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} relates the electric field to the charge density
    • Gauss's law for magnetic fields B=0\nabla \cdot \mathbf{B} = 0 states that magnetic fields have no sources or sinks (no magnetic monopoles exist)
    • Faraday's law ×E=Bt\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} describes how changing magnetic fields induce electric fields (electromagnetic induction)
    • Ampère's law with Maxwell's correction ×B=μ0J+μ0ε0Et\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} relates the magnetic field to the current density and changing electric fields (displacement current)
  • Maxwell's equations in integral form provide a macroscopic description of electromagnetic phenomena
    • Gauss's law for electric fields SEdA=Qε0\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q}{\varepsilon_0} states that the electric flux through a closed surface is proportional to the enclosed charge
    • Gauss's law for magnetic fields SBdA=0\oint_S \mathbf{B} \cdot d\mathbf{A} = 0 indicates that the magnetic flux through a closed surface is always zero (no magnetic monopoles)
    • Faraday's law CEdl=ddtSBdA\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{A} relates the electromotive force (EMF) around a closed loop to the negative rate of change of the magnetic flux through the loop
    • Ampère's law with Maxwell's correction CBdl=μ0I+μ0ε0ddtSEdA\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I + \mu_0 \varepsilon_0 \frac{d}{dt} \int_S \mathbf{E} \cdot d\mathbf{A} equates the magnetic field circulation around a closed loop to the sum of the current and the displacement current through the loop

Wave equation from Maxwell's equations

  • The wave equation for electromagnetic waves can be derived from Maxwell's equations
    1. Start with Faraday's law ×E=Bt\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} and Ampère's law with Maxwell's correction ×B=μ0J+μ0ε0Et\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
    2. Take the curl of both sides of Faraday's law ×(×E)=t(×B)\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B})
    3. Substitute Ampère's law into the right-hand side ×(×E)=μ0Jtμ0ε02Et2\nabla \times (\nabla \times \mathbf{E}) = -\mu_0 \frac{\partial \mathbf{J}}{\partial t} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}
    4. Use the vector identity ×(×E)=(E)2E\nabla \times (\nabla \times \mathbf{E}) = \nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} and Gauss's law for electric fields to simplify 2Eμ0ε02Et2=μ0Jt\nabla^2 \mathbf{E} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = \mu_0 \frac{\partial \mathbf{J}}{\partial t}
    5. In a source-free region (no charges or currents), the wave equation for the electric field becomes 2Eμ0ε02Et2=0\nabla^2 \mathbf{E} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0
  • A similar derivation can be done for the magnetic field, resulting in the wave equation for the magnetic field 2Bμ0ε02Bt2=0\nabla^2 \mathbf{B} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0
  • The wave equations demonstrate that electromagnetic fields propagate as waves in space and time, with a speed determined by the permittivity ε0\varepsilon_0 and permeability μ0\mu_0 of free space

Speed of electromagnetic waves

  • The speed of electromagnetic waves in vacuum is a universal constant denoted by cc and can be calculated from the permittivity ε0\varepsilon_0 and permeability μ0\mu_0 of free space c=1μ0ε03×108m/sc = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3 \times 10^8 \, \text{m/s}
  • In a medium, the speed of electromagnetic waves vv is given by v=1μεv = \frac{1}{\sqrt{\mu \varepsilon}}, where μ\mu and ε\varepsilon are the permeability and permittivity of the medium
    • The refractive index of a medium nn is defined as n=cv=μεμ0ε0n = \frac{c}{v} = \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}}
    • For non-magnetic materials (μμ0\mu \approx \mu_0), the refractive index can be approximated as nεε0n \approx \sqrt{\frac{\varepsilon}{\varepsilon_0}}
  • The speed of electromagnetic waves in a medium is always less than the speed of light in vacuum v=cnv = \frac{c}{n} (light slows down in matter)

Properties of electromagnetic waves

  • Electromagnetic waves are transverse waves, with the electric field E\mathbf{E} and magnetic field B\mathbf{B} perpendicular to each other and to the direction of propagation (wave vector k\mathbf{k})
  • The speed of electromagnetic waves in vacuum is given by c=1μ0ε0c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} and is approximately 3×108m/s3 \times 10^8 \, \text{m/s}
  • Electromagnetic waves carry energy and momentum, with the Poynting vector S=E×H\mathbf{S} = \mathbf{E} \times \mathbf{H} representing the power density and direction of energy flow (H\mathbf{H} is the magnetic field intensity)
  • Electromagnetic waves exhibit polarization, which refers to the orientation of the electric field vector
    • Linear polarization: electric field oscillates in a single plane (horizontal or vertical)
    • Circular polarization: electric field vector rotates in a circular path (right-handed or left-handed)
    • Elliptical polarization: electric field vector traces an elliptical path (combination of linear and circular)

Wave Properties

Explain the concepts of polarization, phase velocity, and group velocity in the context of electromagnetic waves

  • Polarization refers to the orientation of the electric field vector in an electromagnetic wave
    • Linear polarization: electric field oscillates in a single plane perpendicular to the direction of propagation (e.g., horizontal or vertical polarization)
    • Circular polarization: electric field vector rotates in a circular path as the wave propagates, either clockwise (right-handed) or counterclockwise (left-handed)
    • Elliptical polarization: electric field vector traces an elliptical path as the wave propagates, combining aspects of linear and circular polarization (most general case)
  • Phase velocity vpv_p is the speed at which a particular phase of the wave (e.g., a crest or trough) travels through space
    • For electromagnetic waves in vacuum, the phase velocity is equal to the speed of light vp=ωk=cv_p = \frac{\omega}{k} = c, where ω\omega is the angular frequency and kk is the wavenumber
    • In dispersive media, the phase velocity depends on the frequency of the wave vp(ω)=ωk(ω)v_p(\omega) = \frac{\omega}{k(\omega)} (different frequencies travel at different speeds)
  • Group velocity vgv_g is the speed at which the envelope of a wave packet (a superposition of waves with slightly different frequencies) travels through space
    • It represents the speed at which energy and information are transported by the wave vg=dωdkv_g = \frac{d\omega}{dk}
    • In non-dispersive media, the group velocity is equal to the phase velocity vg=vp=cv_g = v_p = c (wave packet maintains its shape)
    • In dispersive media, the group velocity can differ from the phase velocity, leading to phenomena such as pulse broadening and chirping (wave packet changes shape as it propagates)


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.