1.2 Maxwell's equations and electromagnetic waves

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

Top images from around the web for Maxwell's equations: forms and significance

• 

  Maxwell Equations [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Maxwell's equations - Wikipedia View original

  Is this image relevant?

• 

  Faraday’s Law — Electromagnetic Geophysics View original

  Is this image relevant?

• 

  Maxwell Equations [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Maxwell's equations - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Maxwell's equations: forms and significance

• 

  Maxwell Equations [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Maxwell's equations - Wikipedia View original

  Is this image relevant?

• 

  Faraday’s Law — Electromagnetic Geophysics View original

  Is this image relevant?

• 

  Maxwell Equations [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Maxwell's equations - Wikipedia View original

  Is this image relevant?

1 of 3

• Maxwell's equations in differential form encapsulate the fundamental laws of electromagnetism
  • Gauss's law for electric fields $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$ relates the electric field to the charge density
  • Gauss's law for magnetic fields $\nabla \cdot \mathbf{B} = 0$ states that magnetic fields have no sources or sinks (no magnetic monopoles exist)
  • Faraday's law $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ describes how changing magnetic fields induce electric fields (electromagnetic induction)
  • Ampère's law with Maxwell's correction $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$ 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 $\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q}{\varepsilon_0}$ states that the electric flux through a closed surface is proportional to the enclosed charge
  • Gauss's law for magnetic fields $\oint_S \mathbf{B} \cdot d\mathbf{A} = 0$ indicates that the magnetic flux through a closed surface is always zero (no magnetic monopoles)
  • Faraday's law $\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{A}$ 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 $\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I + \mu_0 \varepsilon_0 \frac{d}{dt} \int_S \mathbf{E} \cdot d\mathbf{A}$ 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 $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ and Ampère's law with Maxwell's correction $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$
  2. Take the curl of both sides of Faraday's law $\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B})$
  3. Substitute Ampère's law into the right-hand side $\nabla \times (\nabla \times \mathbf{E}) = -\mu_0 \frac{\partial \mathbf{J}}{\partial t} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}$
  4. Use the vector identity $\nabla \times (\nabla \times \mathbf{E}) = \nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$ and Gauss's law for electric fields to simplify $\nabla^2 \mathbf{E} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = \mu_0 \frac{\partial \mathbf{J}}{\partial t}$
  5. In a source-free region (no charges or currents), the wave equation for the electric field becomes $\nabla^2 \mathbf{E} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$
• A similar derivation can be done for the magnetic field, resulting in the wave equation for the magnetic field $\nabla^2 \mathbf{B} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0$
• The wave equations demonstrate that electromagnetic fields propagate as waves in space and time, with a speed determined by the permittivity $\varepsilon_0$ and permeability $\mu_0$ of free space

Speed of electromagnetic waves

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

Properties of electromagnetic waves

• Electromagnetic waves are transverse waves, with the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ perpendicular to each other and to the direction of propagation (wave vector $\mathbf{k}$)
• The speed of electromagnetic waves in vacuum is given by $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$ and is approximately $3 \times 10^8 \, \text{m/s}$
• Electromagnetic waves carry energy and momentum, with the Poynting vector $\mathbf{S} = \mathbf{E} \times \mathbf{H}$ representing the power density and direction of energy flow ($\mathbf{H}$ 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 $v_p$ 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 $v_p = \frac{\omega}{k} = c$, where $\omega$ is the angular frequency and $k$ is the wavenumber
  • In dispersive media, the phase velocity depends on the frequency of the wave $v_p(\omega) = \frac{\omega}{k(\omega)}$ (different frequencies travel at different speeds)
• Group velocity $v_g$ 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 $v_g = \frac{d\omega}{dk}$
  • In non-dispersive media, the group velocity is equal to the phase velocity $v_g = v_p = c$ (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)