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Mathematics for Physics

286 Maxwell’s Equations

Maxwell’s equations express, formally, the properties of electromagnetic waves.

  1. Gauss’s Law for Electricity \[ \nabla \cdot \mathbf{D} = \rho_v \]

  2. Gauss’s Law for Magnetism \[ \nabla \cdot \mathbf{B} = 0 \]

  3. Faraday’s Law \[ \nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}}{\partial t} \]

  4. Ampere-Maxwell Law \[ \nabla \times \mathbf{H} = \mathbf{J} + \dfrac{\partial \mathbf{D}}{\partial t} \]

where

  • \(\nabla\) = vector differential operator.
  • \(\mathbf{D}\) = electric flux density.
  • \(\mathbf{B}\) = magnetic flux density.
  • \(\mathbf{E}\) = electric field intensity.
  • \(\mathbf{H}\) = magnetic field intensity.
  • \(\rho_v\) = volume charge density.
  • \(\mathbf{J}\) = current density.
  • \(\dfrac{\partial \mathbf{D}}{\partial t}\) = displacement current density.