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15.8: Summary of Maxwell's and Poisson's Equations

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  • Maxwell's Equations

    Maxwell's equations:

    \[\boldsymbol{\nabla} \cdot \textbf{D} = \rho. \tag{15.8.1} \label{15.8.1}\]

    \[ \boldsymbol{\nabla} \cdot \textbf{B} = 0. \tag{15.8.2} \label{15.8.2}\]

    \[\boldsymbol{\nabla} \times \textbf{H} = \dot {\textbf{D}} + \textbf{J}. \tag{15.8.3} \label{15.8.3}\]

    \[\boldsymbol{\nabla} \times \textbf{E} = - \dot{ \textbf{B}}. \tag{15.8.4} \label{15.8.4}\]

    Sometimes you may see versions of these equations with factors such as \(4\pi \) or \(c\) scattered liberally throughout them. If you do, my best advice is to white them out with a bottle of erasing fluid, or otherwise ignore them. I shall try to explain in Chapter 16 where they come from. They serve no scientific purpose, and are merely conversion factors between the many different systems of units that have been used in the past.

    Poisson's Equation

    Poisson's equation for the potential in an electrostatic field:

    \[ \nabla^2 V = - \dfrac{\rho}{\epsilon} \tag{15.8.5} \label{15.8.5}\]

    The equivalent of Poisson's equation for the magnetic vector potential on a static magnetic field:

    \[ \nabla^2 \textbf{A} = - \mu \textbf{J} \tag{15.8.6} \label{15.8.6}\]