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The Maxwell Equations

One the the most important applications of the concepts you have learned are the equations which describe classical electricity and magnetism. When we set \mathbf{E} \ as the electric field,  \mathbf{B} \ as the magnetic field, \mathbf{D} \ as the electric displacement field, and \mathbf{H} \ as the magnetizing field, then the equations describing these quantities are

\nabla \cdot \mathbf{D} = \rho

\nabla \cdot \mathbf{B} = 0

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}

\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}

We can integrate all these equations to get the integral forms:

\oint_S  \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V

\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0

\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l}  = -  \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}

\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
 \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}

The four laws are called, respectively, Coulombs/Gauss' Law, absence of magnetic monopoles, Faraday's/Induction Law, and Ampère's Law. Together, they are called the Maxwell equations, after James Clerk Maxwell.

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