date: 2025-04-16

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Maxwell’s Equations

Adapted from ¹,²

Let \(\vec{E}\) be an electric field, \(\vec{B}\) a magnetic field.

Gauss for electricity

\[div\ \vec{E} = \oint \vec{E} \cdot d\vec{A} = \frac{\rho}{\varepsilon_o}\]

Where \(\rho\) is electric charge density, \(\varepsilon_o\) vacuum permittivity.

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Faraday

\[curl\ \vec{E} = \oint \vec{E} \cdot d\vec{s} = -\frac{\partial{\vec{B}}}{\partial t}\]

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Gauss for magnetism

\[div\ \vec{B} = \oint \vec{B} \cdot d\vec{A} = 0\]

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Ampère-Maxwell

\[curl\ \vec{B} = \oint \vec{B} \cdot d\vec{s} = \mu_o \vec{J} + \mu_o \varepsilon_o \frac{\partial{\vec{E}}}{\partial t}\]

Where \(\mu_o\) is vacuum permeability, \(\vec{J}\) electric charge density

• \(\mu_o = 4\pi \times 10^{-7}\) H/m
• \(\varepsilon_o = 8.85 \times 10^{-12}\) F/m
• \(\vec{J}\) ?

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Equations

Adapted from ³,⁴

Lenz Law - Eddy Currents

Assume there is a conducting material in shape of a closed circuit.

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