Contents
Breakdown of Maxwells Equations
Gauss's Law
The amount of fish passing through a net is the same as the number of fish already in the net (For differing materials, a permittivity constant exists)
The amount of electric flux going through a surface is equal to the amount of charge in the surface.
Differential and integral forms of these given below
Gauss's Law for Magnetism
A year must contain both summer and winter, but the hot and cold from the seasons cancel each other out to make 0.
The magnetic flux going through any surface is 0.
Faraday's Law of Induction
A twig sawing quickly on a stationary one creates fire
A moving magnetic field creates an electric field.
Ampère's law with Maxwell's correction
$ \,D=\epsilon E $
$ \,B=\mu_0 H $
Table 1: Formulation in terms of free charge and current(any material)
| Name | Differential form | Integral form |
|---|---|---|
| Gauss's law: | $ \nabla \cdot \mathbf{D} = \rho_f $ | $ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A = Q_{f}(V) $ |
| Gauss's law for magnetism: | $ \nabla \cdot \mathbf{B} = 0 $ | $ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0 $ |
| Maxwell-Faraday equation (Faraday's law of induction): |
$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} $ | $ \oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t} $ |
| Ampère's circuital law (with Maxwell's correction): |
$ \nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t} $ | $ \oint_{\partial S} \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,S} + \frac {\partial \Phi_{D,S}}{\partial t} $ |
Table 2: Formulation in terms of total charge and current(free space, or no given material)
For the total quantities one has an analogous table; H and D no longer are present:
| Name | Differential form | Integral form |
|---|---|---|
| Gauss's law: | $ \nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0} $ | $ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf E\;\cdot\mathrm{d}\mathbf A = \frac{Q(V)}{\varepsilon_0} $ |
| Gauss's law for magnetism: | $ \nabla \cdot \mathbf{B} = 0 $ | $ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0 $ |
| Maxwell-Faraday equation (Faraday's law of induction): |
$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} $ | $ \oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t} $ |
| Ampère's circuital law (with Maxwell's correction): |
$ \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ $ | $ \oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_S + \mu_0 \varepsilon_0 \frac {\partial \Phi_{E,S}}{\partial t} $ |
