Electric Fields

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The Main Idea

Gauss' Law or Gauss' Theorem can be derived from the Coulomb's Law. Coulomb's law is

[math]\displaystyle{ E = \frac{1}{4 \pi \epsilon_0)\ \frac{Q}{r^2}\ }[/math]

According to Gauss' Law for Electricity, the electric flux out of any closed surface is directly proportional to the total charge enclosed within the surface.

A Mathematical Model

What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.

Electric fields are caused by electric charges or varying magnetic fields The former effect is described by Gauss's law, the latter by Faraday's law of induction, which together are enough to define the behavior of the electric field as a function of charge repartition and magnetic field. However, since the magnetic field is described as a function of electric field, the equations of both fields are coupled and together form Maxwell's equations that describe both fields as a function of charges and Electric current|currents.

In the special case of a steady state (stationary charges and currents), the Maxwell-Faraday inductive effect disappears. The resulting two equations (Gauss's law [math]\displaystyle{ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} }[/math] and Faraday's law with no induction term [math]\displaystyle{ \nabla \times \mathbf{E} = 0 }[/math]), taken together, are equivalent to Coulomb's law, written as [math]\displaystyle{ \boldsymbol{E}(\boldsymbol{r}) = {1\over 4\pi\varepsilon_0}\int d\boldsymbol{r'} \rho(\boldsymbol{r'}) {\boldsymbol{r} - \boldsymbol{r'} \over |\boldsymbol{r} - \boldsymbol{r'}|^3} }[/math] for a charge density [math]\displaystyle{ \mathbf{\rho}(\mathbf{r}) }[/math] ([math]\displaystyle{ \mathbf{r} }[/math] denotes the position in space). Notice that [math]\displaystyle{ \varepsilon_0 }[/math], the permittivity of vacuum, must be substituted if charges are considered in non-empty media.

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

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See also

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