Electric Fields

This page has been claimed by YongHui Cho

The Main Idea

First of all, Gauss' law is a quantitative relationship between measurements of electric field on a closed surface and the amount and sign of the charge inside that closed surface. In which, 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.

To understand Gauss' law, understanding the concept of "flux" in context of Gauss' law is very important.

Flux is the quantitative measure of the amount and direction of electric field over an entire surface.

Flux has 3 properties:

1. Direction of Electric Field : In relation to the surface, if the electric field is directed outward from the surface the electric flux is positive. If the electric field is directed inward toward the surface the electric flux is negative. When in parallel with the surface, the electric flux is zero. Therefore, the electric flux is related to the angle the electric field makes with the surface.

2. Magnitude of Electric Field : The electric flux is directly proportional to Electric field and the angle [math]\displaystyle{ \cos \theta }[/math]. Therefore, the definition of electric flux contains the product of [math]\displaystyle{ E \cos \theta }[/math].

3. Surface Area: The electric flux is affected by the changing size of the surface. Taking account of the electric field on the surface, the surface area required to calculate flux.

Now we can now define Electric Flux on a surface as

[math]\displaystyle{ \Phi_el = \sum_{surface} \vec{E} \cdot \hat{n} \Delta A }[/math]

[math]\displaystyle{ \vec{E} }[/math] represents the Electric Field, [math]\displaystyle{ \hat{n} }[/math] represents the direction, [math]\displaystyle{ \Delta A }[/math] represents the surface area.

Now, back to the definition of Gauss' law - Gauss' law is a quantitative relationship between measurements of electric field on a close surface and the amount and sign of the charge inside that closed surface

This simply means that Gauss' law is a sum of flux in a closed surface.

Since

[math]\displaystyle{ \text{electric flux on a surface} = \sum_{surface} \vec{E} \cdot \hat{n} \Delta A }[/math]

it can also be written as

[math]\displaystyle{ \text{electric flux on a surface} = \Phi_{el} = \oint \vec{E} \cdot \hat{n} \Delta A }[/math]

A Mathematical Model

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

[math]\displaystyle{ E = {1\over 4\pi\varepsilon_0}{Q\over r^2} }[/math]

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

Be sure to show all steps in your solution and include diagrams whenever possible

Connectedness

1. How is this topic connected to something that you are interested in?
2. How is it connected to your major?
3. Is there an interesting industrial application?

History

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

See also

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

Further reading

Books, Articles or other print media on this topic

External links

Internet resources on this topic

References

This section contains the the references you used while writing this page