Electric Fields

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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} = \int \vec{E} \cdot \hat{n} \Delta A }[/math]

Then

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

The quantitative relationship between measurements of electric field is stated, but the amount of charge inside the closed surface is not described.

Recall that Coulomb's law is

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

Consider that we have a sphere with charges going outward. The sum of the total flux will be

[math]\displaystyle{ \sum_{\text{closed surface}} \vec{E} \cdot \hat{n} \Delta A = {1\over 4\pi\varepsilon_0}{Q\over r^2}(+1)(4\pi r^2) = {Q\over \varepsilon_0} }[/math]

having

[math]\displaystyle{ \vec{E}={1\over 4\pi\varepsilon_0}, \hat{n} = (+1), \Delta A (4\pi r^2) \text{ (surface area of sphere)} }[/math]

Now in applying Gauss' law to the electric field of a point charge, The Gauss' law can be stated as

[math]\displaystyle{ \sum_{\text{closed surface}} \vec{E} \cdot \hat{n} \Delta A = {\sum q_{inside} \over \varepsilon_0} }[/math]

or

[math]\displaystyle{ \oint \vec{E} \cdot \hat{n} \Delta A = {\sum q_{inside} \over \varepsilon_0} }[/math]

Further Description

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

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

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