Electric Flux
The Main Idea
Electric flux through an area is the electric field multiplied by the area of a plane that is perpendicular to the field. Gauss's Law relates the electric flux through an area to the amount of charge enclosed in that area. Gauss's law can be applied to any closed surface and calculates the amount of charge enclosed based on the electric field at that closed surface.
A Mathematical Model:
[math]\displaystyle{ \text{Electric Flux:} Φelectric ={ Q \over ε_0} }[/math]
[math]\displaystyle{ \text {Electric Flux:} Φelectric= \int \ {\vec{E}cosθdA} }[/math]
Where theta is the angle between the electric field vector and the surface normal.
Combining these two equations gives:
[math]\displaystyle{ \text{Gauss's Law for Electric Fields:} \oint{ \vec{E} \cdot d\vec{A} } ⃗= {Q\over ε_0} }[/math]
Examples
Simple: A box with a height of 2m, a width of 3m, and a length of 4m has an Electric field of (0, -1400, 0) N uniformly covering it. What is the total electric flux of the box?
Electric flux= [math]\displaystyle{ Φelectric= \int \ {\vec{E}cosθdA} }[/math]
The electric field and area of each side are constant, so they can be pulled out of the integral to give: [math]\displaystyle{ Φelectric= E*cosθ*A }[/math]
For the front, back, left and right sides the angle between the normal and the electric field is 90 degrees, and cos(90)=0. For each of those sides, [math]\displaystyle{ Φelectric= E*0*A }[/math] , so there is no electric flux through those sides.
On the top, the normal points up and the electric field points down, so the angle between them is 180. cos(180)= -1. Electric field and the dimensions are plugged in to give: [math]\displaystyle{ Φtop= 1400*-1*3*4 = -16,800 }[/math]
On the bottom face, the normal points down and the electric field vector also points dowm, so the angles between them is 0. cos(0)=1. When the electric field and dimensions are plugged in the electric flux equals: [math]\displaystyle{ Φbottom= 1400*1*3*4 = 16,800 }[/math]
The total electric flux is the sum of all of the fluxes.
[math]\displaystyle{ Φtotal= Φtop + Φbottom + Φright + Φleft + Φfront + Φback }[/math] [math]\displaystyle{ Φtotal= -16800 +16800 +0 +0 +0 +0 = 0 }[/math]
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Connectedness
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History
Carl Gauss discovered this relation in 1835 and the equation was published in 1867. It is considered to be one of the four equations that are the basis for electrodynamics.
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