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:} }[/math]

[math]\displaystyle{ Φelectric ={ Q \over ε_0} }[/math]

[math]\displaystyle{ 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 all sides. 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]

Middle:

A box with a height of 2 m, a width of 3 m, and a length of 4 m has an electric field, E1, of magnitude 900 N that points in the positive x and y with a 30 degree angle to the top surface and electrics field, E2, of magnitude 900 N in the negative x and y with a 30 degree angle to the bottom surface. How much charge is in the box?

[math]\displaystyle{ Φelectric= \int \ {\vec{E}cosθdA} }[/math]

Pulling out the constant E and theta, and integrating dA give:

[math]\displaystyle{ Φelectric= E*A*cos(θ) }[/math]

The angle between the surface normal and the electric field is not given. Making a right triangle using the normal and the electric field. The angles in a triangle must add up to 180 and two of them, 30 and 90, are known. The angle between the field and the normal must be 180- 90-30= 60.

[math]\displaystyle{ Φtop= 900*cos(60)*3*4= 5400 }[/math]

The bottom surface is calculated the same way, and the angle between the electric field and the normal is found to be 60 degrees, so:

[math]\displaystyle{ Φbottom= 900*cos(60)*3*4= 5400 }[/math]

[math]\displaystyle{ Φtotal= Φtop + Φbottom + Φright + Φleft + Φfront + Φback }[/math]

[math]\displaystyle{ Φtotal= 5400 +5400 = 10800 }[/math]

[math]\displaystyle{ \text{Gauss's Law for Electric Fields:} \oint{ \vec{E} \cdot d\vec{A} } ⃗= {Q\over ε_0} }[/math]

[math]\displaystyle{ 10800 = {Q\over ε_0} }[/math]

[math]\displaystyle{ ε_0 = 8.84e-12 }[/math]

Solving for Q gives: [math]\displaystyle{ Q= {10800*8.84e-12}= 9.56e-8 }[/math] C

Difficult:

2 m away from an unknown particle there is an electric field of magnitude 700 N pointing outward in all directions. What is the charge of the particle?

Because the field is radiating outward and is constant at a given location from the particle, the closed surface used for Gauss's law should be a spherical shell of radius 2 m with the particle at the center. This allows the field to be perpendicular to the surface at any given location and constant over the closed surface.

[math]\displaystyle{ SA_sphere = 4\pi r^2 }[/math]

[math]\displaystyle{ \text{Gauss's Law for Electric Fields:} \oint{ \vec{E} \cdot d\vec{A} } ⃗= {Q\over ε_0} }[/math]

[math]\displaystyle{ E*cos(θ)*4\pi r^2 = 700* cos(90)*4\pi r^2 = 0 }[/math]

[math]\displaystyle{ 0 = {Q\over ε_0} }[/math]

[math]\displaystyle{ Q= 0 }[/math]

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.

References

[1] [2]