# Electric Flux, Gauss's Law Problem

• henryli78
In summary, the conversation revolves around finding the magnitude of the electric flux through a square with a proton located above its center. The equations used include the electric flux equation, Coulomb's constant, and the relationship between electric flux and enclosed charge. The problem is solved using two different methods, with the textbook solution using the assumption of a cube with the square as one of its faces. The flaw in the logic of the first method is that the electric field is not constant along the surface, leading to a different answer.

## Homework Statement

A proton is a distance d/2 directly above the center of a square of side d. What is the magnitude of the electric flux through the square?

## Homework Equations

1. Electric flux, $\Phi_{net}$ = $\oint \vec{E}\cdot d\vec{A}$
2. $\Phi_{net}$ = $\frac{q_{enc}}{\epsilon_{0}}$
3. $\vec{E}$ = $\frac{kQ}{r^{2}}$

## The Attempt at a Solution

I tried to use Equation - 3 first to calculate the net electric field and then from there, use the value of the electric field and multiply it by the area of the square.

I solved for the electric field from Equation 3 to be $\frac{4ke}{d^{2}}$ where k is Coulomb's constant and $e\ =\ 1.602176462(63)\ \times\ 10^{-19}\ C$

Then I used Equation - 1 and because there is only one surface, the area is just equal to ${d^{2}}$. Thus, I calculated $\Phi_{net} \ =\ \frac{4ke}{d^{2}}\times {d^{2}}\ =\ 4ke\ = 5.8 \times 10^{-9} N*m^2/C$.

However, the answer in my textbook says it is actually $3.01 \times10^{-9} N*m^2/C$. They used equation two and assumed that if the proton was contained in a cube and the square was one of the faces of the cube, the net electric flux of the cube would be $\Phi_{net}$ = $\frac{1.6 \times 10^-19}{\epsilon_{0}}$ and thus the electric flux for the square is 6 times less that.

Can someone point out how my method is flawed and where I went wrong in my logic? Thanks!

Ahh I forgot to search first. But is it possible if someone could point out where I have a flaw in my logic? I understand the solution's concept but I get why my answer is different.

The electric field changes from point to pint along the surface. It is not a constant, as you assumed.
This is also discussed in that thread.

Ok thank you very much! I had a hard time really understanding my error but now I get it. Thanks!