Finding the electrostatic potential of a square sheet.

[majormuss]

Homework Statement

Consider a uniform surface charge density σ on a square of unit area.
(a) Compute the electrostatic potential Φ along the line normal to the center of the square.

My current attempt at a solution (image attached) is either incomplete or is simply wrong but I am unable to figure out where I went wrong. I am told the final answer involves an arc-coth (hyperbolic inverse cotangent) term and an inverse tangent term. I have one of those but not the other and I am now at a loss. Any help is highly appreciated thanks.

Homework Equations

https://bit.ly/2SGFpUM Also attached

The Attempt at a Solution

See attached picture.

 

[TSny]

You will need to integrate with respect to $y$ as well as $x$.
$r = \sqrt{z^2 + x^2}$ only if $y = 0$.

Posting images of your work can make it difficult for helpers to quote specific parts of your work. So, we ask that you type your work if possible.
See item #5 here: https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686781/

 

[majormuss]

You will need to integrate with respect to $y$ as well as $x$.
$r = \sqrt{z^2 + x^2}$ only if $y = 0$.

Posting images of your work can make it difficult for helpers to quote specific parts of your work. So, we ask that you type your work if possible.
See item #5 here: https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686781/

Hi TSny,
Thanks for your suggestion! The thought about adding the 'y' did cross my mind many times but I was under the impression that since the dx slices, when integrated from x=-1/2 to x=1/2, accounted for the entire surface of the square, hence, making it unnecessary to integrate the y-axis? I know my reasoning sounds flawed but I am having a hard time seeing why a 3D Pythagorean theorem is necessary here since, as my sketch shows if infinite slices of the sheet are used, then the entire surface would be accounted for. What do you think?

Sorry, I didn't know about that rule. I don't know of any relatively easy ways of typing up my attempt. Would cleanly and more orderly written out (also numbered) steps help? That could make referencing specific steps a lot easier.

 

[TSny]

I was under the impression that since the dx slices, when integrated from x=-1/2 to x=1/2, accounted for the entire surface of the square, hence, making it unnecessary to integrate the y-axis?

When you do this, you have not taken into account that different points along your red strip are at different distances from the point $P$ on the z-axis where you are finding $\Phi$. You can't lump all of the charge of the red strip as a single point charge located on the x-axis. An element of charge on the red strip that is near one end of the strip is farther from $P$ than an element of charge on the red strip that is close to the x-axis. So, different elements of charge along the strip contribute differently to the potential at $P$. So, you need to sum up the contributions to $\Phi$ from all the different elements of charge on the red strip. That is, integrate along the red strip.

Sorry, I didn't know about that rule. I don't know of any relatively easy ways of typing up my attempt. Would cleanly and more orderly written out (also numbered) steps help? That could make referencing specific steps a lot easier.

It is best to try to type in your work. The best way is to use LaTex formatting. See
https://www.physicsforums.com/help/latexhelp/

However, if you are not familiar with LaTex, then you can try using the formatting tools on the toolbar at the top of the window where you type in your posts. In particular, you will find mathematical symbols by clicking on the $\Sigma$ icon on the toolbar