How to Find the Electric Field Above a Square Loop with Uniform Line Charge?

[Niles]

Homework Statement

Find the electric field a distance z above the center of a square loop (side s) carrying a uniform line charge λ (see Figure 1) - it's problem 2 on this page:

http://teacher.pas.rochester.edu/PHY217/Homework/HomeworkSet3/HomeworkSet3.html

The Attempt at a Solution

Ok, first I use symmetry, so the horizontal components must cancel for the 4 edges. The vertical component of one edge is given by:

$$\frac{2}{{4\pi \varepsilon }}\int\limits_{x1 }^{x2 } {\frac{\lambda }{{\left( {x^2 + z^2 } \right)}}} \cos \phi dx$$,

where cos(\phi) = z / sqrt(x^2+z^2) so x1 = s/2 and x2=s/sqrt(2) since these are the minimal and maximal values of x.

Pay attention to that I have multiplied with 2, so we find the one whole edge, not just the half of one edge. I integrate, and multiply it all with 4 to give the correct result.

Am I correct?

 

[Jhero]

I would like to commend you on your use of symmetry to simplify the problem. Your approach seems logical and your integration process is correct. However, I would suggest double-checking your limits of integration for the vertical component. Since the loop is centered at the origin, the limits should be from -s/2 to s/2, rather than s/2 to s/sqrt(2). Additionally, I would recommend including units in your equations to ensure accuracy in your final result. Overall, your attempt at a solution is on the right track, but make sure to double-check your limits and units for a more accurate answer.