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Extended one-dimensional rod of length L and charge Q

I'm taking the E&M Class of a first-year Physics with Calculus series. We're working
with determining the electric fields of extended objects with integration, and I'm having trouble with properly modeling the problems. If someone could show me a sample solution to one of these, I feel it could really help. Here goes:

Consider an extended one-dimensional rod of length L and charge Q uniformly distributed. Determine the field from the rod at a point P a distance y from its center perpendicular to the rod.

(It asks for the answer in terms of the given variables, and any others you must define for the problem)

Solution Preview

In this formulation of this problem you are supposed to assume that the rod is infinitely thin.

Let us set up some coordinates (x,y,z):
Suppose the rod lies on the X-axis from xmin = -L/2 to xmax = +L/2 and the point P has coordinates P = (0,y,0).

Let us calculate the electric potential F at pount P:
Each small element of length dx of the rod will contribute its part equal dF = dQ / sqrt(x^2+y^2), where the charge on this rod element id dQ = (Q/L)dx.

Now we can integrate ...