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Axis Rod

How do i find the proper integral of this uniformly charged rod?

A uniformly charged rod is places along the x-axis from x=0 to x= L. carefully set up, but do not solve the proper integral to determine the x component of the electric field at the point (L,a)

i know for a line dq=λdl
i think its something along the lines of
Ey=Kλy= ∫ dx/(x^2y^2)^(3/2) from 0 to L

I have no idea if this is right. I am using my notes to guide me.

Choose an arbitrary charge element, dq = Ddx , a distance “x” from the origin and distance (L-x) from the other end. (D is the charge density)
Draw position vector “r” from this “dq” to point “a” above the end of the rod. “r” will make angle “theta” with x-axis.
The field at “a” (due to point charge “dq”) is then;

dE = kDdx/r^2

To Integrate you have to get components;

dEx = dECos(theta) = dE(L-x)/r

dEy = dESin(theta) = dEa/r

Since dE is in terms of “dx” you need to write “r” in terms of “x” as;

r = [(L-x)^2+a^2]^(1/2)

Put all this together to get;

dEx = kD(L-x)dx/[(L-x)^2+a^2]^3/2

dEy = kDadx/[(L-x)^2+a^2]^3/2

Integrate from x=0 to x=L , (and D=Q/L if you’re given Q rather then D)

This can be simplified somewhat by letting, z=L-x , then dx = – dz

dEx = – kDzdz/[z^2+a^2]^3/2

dEy = -kDadz/[z^2+a^2]^3/2

Integrate from z = L to z = 0

This last form is equivalent to putting your origin at the right end of the rod, but I prefer to set it up conventionally and then look for ways to simplify it later.

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