Find the Electric field due to two curved rods

1. The problem statement, all variables and given/known data

Two curved plastic rods, together form a circle of radius r. The top half of the circle is the first rod with charge -q, the bottom half of the circle is the other rod with charge q. Find the magnitude and direction of the electric field at point P, at the center of the circle.





2. Relevant equations

dE=k∫dq/r²





3. The attempt at a solution

I'm wondering if my assumptions are correct.

So since the top half of the circle has charge -q, the electric field would go towards the rod, away from the center so it would not contribute.

The bottom half of the circle has charge +q, so the electric field would be directed away from the rod, towards the center of the circle.

By symmetry the x-components of the field would cancel and we would have only the y-components.

Therefore dE=dE_y=dEsinθ=k∫dqsinθ/r²

we know that λdx=dq

but we know that dx=rdθ, so by substitution I would have, dq=λrdθ

also λ=q/x=q/(rθ)

so dq=qrdθ/(r*pi)

for my limits of integration I would go from pi to 2pi

does all of this seem reasonable? or is it completely off?