Electric field on the axis of a ring-shaped charged conductor

[yashboi123]

Homework Statement

A ring-shaped conductor with radius a = 2.20 cm has a total positive charge Q = 0.130 nC uniformly distributed around it

Relevant Equations

E = F/q

Hello. I was wondering why do we not multiply cos(alpha) by 2. I believe we should do this since the y-components of the electric field cancel out, meaning there would be 2 x-components of the electric field(at least I think so). Currently, this derivation/answer only considers one horizontal component, not the other half.

 

[PhDeezNutz]

How does it only consider one horizontal component when the integration is from $0$ to $2 \pi$?

The integration bounds take care of this “doubling up effect”.

If you did it from $0$ to $\pi$ you would certainly have to multiply by 2 at the end.