How Is the Radial Component of an Electric Field Proven to Cancel Out in a Ring?

[rick_2009]

This is about the electric field of a ring with radius r, at a distance z from center, along the axis of the ring. The ring carries a uniform line charge $$\lambda$$. We always say that the radial component of the field cancels out due to symmetry. Can somebody tell how to prove it mathematically (using cylindrical coordinate system only)?

$$dE_{rad}=-\frac{1}{4\pi\epsilon_0}\frac{r\lambda d\theta}{(r^2+z^2)}\hat{\textbf{r}}$$
$$E_{rad}=-\frac{1}{4\pi\epsilon_0}\frac{r\lambda}{(r^2+z^2)}\int_0^{2\pi} \hat{\textbf{r}} d\theta$$

?

 

[Born2bwire]

First thing you need to do is convert the directional vector that is local to the current element to a directional vector at the point of observation.

So the electric field element points in the r-hat direction, convert that to cylindrical coordinates and then also provide the appropriate translation since you will be observing it at a point offset from the origin of the coordinate system of the electric field element.

 

[MinaKaiser]

I don't get your answer