I have a question as follows:
"Show that $E_x $ on the axis of a ring charge of radius $a$ has its maximum value at $x = \pm a /\sqrt{2} $. Sketch $E_x$ versus x for both positivve and negative values of $x$"
This is a bit more abstract than my other questions "Given some givens, find the electric field at this point". As I understand I need to find a function $E_x (x)$ along all x from $-a$ to $a$, and confirm the maxes by taking its derivative that $E_x '(x) = 0$ at the given maxes.
I take a point on $x$ axis, I call it $x$. Considering all $dQ$ along the circle, all of the electric field should be along the x axis due to symmetry. That's nice I guess. The line towards $dQ$ forms an unknown length I'll call $w$. The line projected down from this point on $dQ$ I'll call $u$, and the point from the end of the projection back to $x$ is $v$. I don't have the rep to draw a pretty picture and share it here unfortunately.
The direction I'm obviously going is some trigonometry trickery, but I have no idea where to go. I have $u,v,w$ as triangle sides and are unknowns and $w$ is needed for the distance part of the coulumb's equation. there is the triangle with $a,x,w$ but isn't a right triangle and I'm unsure of any useful properties to work with there. Perhaps there's some actual physics I have overlooked. Thanks for any help.