I've been presented with a problem in which an electron is placed a certain distance x from the center of a positively charged ring and allowed to move freely. The ring has a known charge density λ. I am tasked with finding the velocity of the electron as it passes through the center of the ring.
First of all, I know that when the electron is initially positioned it will have a certain electric potential energy and that as it moves in towards the ring that potential energy will be converted into kinetic energy until it reaches the center, at which point all of the potential energy will be converted into kinetic energy.
The best way that I can think of to find the initial potential energy is to first find the electric potential at that point and then translate that into potential energy via the equation $V = \frac{U_e}{q}$. However, I am not certain what charges should be used in each equation. I have tried using the total charge of the ring (extrapolated from λ) in the electric potential calculation, and the dividing the charge of the electron out of that expression to produce the electric potential energy. Which produces the following equations:
$$ V = k\int\frac{dq}{r} = k\int\frac{dq}{\sqrt{R^2 + x^2}} = \frac{kQ}{\sqrt{R^2 + x^2}} $$
$$ U_e = \frac{V}{q_e} = \frac{kQ}{q_e\sqrt{R^2 + x^2}} $$
$$ U_e = \frac{1}{2}mv^2 $$
$$ velocity = \sqrt{\frac{2U_e}{m_e}} = \sqrt{\frac{2kQ}{m_eq_e\sqrt{R^2+x^2}}} $$
I'm told that this is an incorrect solution. Could somebody please explain to me where I went wrong?