Drift of a Positron's Position Between two Plates
I am writing code for a 1D case where a positron with charge $e$ and mass $m$ is placed between a charged plate, and a grounded plate in the positive $x$ direction, a distance $d$ away . Let the initial velocity and position of the electron be zero, i.e., it is at rest on the charged plate (non physical, I know, but lets just run with it). Say the charged plate is supplied by a potential $V = V_0 \sin(\omega t) $. Since the electric field $E = \frac{V}{d}$ , then, $$\frac{dv}{dt} = \frac{eV_0 \sin(\omega t)}{md}$$ Integrating this for velocity gives, $$\frac{dx}{dt} = -\frac{eV_0 \cos(\omega t)}{\omega md}+C$$ As the initial velocity is $0$, $C = \frac{eV_0}{\omega md} $, thus, $$\frac{dx}{dt} = \frac{eV_0}{\omega md}(1-\cos(\omega t)) $$ Integrating again gives position $x$,
$$x = \frac{eV_0}{\omega m d}(t-\frac{\sin(\omega t)}{\omega} + D)$$ Where $D$ is some constant. In fact, $D = 0$ since the initial position $=0$ .
So, finally we have that $$x = \frac{eV_0}{\omega m d}(t-\frac{\sin(\omega t)}{\omega})$$
This might look fine, but when I plot $x$ vs $t$, I obtain this:
Note that I initialised the positron at $\frac{d}{2}$, not $x = 0$, but the same maths follows. As can be seen, the positron drifts over time. 402
This is my problem, I can't seem to understand why it does that. More intriguing still, if I complete the derivation again, this time for $V = V_0 \cos (\omega t)$, I obtain equations of simple harmonic motion, and the position does not drift over time. Yet surely the relationship between $\cos$ and $\sin$ are just a phase shift for the position. I'm not looking for you all to just solve my workings, I entered my workings for some context. I would just like some advice on how to go about visualising why drift occurs, given logic would state that the positron would accelerate away from the positive plate, reaching a maximum acceleration before accelerating at a lessening rate. It would then surely start being attracted and return to the original starting position for each full cycle. 766