Influence of distance between capacitor plates

We have discoverd:$$y(x) \sim V_\text{p}$$For the influence of the capacitor voltage $V_\text{p}$ on the deflection y(x) does the distance d between the two capacitor plates matter. The distance is important for the strength of the electric field between the plates.

The electric field strength $E$ is given by: $$E=\frac{V_{\text p}}{\text d}$$ Because of the distance is non-varying in our experiment it also yields $$y(x) \sim \frac{V_{\text p}}{\text d}\qquad \text{resp.}\qquad y(x) \sim E$$ In the electron deflection tube used in our experiment the distance d between the capacitor plates is $$d=5{.}4\,\rm{cm}$$.