After reading this very interesting post about the electric field and the electric potential of a point charge in 2D and 1D, I've understood that, for the $2D-$case, the following formulas hold: $$ \Phi_{\operatorname{2-d}}(r) = -\frac{\lambda}{2\pi \epsilon_0} \ln(r) $$
$$ \vec{E}_{\operatorname{2-d}}(r) = \frac{\lambda}{\epsilon_0} \left(\frac{\hat{r}}{2\pi r}\right) $$
Nevertheless, I haven't fully understood how the dimensional analysis and the numerical values of the quantities that come into play (namely, the electric charge $\lambda$ and the vacuum permittivity $\epsilon_0$) change due to the reduced dimensionality of the system.
In other words:
- Is the value of the vacuum permittivity still $8.85 \, 10^{-12}\, F/m$, as in the usual 3D world?
- Is the value of a single electric charge still $1.60 \, 10^{-19} C$, as in the 3D world?