How to calculate Electric Field near a charged conducting surface without Gauss' law?

Question

I have two problems :

1. In every textbook I find the use of Gauss' law in calculation of Electric Field near a charged conducting surface. Can it be calculated without Gauss' law?
2. Suppose while using Gauss' law to calculate field near a charged surface, the Gaussian surface (cylindrical) that we take is so long that the other side of the charged conductor lies within the cylinder, then how is the situation explained?

Answer 1

Usually, applying Gauss's law to a problem with $$ \int_A \vec{E} \cdot d\vec{A} \propto Q$$ is only suitable, if one knows, that the electric field is perpendicular to the surface $A$ and is constant in magnitude over the whole surface. This leaves: $$ E \propto \frac{Q}{A}$$ On can conclude such statements if the problem is symmetric (e.g. spherical symmetric $\rightarrow$ choose $A$ as a surface of sphere).

However, if the problem shows no obvious symmetries, one uses the solution of Poisson's equation for a vanishing potential at infinity or something similar in Lorenz-gauge: $$ \varphi(\vec{r},t) \propto \int \frac{\varrho(\vec{r}\ ', t_r)}{|\vec{r} - \vec{r}\ '|} \ d^3r\ '$$ with $\varrho$ charge distribution and $t_r = t - \frac{|\vec{r} - \vec{r}\ '|}{c}$