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

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?

## 1 Answer

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 $t_r = t - \frac{|\vec{r} - \vec{r}\ '|}{c}$