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Cylindrical and planar symmetries yield two of the most important results in electrostatics.

Infinite line charge (linear charge density lambda): Gaussian surface: coaxial cylinder of radius r and length L. The curved surface has E perpendicular to it (and constant), while E is parallel to the flat end caps (zero flux contribution). Gauss's law: E * 2pirL = lambdaL/$epsilon_0$. Result: E = $\frac{lambda}{2*pi*epsilon_0*r}$. The 1/r dependence is characteristic of cylindrical symmetry.

Uniformly charged infinite cylinder (volume charge density rho, radius R): Inside (r < R): E = rho*$\frac{r}{2*epsilon_0}$. Field increases linearly. Outside (r > R): E = rhoR^$\frac{2}{2*epsilon_0*r}$ = $\frac{lambda}{2*pi*epsilon_0*r}$ where lambda = rhopi*$R^2$. Same as a line charge.

Infinite plane sheet (surface charge density sigma): Gaussian surface: pillbox (thin cylinder) straddling the sheet, with cross-sectional area A. Only the two flat faces contribute flux: 2EA = sigma*A/$epsilon_0$. Result: E = $\frac{sigma}{2*epsilon_0}$, uniform and independent of distance. This remarkable result means the field is the same whether you are 1 mm or 1 km from an infinite sheet.

Two parallel sheets with opposite charges (+sigma, -sigma): Between the sheets, fields add: E = $\frac{sigma}{epsilon_0}$ (from + to -). Outside, fields cancel: E = 0. This creates the uniform field of a parallel plate capacitor.

Conductor surface: Using a pillbox with one face inside (E = 0) and one outside: EA = sigmaA/$epsilon_0$. Result: E = $\frac{sigma}{epsilon_0}$ (not sigma/2*$epsilon_0$, because the field exists only on one side).