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# Figure shows a section of an infinite rod of charge having linear charge density $\lambda$ which is constant for all points on the line. Find electric field E at a distance r from the line.

From symmetry, $\bar {\bar {E}}$ due to a uniform linear charge can only be radially directed. As a Gaussian surface, we can choose a circular cylinder of radius r and length l, closed at each end by plane caps normal to the axis.
$\in_o \int \overrightarrow {E} d \overrightarrow {s} =q _{in}$
$\in_0 [ \int \overrightarrow {E} . d\overrightarrow {s} +\int \overrightarrow {E} d \overrightarrow {s} ]=q_{in}$
$E= \large\frac{\lambda l}{\in_0 2 \pi r l}$
$\quad= \large\frac{\lambda}{2 \pi \in_0 r}$
The direction of $\overrightarrow {E}$ is radially outward for a line of positive charge.