A long cylindrical volume contains a uniformly distributed charge of density ρ. Find the electric field at a point P inside the cylindrical volume at a distance x from its axis (see the figure).
Volume charge density inside the cylinder = ρ
Let the radius of the cylinder be r.
Let charge enclosed by the given cylinder be Q
Consider a Gaussian cylindrical surface of radius x and height h.
Let charge enclosed by the cylinder of radius x be q'.
The charge on this imaginary cylinder can be found by taking the product of the volume charge density of the cylinder and the volume of the imaginary cylinder. Thus, q' = ρ( π x2 h)
From Gauss's Law,
`oint "E" . "ds" = ("q"_"en")/∈_0`
`"E" . 2 pi "xh" = (ρ( pi "x"^2 "h"))/ ∈_0`
`"E" = (ρ"x")/(2∈_0)`