Question

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).

Answer 1

Given:-

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)`