Question

A long cylindrical volume contains a uniformly distributed charge of density $\mathrm{\rho }$. Find the electric field at a point P inside the cylindrical volume at a distance x from its axis (figure 30-E5).

Answer 1

Given:
Volume charge density inside the cylinder = $\mathrm{\rho }$
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 $q^{\text{'}}$.


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,
$q^{\text{'}}$=$\rho \left(\pi x^{\mathit{2}}h\right)$
From Gauss's Law,

$$\begin{gathered} \oint E\mathit{.}ds=\frac{q_{\mathrm{en}}}{{\in }_0} \\ E.2\mathrm{\pi xh}=\frac{\rho (\pi x^{2}h)}{{\in }_0} \\ \mathrm{E}=\frac{\rho x}{2{\in }_0} \end{gathered}$$