91Ó°ÊÓ

A hollow cylindrical shell with a radius of R and a length of L has a uniform charge distribution along its length, resulting in a total charge of Q.

The electric potential at a distance r from a point charge q is calculated as follows:

$V\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{r}$

Potential along the central axis of a ring of radius R having charge q, at a distance $X$ from the center,

$$\begin{gathered} V=\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{q}{\sqrt{R^2+x^2}} \\ ∫\frac{dx}{\sqrt{R^2+x^2}}=\ln \left(x+\sqrt{x^2+R^2}\right) \end{gathered}$$

Consider a ring with a width of $dx$and a distance of x from the cylinder's center. The charge per square meter of cylinder area,

$\mathrm{σ}=\frac{Q}{2\mathrm{π}RL}$

The ring element's charge,

$\begin{array}{rcl}dq& =& \mathrm{σ}×2\mathrm{π}R×dx\\ & =& \frac{Q}{L}dx\end{array}$

Therefore, the potential at the center of the cylinder due to ring element

$\begin{array}{rcl}dV& =& \frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{dq}{\sqrt{R^2+x^2}}\\ & =& \frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{Q}{L}\frac{dx}{\sqrt{R^2+x^2}}\end{array}$

As a result of the whole charge distribution, the potential at the cylinder's core,

$\begin{array}{rcl}V& =& ∫dV\\ & =& \frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{Q}{L}{∫}_{-4/2}^{+1/2}\frac{q}{\sqrt{R^2+x^2}}\\ & =& \frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{Q}{L}\ln \left(\frac{\sqrt{L^2+4R^2}+L}{\sqrt{L^2+4R^2}-L}\right)\end{array}$

The potential at the center of the cylinder is found to be

$V=\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{Q}{L}\ln \left(\frac{\sqrt{L^2+4R^2}+L}{\sqrt{L^2+4R^2}-L}\right)$