Question

The quarter disc of radius $R$ (see figure) has a uniform surface charge density $\sigma$ . Find electric potential at a point $\left(O,O,Z\right)$ .

• A. $\frac{\sigma }{8{ϵ}_0}\left[1-\frac{Z}{\sqrt{R^2+Z^2}}\right]$
• B. $\frac{\sigma }{4{ϵ}_0}\left[\sqrt{R^2+Z^2}+Z\right]$
• C. $\frac{\sigma }{8{ϵ}_0}\left[\sqrt{R^2+Z^2}-Z\right]$
• D. $\frac{\sigma }{8{ϵ}_0}\left[1+\frac{Z}{\sqrt{R^2+Z^2}}\right]$

Answer 1

Answer: (C)

$V={\int }_0^R\frac{k\alpha \pi rdr}{\alpha \sqrt{r^2+z^2}}=\frac{\sigma }{8{ϵ}_0}{\int }_0^R\frac{rdr}{\sqrt{r^2+z^2}}$
$t^2=r^2+z^2$
$tdt=rdr$
$V=\frac{\sigma }{8{ϵ}_0}{\int }_z^{\sqrt{2^2+R^2}}dt=\frac{\sigma }{8{ϵ}_0}\left(\sqrt{R^2+z^2}-z\right)$