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. $\frac{\sigma}{8 \epsilon_0}\left[1-\frac{Z}{\sqrt{R^2+Z^2}}\right]$
• 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)

$V=\,\int\limits _{0}^{R} \frac{k \alpha \pi r d r}{\alpha \sqrt{r^{2} + z^{2}}}=\frac{\sigma }{8 ϵ_{0}}\,\int\limits _{0}^{R} \frac{r d r}{\sqrt{r^{2} + z^{2}}}$
$t^{2}=r^{2}+z^{2}$
$tdt=rdr$
$V=\frac{\sigma }{8 ϵ_{0}}\,\int\limits _{z}^{\sqrt{2^{2} + R^{2}}} d t=\frac{\sigma }{8 ϵ_{0}}\left(\sqrt{R^{2} + z^{2}} - z\right)$