Question

Consider a sphere of radius $R$ with unform charged density and total charge Q. The electrostatic potential distribution inside the sphere is given by $\varphi (r)=\frac{Q}{4\pi {\epsilon }_{0}R}\left(a+b(r/R{)}^c\right)$. Note that the zero of potential is at infinity. The values of $(a,b,c)$ are :-

• A. $\left(\frac{1}{2},-\frac{3}{2},1\right)$
• B. $\left(\frac{3}{2},-\frac{1}{2},2\right)$
• C. $\left(\frac{1}{2},\frac{1}{2},1\right)$
• D. $\left(\frac{1}{2},-\frac{1}{2},2\right)$

Answer 1

Answer: (B)

Potential inside uniformly charged solid sphere is given by
$V=\frac{kQ}{2R^3}\left[3R^2-r^2\right]$
$=\frac{kQ}{R}\left[\frac{3R^2}{2R^2}-\frac{r^2}{2R^2}\right]$
$=\frac{Q}{4\pi {\in }_{0}R}[\frac{3}{2}-\frac{1}{2}{\left(\frac{r}{R}\right)}^2$
Compare with given formula, i.e,
$\frac{Q}{4\pi {ϵ}_{0}R}\left[a+b{\left(\frac{r}{R}\right)}^C\right]$
$a=\frac{3}{2},b=-\frac{1}{2},c=2$