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 $\phi( r )=\frac{ Q }{4 \pi \varepsilon_{0} R }\left( a + b ( r / R )^{ c }\right)$. 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
$V =\frac{ kQ }{2 R ^{3}}\left[3 R ^{2}- r ^{2}\right]$
$=\frac{ kQ }{ R }\left[\frac{3 R ^{2}}{2 R ^{2}}-\frac{ r ^{2}}{2 R ^{2}}\right]$
$=\frac{ Q }{4 \pi \in_{0} R }\left[\frac{3}{2}-\frac{1}{2}\left(\frac{ r }{ R }\right)^{2}\right.$
Compare with given formula, i.e,
$\frac{ Q }{4 \pi \epsilon_{0} R }\left[ a + b \left(\frac{ r }{ R }\right)^{ C }\right]$
$a =\frac{3}{2}, b =-\frac{1}{2}, c =2$