Consider a sphere of radius $R$ with uniform charge density and total charge $Q$. The electrostatic potential distribution inside the sphere is given by $\theta_{(r)}=\frac{Q}{4 \pi \varepsilon_{0} R}\left(a+b(r / R)^{C}\right)$. Note that the zero of potential is at infinity. The values of $(a, b, c)$ are
KVPY 2020, Advanced
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$(b)$ Potential inside a uniformly charged sphere,
$V=\frac{K Q}{2 R^{3}}\left(3 R^{2}-r^{2}\right)$
$\,\,\,\,=\frac{Q}{4 \pi \varepsilon_{\varepsilon} R}\left[\frac{3}{2}-\frac{1}{2}\left(\frac{r}{R}\right)^{2}\right]$
Comparing with given value, we get
$a=\frac{3}{2}, b=-\frac{1}{2}$ and $c=2$
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