[YMP/EM-03007] Potential Due to a Thick Shell

This is very similar to the full sphere having uniform charged density. The thick shell is divided into thin shells lying between \(x\) and \(x+dx\). Potential at the given point P due to a thin shell of radius $x>R$ is$\phi_1$:

\begin{eqnarray}
\phi_1&=&\frac{4\pi x^2dx\rho}{4\pi\epsilon_0x}
=\frac{\rho xdx}{\epsilon_0} \label{eq34}
\end{eqnarray}

Therefore potential at a point \(r\), due to shells between \(a\) and \(b\), is $\phi_2$

\begin{eqnarray}
\nonumber\phi_2&=&\int_a^b\frac{\rho x^2dx}{\epsilon_0x}\\
&=&\frac{\rho}{2\epsilon_0}(R^2-r^2) \label{eq35}
\end{eqnarray}

Therefore, the total electric potential inside the sphere is $\phi$:

\begin{eqnarray}
\nonumber\phi&=&\phi_1+\phi_2\\
\nonumber&=&\frac{\rho r^2}{3\epsilon_0}+\frac{\rho}{2\epsilon_0}(b^2-a^2)\\
\nonumber&=&\frac{\rho}{6\epsilon_0}(3R^2-r^2)\\
&=&\frac{Q}{8\pi\epsilon_0R^3}(3R^2-r^2) \label{eq36}
\end{eqnarray}