Problem $\backslash$#$1$

A spherical capacitor is located at the origin of a cartesian coordinate system. It consists of two concentric spherical conductors with homogeneous dielectric material $\backslash (\backslash$epsilon $\backslash )$ in between the conductors, as shown in the figure below, left.

The inner conductor has radius $\backslash ($ a $\backslash ),$ and the outer conductor has radius $\backslash ($ b $\backslash ),$ as shown in the figure below, right. Equal charges $\backslash (+$Q $\backslash )$ and $\backslash (-$Q $\backslash )$ are placed on the inner and outer conductors, respectively.

$($a$)$ Apply the Laplace equation for the following electrostatic boundary problem, determine the capacitance of the spherical capacitor.

$($b$)$ With the expression you found in part $($a$),$ find the capacitance of the spherical shell if the inner radius is $\backslash ($ a$=2\backslash$mathrm$\{$~cm$\},\backslash$mathrm$\{$~b$\}=2\mathrm{.}5\backslash$mathrm$\{$~cm$\}\backslash ),$ and the dielectric medium has relative permittivity $\backslash (\backslash$epsilon$\_\{$r$\}=12\mathrm{.}25\backslash )$