Using example $4\mathrm{.}7$ as a reference. Consider if you were given symmetrical object with linear dielectric and it were placed in an otherwise uniform electric field where the potential inside and outside are given as:

$V_{in}(s,)={\displaystyle \underset{k=1}{\overset{}{}}}{s}^k(a_{k}cosk+b_{k}sink)$

$V_{\text{o}\text{u}\text{t}}(s,)=-E_0$scos$+{\displaystyle \underset{k=1}{\overset{}{}}}{s}^{-k}(c_{k}cosk+d_{k}sink)$

Apply the boundary conditions below to find the electric field inside the object

$($i$)V_{in}=V_{\text{o}\text{u}\text{t}},$ at $s=a,$

$($ii$)lon\frac{V_{in}}{\text{d}\text{e}\text{l}\text{s}}=lo{n}_0\frac{del{V}_{\text{o}\text{u}\text{t}}}{\text{d}\text{e}\text{l}\text{s}},$ at $s=a,$

$($iii$)V_{\text{o}\text{u}\text{t}}^{\text{d}\text{e}\text{l}\text{s}}-E_0^{\text{d}\text{e}\text{l}\text{s}}$scos$$ for $sa.$