Question $1$:

$[10$ pts$]$

Find the electric field and potential in cases $($a$)$ and $($b$).$

$($a$)$ As shown in Figure $1($a$),Q_1,Q_2,$ and $Q_3$ are charged at $r_1,r_2,$ and $r_3$ of the concentric conductor spherical shell, respectively, and the dielectric permittivity $lo{n}_1,lo{n}_2,$ and $lo{n}_3$ are packed. Find the electric field at an arbitrary position $r$ and the potential at $r_1,r_2,$ and $r_3[5$ points$].$

$($b$)$ Electric permittivity $lon(r)=\frac{k+r}{r}$ where $k$ is constant is given between conductor shells of radius $r=$a and $r=b.$ When $Q$ is given at $r=a,$ find the electric field and potential at an arbitrary point $r,$ which is a distance from the center. Suppose that it is grounded at $r=b[5$ points$].$

$($a$)$

$($b$)$