I would like some help with my solution attempt.
I have a conducting shell with radius $R1$, surronded by a dielectric shell with $\varepsilon_1$ and radius $R2$, and on the outside i have a $POINT$ $charge$ $q$ from distance $L$, in Vacuum so $\varepsilon_0$. So its $R1<R2<L$.
The question is the potentials, $\Phi_1,\Phi_2,\Phi_3$.
Edit: My question is how should i choose the coefficents to satisfy the following things:
1.Inside the inner circle the potential should be zero because its a conductor, so there is only potential on its side where $r=R1$, and its a constant. Usually we choose the $B_l r^{-(l+1)}$ part to be zero. But i dont know which one should be zero in a case of conductor with no charge inside
2.The second shell is dielectric, but it doesn't contain the origo it goes from $[R_1,R_2]$, so i also dont know which coefficents to have inside here. \If there would just be a dielectric Sphere with outside charge inside would be the part where $C_l r^{l}$, and in the inside there would be the $D_l r^{-(l+1)}$ part with the point charge $\dfrac{q}{4\pi \varepsilon_0}\dfrac{1}{|\mathbf{r-L}|}$
3.On the outside we have the point charge $$\dfrac{q}{4\pi \varepsilon_0}\dfrac{1}{|\mathbf{r-L}|}=\sum_{n=0}^\infty \dfrac{r^l}{L^{l+1}}P_l{cos\vartheta}+Outer terms from the two shells $$
4.I also have the two types of boundary conditions, which should satisfy
And this is how i attemted it: I write up three cylindrical Laplacians for each section:
$$\Phi_1(r,\vartheta)=\sum_{n=0}^\infty (A_l r^l+B_lr^{-(l+1)})P_l(\cos \vartheta) \\ \Phi_2(r,\vartheta)=\sum_{n=0}^\infty (C_l r^l+D_lr^{-(l+1)})P_l(\cos \vartheta) \\ \Phi_3(r,\vartheta)=\sum_{n=0}^\infty (E_l r^l+F_lr^{-(l+1)})P_l(\cos \vartheta)$$
$B_l$ is zero because $r^{-(l+1)}$ diverges there. I'm not sure if $A_l$ should be zero or not. Because $\Phi_1(r<R_1)=0$, this should give me that $A_l=0$, but on the surface: so $\Phi_1(r=R_1)$, should be a constant value. $\Phi_1(r=R1)=\Phi_2(r=R1)=V1(\text{constans})$.
And there should also be another boundary condition here $$\varepsilon_0 \dfrac{\partial \Phi_1}{\partial r} \bigg|_{r=R_1}=\varepsilon_1 \dfrac{\partial \Phi_2}{\partial r} \bigg|_{r=R_1}$$
When $r=R_1$, the Potential is a constant $V_1$.
So i have these : $$\Phi_2(r=R,\vartheta)=\sum_{n=0}^\infty (C_l R^l+D_lR^{-(l+1)})P_l(\cos \vartheta)=V1$$
$$\sum_{n=0}^\infty (A_l l r^{l-1})P_l(\cos \vartheta)=\sum_{n=0}^\infty (C_l l r^{l-1}+D_l(-l-1)r^{-(l+2)})P_l(\cos \vartheta)$$
I'm not sure in the part where $r\in [R_1,R_2]$. So $\Phi_2$ there is :$$\Phi_2(r,\vartheta)=\sum_{n=0}^\infty (C_l r^l+D_lr^{-(l+1)})P_l(\cos \vartheta) $$. We leave both of the coefficients because there is no divergent part, because it $r$ doesn't go to zero here.
We have the boundary where $r=R2$ and the conditions are: $$\Phi_2(r=R_2)=\Phi_3(r=R_2)\\ \varepsilon_1 \dfrac{\partial \Phi_2}{\partial r} \bigg|_{r=R_2}=\varepsilon_0 \dfrac{\partial \Phi_3}{\partial r} \bigg|_{r=R_2}$$
$$\sum_{n=0}^\infty (C_l R_2^l+D_lR_2^{-(l+1)})P_l(\cos \vartheta) =\sum_{n=0}^\infty (E_l R_2^l+F_lR_2^{-(l+1)})P_l(\cos \vartheta)$$
$$\sum_{n=0}^\infty (C_l l R_2^l+D_l(-l-1)R_2^{-(l+2)})P_l(\cos \vartheta) =\sum_{n=0}^\infty (E_l l R_2^l+F_l(-l-1)R_2^{-(l+2)})P_l(\cos \vartheta)$$