Laplace Eq. in Cylindrical coordinates (no origin)

[chimay]

Hi,
I need to solve Laplace equation:$\nabla ^2 \Phi(x,r)=0$ in cylindrical domain $r_1<r<r_2$, $0<z<+\infty$. The boundary conditions are:
$\left\{ \begin{aligned} &\Phi(0,r)=V_B \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_2} \\ &\frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_1}=0 \\ &\lim_{z\rightarrow \infty} \Phi(z,r)=0 \end{aligned} \right.$
By separation of variables, I can write the solution as:
$\Phi_(z,r)=e^{-\lambda z}(A J_0(\lambda r)+BY_0(\lambda r))$
By applying the first boundary condition I get:
$\sum_{m=1}^{\infty} A_m J_0(\lambda_m r)+B_m Y_0(\lambda_m r)=V_B$
Now I can compute a first relation between $A_m$ and $B_m$ by exploiting orthogonality between different scaled Bessel functions ($i \ne m$):
$\left\{ \begin{aligned} & \int_{r_1}^ {r_2} r J_0(\lambda_i r)J_0(\lambda_m r) dr =0 \\ & \int_{r_1}^ {r_2} r J_0(\lambda_i r)Y_0(\lambda_m r) dr =0 \\ & \int_{r_1}^ {r_2} r Y_0(\lambda_i r)Y_0(\lambda_m r) dr =0 \end{aligned} \right.$
by virtue of the second and the third boundary conditions, as it is explained here (http://www.hit.ac.il/staff/benzionS/Differential.Equations/Orthogonality_of_Bessel_functions.htm)
and it was pointed out also here
(https://www.physicsforums.com/threads/laplace-eq-in-cylindrical-coordinates.915420/)

My question is the following:
Assume the second boundary condition was ${C^{'}}_{ox} (V_G - \Phi(x,r_2))=C_0 \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_2}$, $V_G=constant$. It seems to me that in this case the orthogonality conditions are not valid anymore, because of the presence of the constant term $V_G$.
1) Is this true ?
2) How can I proceed in this case?

Thank you all

Edit: I am having some doubts also about the integrals concerning $Y_0$. Are they the same as the ones that involve $J_0$?

 

[AndyW]

it is important to approach this problem in a systematic and logical manner. Here are some steps you can take to solve this problem:

1. Start by clearly defining the problem and the given boundary conditions. This will help you understand the problem better and identify any potential issues or discrepancies.

2. Next, review the approach you have taken so far and make sure it is correct. It is always a good idea to double-check your work and make sure you have not made any mistakes.

3. If you are unsure about the validity of the orthogonality conditions in the new boundary condition, you can try to derive them yourself using the new boundary condition. This will help you understand if and how the orthogonality conditions have changed.

4. It may also be helpful to look for similar problems or examples where a similar boundary condition has been used and see how the solution was obtained in those cases.

5. If you are still unsure about how to proceed, you can consult with colleagues or experts in the field for their insights and suggestions.

Remember, as a scientist, it is important to be thorough and meticulous in your approach to problem solving. Good luck with your research!