# A Laplace Eq. in Cylindrical coordinates (no origin)

1. Jun 8, 2017

### 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

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 $J_0$?

Last edited: Jun 8, 2017
2. Jun 13, 2017

### PF_Help_Bot

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