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I need to solve Laplace equation ##\nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ##r_1<r<r_2##, ##0<z<L##.

The boundary conditions are:

##

\left\{

\begin{aligned}

&\Phi(0,r)=V_B \\

&\Phi(L,r)=V_P \\

& -{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 \\

\end{aligned}

\right.

##

By separation of variable I obtain:

##

\Phi_(z,r)=(A e^{-\lambda z} + B e^{+\lambda z})(C J_0(\lambda r) + D Y_0(\lambda r))

##

##J_0## and ##Y_0## being zero order first type and second type Bessel functions.

The constants I have to determine are : ##A, B, C, D## and ##\lambda##; they are ##5## constants, but only ##4## equations are available coming from the behaviour of ##\Phi## at the boundaries of the domain, so I do not know how to proceed.

Is there any mistake in my reasoning?

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# I Laplace equation boundary conditions

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