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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 variables 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 general solution is:

##\Phi_{tot}= \sum_m (A_m e^{-\lambda_m z} + B_m e^{+\lambda_m z})(C_m J_0(\lambda_m r) + D_m Y_0(\lambda_m r)) ##

and all the constants che can be calculated exploiting the boundary conditions.

Now assume ##L## is not fixed, but it can vary in a certain range ##0<L_1<L<L_2##. What I am thinking about is: is it possible to compute ##L## such that [tex] \frac{\partial \Phi_{tot}}{\partial z} \rvert_{z=L} =0[/tex]?

Thank you!

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# A Laplace equation- variable domain

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