- #1
chimay
- 81
- 7
Hi,
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!
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!