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Madoro
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I have an electrostatics problem (shown here: https://www.physicsforums.com/showthread.php?t=654877) which leads to the following system of differential equations:
[itex]\frac{\partial E_z}{\partial z}=\frac{\rho}{\epsilon_0}[/itex] (1)
[itex]Z_i E_r \frac{\partial \rho}{\partial r}+(u_g+ Z_i E_z) \frac{\partial \rho}{\partial z} + \rho Z_i \frac{\partial E_z}{\partial z}=0[/itex] (2)
Substituting eq. (1) into eq. (2):
[itex]Z_i E_r \frac{\partial \rho}{\partial r}+(u_g+ Z_i E_z) \frac{\partial \rho}{\partial z} + \frac{\rho^2 Z_i}{\epsilon_0}=0[/itex] (3)
Therefore I have a system of 2 equations (1 & 3) with 2 unknowns, the axial field [itex]E_z[/itex] and the charge density [itex]\rho(z,r)[/itex]. The rest of the variables are known so they can be supposed as constants.
I'm not sure on how to solve it, I'm considering two options:
- derivate eq. (3) with respect to [itex]z[/itex] to substitute in eq. (1), but I don´t get rid of [itex]E_z[/itex] and the eq. (3) becomes more complicated.
- Solve by semi-implicit method, considering that [itex]z=du_z/dt[/itex], but since is an equation in partial derivatives I'm not sure on how to manage the term in [itex]r[/itex]
I'm totally stuck on this, I'm asking for a direction of solving it, not for a solution, so any help would be grateful.
Thanks in advance.
[itex]\frac{\partial E_z}{\partial z}=\frac{\rho}{\epsilon_0}[/itex] (1)
[itex]Z_i E_r \frac{\partial \rho}{\partial r}+(u_g+ Z_i E_z) \frac{\partial \rho}{\partial z} + \rho Z_i \frac{\partial E_z}{\partial z}=0[/itex] (2)
Substituting eq. (1) into eq. (2):
[itex]Z_i E_r \frac{\partial \rho}{\partial r}+(u_g+ Z_i E_z) \frac{\partial \rho}{\partial z} + \frac{\rho^2 Z_i}{\epsilon_0}=0[/itex] (3)
Therefore I have a system of 2 equations (1 & 3) with 2 unknowns, the axial field [itex]E_z[/itex] and the charge density [itex]\rho(z,r)[/itex]. The rest of the variables are known so they can be supposed as constants.
I'm not sure on how to solve it, I'm considering two options:
- derivate eq. (3) with respect to [itex]z[/itex] to substitute in eq. (1), but I don´t get rid of [itex]E_z[/itex] and the eq. (3) becomes more complicated.
- Solve by semi-implicit method, considering that [itex]z=du_z/dt[/itex], but since is an equation in partial derivatives I'm not sure on how to manage the term in [itex]r[/itex]
I'm totally stuck on this, I'm asking for a direction of solving it, not for a solution, so any help would be grateful.
Thanks in advance.