I have an electrostatics problem (shown here: https://www.physicsforums.com/showthread.php?t=654877) wich leads to the following system of differential equations:(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# System of non-linear partial differential eqs from electrostatics

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