- #1
peterjaybee
- 62
- 0
Hello,
In my plasma physics book, poissons equation in relation to a plasma made up of electrons and protons is given as
[tex]\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\varphi\left(r\right)}{dr}\right) = \frac{2}{\lambda_{D}}\varphi\left(r\right)[/tex]
The solution of this equation when phi(r) tends to 0 for r tending to infinity is
[tex]\varphi\left(r\right) = \frac{e}{4\pi\epsilon_{0}r}exp\left(-\frac{\sqrt{2}r}{\lambda_{D}}\right)[/tex]
How do you get to this solution? I have tried multiplyin both sides by r^2, then integrating w.r.t. r, by parts, but I cannot get any further than that.
Regards,
Peter
In my plasma physics book, poissons equation in relation to a plasma made up of electrons and protons is given as
[tex]\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\varphi\left(r\right)}{dr}\right) = \frac{2}{\lambda_{D}}\varphi\left(r\right)[/tex]
The solution of this equation when phi(r) tends to 0 for r tending to infinity is
[tex]\varphi\left(r\right) = \frac{e}{4\pi\epsilon_{0}r}exp\left(-\frac{\sqrt{2}r}{\lambda_{D}}\right)[/tex]
How do you get to this solution? I have tried multiplyin both sides by r^2, then integrating w.r.t. r, by parts, but I cannot get any further than that.
Regards,
Peter