Poissons equation for plasma

[peterjaybee]

Hello,

In my plasma physics book, poissons equation in relation to a plasma made up of electrons and protons is given as

$$\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\varphi\left(r\right)}{dr}\right) = \frac{2}{\lambda_{D}}\varphi\left(r\right)$$

The solution of this equation when phi(r) tends to 0 for r tending to infinity is

$$\varphi\left(r\right) = \frac{e}{4\pi\epsilon_{0}r}exp\left(-\frac{\sqrt{2}r}{\lambda_{D}}\right)$$

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

 

[Born2bwire]

The simplest thing to do would be to substitute using:

$$R(r) = r \varphi (r)$$

Then you can rewrite it as:

$$\frac{1}{r}\frac{d}{dr} \left( r^2 \frac{d \varphi}{dr} \right) = \frac{2}{\lambda_d}R$$

You can continue from here by performing the chain rule to place the left hand side in terms of $$R$$.

EDIT: Nevermind, I made a mistake earlier on, fixed now. Sorry for any confusion.

 

[peterjaybee]

I have now got a homogeneous 2nd order differential equation and from that an auxillary equation, which is beginning to take the form of the answer. Thanks