Solve Electromag Question: Find V(x) from p Expression

[coffeem]

Homework Statement

An electron beam with negligible velocity in the x direction is emitted from a heated plane cathode through a vacuum towards a plane anode at x = s, there the potential is V = Vo. At the cathode V(x) = 0 and x = 0. The charge density of the beam is;

p = - (4eoVo)/(9s^(4/3)x^2/3

Show that V(x) takes the form V(x) = Vos^p.x^q

Homework Equations

The Attempt at a Solution

I have worked out that I have to use: divE = p/e0

So when I substitute the expression I have for p into this I ger:

divE = (4Vo)/(9s^(4/3)x^2/3

However I do not know how to get this into a form to find V(x), I am stuck. Would appretiate some helpful hints or even something a bit more forceful. Thanks

 

[gabbagabbahey]

Using Poisson's equation is a much easier way to go here...since V is a function of x only, $$\nabla ^2V= \frac{d^2V(x)}{dx^2}$$ and so applying Possion's equation gives you an ordinary differential equation which you can solve...try it and see what you get.