Electric field between a needle and a plate

[Mniazi]

http://i66.tinypic.com/2yts08m.jpg

I was doing an experiment in which i have a needle perpendicular to a metal plate. I run 210kV between them. I want to find out the electric field between the two, and I also want to know the electric al field density throught the metal plate? Pls help. I found a formula for the electrical field. it was $$E(r) = \frac{V*a}{r}*\frac{1}{1-a/b}$$
where V is the applied voltage, a is the radius of the pin point, b is the distance from the pin point to the grounded plate, and r is the radial distance from the pin point to the grounded plate?

The plate is a circular piece of metal, I want to find the density of the electrical field going through a smaller circular region right below the metal. will I need to put r as the radius of the area?

 

[BvU]

You sure about this expression ? It doesn't even have the right dimension !
Where does this formula apply ? Is it an approximation in a limited region ?
For the field underneath the grounded plate a very detailed calculation will be necessary.

 

[Mniazi]

No I am not sure about this equation. I got it on a website (http://www.afssociety.org/air-filtration/81-what-is-an-electret ) It says it is the electric field distribution from the pin to the plate. If detailed calculation is necessary then can you like guide me on it? I will perfomr the calculations and then update over here.

 

[BvU]

Well, at least that settles the dimensionality:$$E(r) = \frac{V*a}{r^2}*\frac{1}{1-a/b}$$
and it looks like the electric field from a point charge $\ \ \displaystyle q = {4\pi\epsilon_0 V_0 \over {1\over a} - {1\over b} }$

It is unfortunate that they don't clearly define $r$ but I suspect it is the radial distance to the center inside the tip (the center of the sphere with radius a). So surfaces with the same |E| would then be spherical shells and that should work reasonably well if $b \gg a$. For the region between tip and plate. The further sideways, the worse. And not sensible upwards along the rod at all.

Another way to look at this configuration is to consider it as a point charge opposite a grounded plate, for which there is plenty info (see e.g. Errede 2007). You get a dipole field and the surface charge on the plate can easily be determined.

But you are looking for trouble: you want the field underneath a finite circular grounded disc. As Errede says on his page 14, you don't get a solution for the region where the image charge is located.

So all I can think of is a numerical approach. No experience. Perhaps https://www.integratedsoft.com/papers/techdocs/tech_1cx.pdf helps ?

 

[Mniazi]

Forgive my typo in the equation i wrote.
Well I am not trying to find the field under the plate. I am actually studying effects of the electric field on a liquid dielectric on the surface of the plate. So I wanted to find the field in that area of dielectric on the plate. I think taking the pin as a point will simplify stuff, so ill go with that. Ill check out the links you gave as well.