How Does a Cone's Voltage Affect the Electric Field Near a Grounded Plane?

[Xyius]

The problem gives a cone above a conducting grounded plane. (The xy plane) The cone has a voltage of 100V on it. It wants me to find the electric field between the cone and the plane.

The angle the cone makes with the z axis is 10 degrees. And it is at a height of "h".

So my method for attacking this problem is to use Laplace's Equation in Spherical coordinates. Here is my work.. (I hope you can read my hand writing!)

http://imageshack.us/a/img211/1927/coneproblem.jpg

My question is, is what I have so far correct? And is this the correct approach? Also, how would I go about solving for B_n?? I know that the Legendre Polynomials are orthogonal but what confuses me is, I need to plug in theta=10° into them, thus making them a constant. That leaves me with a function of just r^-(n+1) which (to my knowledge) is not orthogonal.

Any help would be appreciated!

EDIT: Unless there is no "r" dependence either. This would make the problem much easier! But since the cone had a finite height, wouldn't there be an r" dependence?

 

[member 553083]

Yes, your approach is correct. To solve for $B_n$, you need to use the orthogonality of the Legendre polynomials. In particular, you can use the orthogonality relation:$$ \int_{-1}^1 P_n (\cos \theta) P_m (\cos \theta) \, d(\cos \theta) = \delta_{nm} $$where $\delta_{nm}$ is the Kronecker delta function. This will allow you to solve for $B_n$ by integrating over the angle $\theta$.