Solving Laplace's Equation Between Two Conical Surfaces

[Tsunoyukami]

"Two co-axial conducting cones (opening angles $\theta_{1} = \frac{\pi}{10}$ and $\theta{2} = \frac{\pi}{6}$) of infinite extent are separated by an infinitesimal gap at $r = 0$. If the inner cone is held at zero potential and the outer cone is held at potential $V_{o}$ find the potential and the electric field between them. [Hint: before you try to solve Laplace's equation for this system, think carefully about which variables V will depends on."I know that I need to solve Laplace's Equation subject to the boundary conditions $V(\theta_{1}) = V(\frac{\pi}{10}) = 0$ and $V(\theta_{2}) = V(\frac{\pi}{6}) = V_{o}$. Because these boundary conditions are best written as functions of the opening angle I suspect that this problem should be approached using spherical coordinates. Is this correct?

However, Laplace's equation can be made much simpler by considering symmetries and determining which (if any) variables it does not depend on. Clearly, the potential does not depends on the angle $\phi$; that is, this conical system has azimuthal symmetry and therefore the problem has been reduced to a 2D problem.Next, I feel like the potential should be independent of r as well because the potential is constant on the surfaces of the cone regardless of how far away from the tip you go. However, I'm not sure if this property necessarily holds true in general for the space between the two cones. Is there any way for me to see whether or not this would be true? (I hope so, because then the problem will be reduced to a rather "simple" 1D problem).

More and more the hint is making me feel like it should be possible to reduce the system to 1D.However, I also considered the system in cylindrical coordinates, where it is clear that the r- and z- components are coupled by the relation $z = r tan(\theta)$. Which leads me to suspect that the potential will depend on both r and $\theta$...any insight on whether or not this problem can be made simpler would be appreciated!

 

[TSny]

The boundary conditions for $V(r, \theta, \phi)$ depend only on $\theta$. This suggests separating variables so that $V(r, \theta, \phi) = R(r)\Theta(\theta)\Phi(\phi)$. Think about what the boundary conditions tell you about $R(r)$ and $\Phi(\phi)$.

 

[Tsunoyukami]

I was able to argue that the potential in the region bounded by the two cones is independent of r by considering the potential difference from one cone to another by traveling along an arbitrary path of constant r connecting the two cones. This potential difference is independent of r and therefore the final expression for the potential is independent of r.