Solving Laplace's Equation Between Two Conical Surfaces

In summary, the problem of finding the potential and electric field between two co-axial conducting cones of different opening angles can be simplified by using spherical coordinates and considering symmetries. The potential can be reduced to a 1D problem and is independent of both the azimuthal angle and the radial distance. The boundary conditions for the potential suggest separating variables and considering the potential difference between the two cones along a path of constant radial distance.
  • #1
Tsunoyukami
215
11
"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!
 
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  • #2
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)##.
 
  • #3
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.
 

Related to Solving Laplace's Equation Between Two Conical Surfaces

1. What is Laplace's equation and why is it important?

Laplace's equation is a second-order partial differential equation that relates to the distribution of potential energy in a physical system. It is important because it is a fundamental equation in many areas of science and engineering, such as electromagnetism, fluid mechanics, and heat transfer.

2. How is Laplace's equation related to conical surfaces?

Laplace's equation can be used to solve for the potential energy distribution between two conical surfaces. This is because the equation describes the behavior of a scalar field in three-dimensional space, which is applicable to the electric potential between two charged conical surfaces.

3. What are the boundary conditions for solving Laplace's equation between two conical surfaces?

The boundary conditions for solving Laplace's equation between two conical surfaces are that the potential energy is constant along the conical surfaces, and the potential energy approaches zero at infinity. Additionally, the potential energy must be continuous and have continuous derivatives at the intersection of the two conical surfaces.

4. What methods can be used to solve Laplace's equation between two conical surfaces?

There are several methods that can be used to solve Laplace's equation between two conical surfaces, including separation of variables, the method of images, and conformal mapping. Each method has its own advantages and limitations, and the choice of method depends on the specific problem at hand.

5. What are the applications of solving Laplace's equation between two conical surfaces?

The applications of solving Laplace's equation between two conical surfaces include modeling the electric potential between charged cones, studying the thermal distribution in conical heat exchangers, and understanding the fluid flow around conical structures. This type of problem also has broader applications in fields such as aerodynamics, geophysics, and astrophysics.

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