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Solved Problem-3

Problem-3
Region between the two coaxial cones is shown in Fig. A potential V1 exists at θ1 and V = 0 at θ2. The cone vertices are insulated at r = 0. Solve Laplace’s equation to get potential at a cone at any angle θ.
Arrangement for problem
 
Solution
Since V varies with θ only, Laplace’s equation can be written as,
Description: 169635.png
 
Integrating,
Description: 219498.png
 
Integrating once again,
Description: 219514.png
 
where, A and B are constants.
 
Applying boundary conditions, we get,
At Description: 219537.png
At Description: 219544.png
 
Solving for A and B,
Description: 169593.png
 
Description: 219552.png
 
Substituting the values of A and B and rearranging, we get,
 
Description: 169577.png
 





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