What is the potential at the center of a sphere with given boundary conditions?

In summary, the problem involves finding the potential of a spherical surface with a given boundary condition and satisfying Laplace's equation. The solution can be found by expressing the angular component in terms of spherical harmonics and then solving the radial ODE with non-singular solutions at r=0.
  • #1
rolandas9999
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Homework Statement


Given potential r=2m on the surface of the sphere which meets the Laplasian equation (triangle)u=0 and is u(2,theta,psi)=5sin(theta)sin(0.25psi). I need to find the potential of sphere center. Can anyone help me?


Homework Equations


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The Attempt at a Solution


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  • #2
Write Laplace's equation in spherical coordinates, plug in your boundary condition and solve.

Note that the angular component has solutions which you can expand in terms of http://en.wikipedia.org/wiki/Spherical_harmonics" . You should first match the angular component with the boundary condition. Then it is a simple matter of solving the radial ODE for the boundary conditions... noting you must have non-singular solutions only at r=0.
 
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What is a sphere center potential?

A sphere center potential is a type of potential energy that exists within a spherical object, such as a planet or a ball. It is the energy that is associated with the position of an object within the gravitational field of the sphere.

How is a sphere center potential calculated?

The sphere center potential is calculated using the formula V = -GM/r, where V is the potential energy, G is the gravitational constant, M is the mass of the sphere, and r is the distance from the center of the sphere.

What factors affect the sphere center potential?

The sphere center potential is affected by the mass of the sphere, the distance from the center of the sphere, and the gravitational constant. It is also affected by the presence of other objects that may create their own gravitational fields.

What is the relationship between sphere center potential and gravitational potential energy?

The sphere center potential is a type of gravitational potential energy. It represents the amount of energy that an object has due to its position within the gravitational field of a sphere. As the object moves closer or further away from the sphere, the sphere center potential and gravitational potential energy change accordingly.

Why is the sphere center potential important in physics?

The sphere center potential is important in physics because it helps us understand the behavior of objects within a gravitational field. It is a fundamental concept in the study of mechanics and is used in many equations and calculations, such as those related to orbital motion and planetary dynamics.

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