Question

FIELD THEORY

Otherwise in the unbounded space, there is a centered origin A-radius spherical shell with a net charge and on its surface, the potential is
ϕs(r=a,θ,φ)=A+Csinθsinφ.

Starting from the general solution and boundary conditions, solve for the potential everywhere. Justify carefully why each constant can be interpreted as zero. The final answer should be written in the simplest form possible; usually the given boundary value helps with this. Use the most common definition of zero level: ϕ → 0 at infinity.

Answer 1

To solve for the potential everywhere in an unbounded space with a centered origin A-radius spherical shell with a net charge, start with the general solution and apply the boundary conditions. The final answer for the potential everywhere is dependent on the specific boundary values given in the problem.

Step-by-step explanation:

To solve for the potential everywhere in an unbounded space with a centered origin A-radius spherical shell with a net charge, we start with the general solution and apply the boundary conditions. In this case, the potential on the surface of the shell is given by ϕs(r=a,θ,φ)=A+Csinθsinφ. To find the potential inside and outside the shell, we can use the fact that for r ≥ R, the potential must be the same as that of an isolated point charge q located at r = 0.

Each constant in the solution can be interpreted as zero based on the given boundary condition that the potential goes to zero at infinity. This is because as the distance from the origin becomes very large, the terms inside the natural log in the potential equation approach one, resulting in the potential approaching zero in this limit.

The final answer for the potential everywhere is dependent on the specific boundary values given in the problem, and the solution should be written in the simplest form possible. It is important to carefully consider the given boundary conditions to determine the appropriate values for the constants.