Electrostatic potential in Legendre polynomials

In summary, the problem involves two spherical shells of different radii centered about the origin and grounded, with a point charge placed between them. The electrostatic potential can be expanded in Legendre polynomials to find the Green function of the problem, with pre-factors explicitly calculated. The potential on the outer shell is q/b and on the inner shell is q/R, and the general expansion of the potential in terms of Legendre polynomials can be used to represent this.
  • #1
technion
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Homework Statement



Two spherical shells of radius ‘a’ and ‘b’ (b>a) are centered about the origin of the
axes, and are grounded. A point charge ‘q’ is placed between them at distance R from the
origin (a<R<b).
Expand the electrostatic potential in Legendre polynomials and find the Green function of
the problem as a series. Explicitly find all pre-factors.


Homework Equations





The Attempt at a Solution



The potential on the outer shell is q/b and the inner shell q/R, right? How do I represent this in legendre polynomials?
 
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  • #2
Hi technion, welcome to PF!:smile:

Is it safe to assume that the the shells are conductors? (You didn't tell us in your problem statement)

technion said:
The potential on the outer shell is q/b and the inner shell q/R, right?

Well, what is your reasoning behind this?

How do I represent this in legendre polynomials?

What is the general expansion of the potential in terms of Legendre polynomials?
 

Related to Electrostatic potential in Legendre polynomials

1. What are Legendre polynomials?

Legendre polynomials are a type of mathematical function that are used to describe the electrostatic potential of a charged particle in a particular system. They are named after the French mathematician Adrien-Marie Legendre, who first introduced them in the early 19th century.

2. How are Legendre polynomials used to describe electrostatic potential?

Legendre polynomials are used in electrostatics to describe the electric potential of a charged particle in a given system. They are used to calculate the potential energy of a charged particle at a specific location in relation to other charged particles in the system.

3. What is the significance of using Legendre polynomials in electrostatic potential?

Legendre polynomials are particularly useful in electrostatics because they can accurately describe the potential energy of a charged particle in a variety of complex systems. They are also important in quantum mechanics, where they are used to describe the wave function of an electron in an atom.

4. How are Legendre polynomials related to other mathematical functions?

Legendre polynomials are a special type of orthogonal polynomial, which means they are perpendicular to each other when plotted on a graph. They are also related to other mathematical functions, such as spherical harmonics, which are used to describe the behavior of charged particles in three-dimensional space.

5. Can Legendre polynomials be used to describe the potential of a magnetic field?

No, Legendre polynomials are specifically used to describe electrostatic potential, which is the potential energy of charged particles in an electric field. For magnetic fields, other mathematical functions, such as spherical Bessel functions, are used to describe the potential energy of charged particles.

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