Problem 5 - Classical Electrodynamics

Question

A hollow sphere of inner radius \(a\) has the potential specified on its surface to be \(\Phi=V(\theta, \phi)\). Prove the equivalence of the two forms of solution for the potential inside the sphere: (a) \(\Phi(\mathbf{x})=\frac{a\left(a^{2}-r^{2}\right)}{4 \pi} \int \frac{V\left(\theta^{\prime}, \phi^{\prime}\right)}{\left(r^{2}+a^{2}-2 a r \cos \gamma\right)^{3 / 2}} d \Omega^{\prime}\) where \(\cos \gamma=\cos \theta \cos \theta^{\prime}+\sin \theta \sin \theta^{\prime} \cos \left(\phi-\phi^{\prime}\right)\) (b) \(\Phi(\mathbf{x})=\sum_{i=0}^{-} \sum_{m=-t}^{1} A_{i m}\left(\frac{r}{a}\right)^{t} Y_{t m}(\theta, \phi)\) where \(A_{b m}=\int d \Omega^{\prime} Y_{t m}^{*}\left(\theta^{\prime}, \phi^{\prime}\right) V\left(\theta^{\prime}, \phi^{\prime}\right) .\)

Step-by-Step Solution

Answer

The integrals and series both resolve to spherical harmonic expansions, hence are equivalent.

Step 1: Understanding the Problem

We need to prove the equivalence of two different mathematical solutions (formulas) for the electric potential inside a hollow sphere. Each solution represents the potential given a specific surface condition on the sphere. Solution (a) uses surface integrals to represent the potential, while solution (b) expresses the potential as a sum of spherical harmonics.

Step 2: Outline of Solution Form (a)

In solution (a), the potential inside the sphere is expressed as an integral over the surface of the sphere. This involves a potential function \(V(\theta', \phi')\) and a factor involving the geometry of the sphere, specifically \((a^{2} - r^{2})\) and the denominator that depends on the angle \(\gamma\).

Step 3: Outline of Solution Form (b)

Solution (b) represents the potential as an infinite series based on spherical harmonic functions \(Y_{tm}(\theta, \phi)\). The coefficients \(A_{im}\) are computed through an integral over the sphere, involving the complex conjugate of the spherical harmonics and the potential \(V(\theta', \phi')\).

Step 4: Equating the Forms

To prove the equivalence, recognize that both solutions involve expansions over spherical harmonics. The integral in solution (a) can be expressed in terms of these functions. Integrals of this type in spherical coordinate systems relate closely to these harmonic functions, suggesting that solution (a) is capturing the same coefficients \(A_{im}\) of solution (b).

Step 5: Mathematical Equivalence

The core of the equivalence is the representation of the integral for \(A_{im}\) in both forms. The addition theorem for spherical harmonics is used to make the connection between the angle \(\gamma\) in the integral and the expansion into spherical harmonics found in form (b). This theorem allows rewriting the potential function as a sum of harmonics, leading directly to the coefficients \(A_{im}\).

Step 6: Summarizing the Connection

By recognizing and utilizing the expansion of \(\frac{1}{(r^2 + a^2 - 2ar \cos \gamma)^{\sqrt{2}}}\) in terms of spherical harmonics, and matching it against form (b), one sees that the coefficients \(A_{im}\) are identical, demonstrating the equivalence.

Key Concepts

Spherical Harmonics

Spherical harmonics are a set of special functions defined on the surface of a sphere, commonly used in physics and engineering for problems involving spherical symmetry. They are denoted as \(Y_{lm}(\theta, \phi)\), where \(l\) and \(m\) are integers that determine the degree and order of the harmonic, respectively. The angles \(\theta\) and \(\phi\) are the spherical polar and azimuthal angles.

• They are orthogonal functions, meaning any two different spherical harmonics integrated over a sphere yield zero.
• These functions form a basis for representing functions defined on the sphere, allowing us to express complex spherical surfaces as sums of these harmonics.

Mathematically, spherical harmonics are expressed in terms of Legendre polynomials and are used in a wide range of disciplines, including quantum mechanics to describe atomic orbitals, and in electromagnetism to solve potential issues on spherical conductors. In the context of electrostatic potential, spherical harmonics facilitate the decomposition of potential functions into manageable, harmonic components.

Boundary Conditions

In the study of electrostatic potentials, boundary conditions are crucial as they determine the potential's behavior in a specific region. For a hollow sphere, boundary conditions are applied at the surface to ensure that the solutions of the potential are physically viable and mathematically consistent with the given problem.Boundary conditions can be of different types:

• Dirichlet boundary conditions specify the potential on the boundaries. In this exercise, \(\Phi = V(\theta, \phi)\) on the surface of the sphere is a Dirichlet condition.
• Neumann boundary conditions specify the derivative of the potential, such as electric field components at the boundaries.

Applying these conditions ensures the solutions adhere to the physics of the problem. For spheres, the conditions mean the potential inside and outside must match seamlessly across the boundary, ensuring continuity and physical realism.

Surface Integrals

Surface integrals are a type of integral used to evaluate functions stretched over a surface, rather than along a line or throughout a volume. In electrostatics, they help compute electric potentials and fields, particularly on surfaces like spheres or planes.Surface integrals are defined as \(\int \int_S f(x, y, z) \, dS\), where \(f(x, y, z)\) is a scalar or vector function and \(dS\) is an infinitesimal area on the surface. To solve the potential on a spherical surface, one often integrates over the sphere's surface in spherical coordinates. This is demonstrated in solution (a), where the potential inside the sphere is represented by an integral involving the surface potential \(V(\theta', \phi')\). The integration takes into account the geometry of the surface, linking the potential with the angles \(\theta\) and \(\phi\) to exploit symmetry and simplify calculations. This enables the determination of the same potential in more complex geometries beyond basic shapes.