Problem 12 - Advanced Engineering Mathematics

Question

A conducting sphere of radius \(c\) is grovnded and placed in a uniform electric field that has intensity \(E\) in the \(z\)-direction. The potential \(u(r, \theta)\) outside the sphere is determined from the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{2}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}+\frac{\cot \theta}{r^{2}} \frac{\partial u}{\partial \theta}=0, \quad r>c, 0<\theta<\pi \\\ &u(c, \theta)=0, \quad 0<\theta<\pi \\ &\lim _{\rightarrow \infty} u(r, \theta)=-E z=-E r \cos \theta . \end{aligned} $$ Show that $$ u(r, \theta)=-E r \cos \theta+E \frac{c^{3}}{r^{2}} \cos \theta $$

Step-by-Step Solution

Answer

The potential is \( u(r, \theta) = -E r \cos \theta + E \frac{c^3}{r^2} \cos \theta \).

Step 1: Identify the Solution Type

The problem is a boundary value problem involving Laplace's equation in spherical coordinates for the potential function \( u(r, \theta) \). The goal is to solve this equation subject to specific boundary conditions. This type of problem suggests using separation of variables.

Step 2: Set Up the General Solution

The solution to Laplace's equation in spherical coordinates can generally be expressed in terms of spherical harmonics. For axisymmetric problems, like this one, the solution can be written as: \[ u(r, \theta) = \sum_{n=0}^{\infty} \left(A_n r^n + \frac{B_n}{r^{n+1}}\right) P_n(\cos \theta) \] where \( P_n(\cos \theta) \) are Legendre polynomials.

Step 3: Apply the Boundary Conditions

Start with the first boundary condition: \( u(c, \theta) = 0 \). Substitute \( r = c \) into the general solution: \[ u(c, \theta) = \sum_{n=0}^{\infty} \left(A_n c^n + \frac{B_n}{c^{n+1}}\right) P_n(\cos \theta) = 0 \] This condition implies that each term \( A_n c^n + \frac{B_n}{c^{n+1}} = 0 \) for all \( n \).

Step 4: Simplify Using the Condition at Infinity

Apply the boundary condition as \( r \to \infty \): \( u(r, \theta) = -E r \cos \theta \). This suggests the leading term for large \( r \) is \( -E r \cos \theta \), which can be represented by \( -E r P_1(\cos \theta) \) since \( P_1(\cos \theta) = \cos \theta \). This condition requires that \( A_1 = -E \) and for all other \( n \), \( A_n = 0 \) since higher order terms would dominate.

Step 5: Solve for Remaining Coefficients

Given \( A_1 = -E \), the condition from Step 3, \( A_n c^n + \frac{B_n}{c^{n+1}} = 0 \) for \( n = 1 \), becomes \[ -E c + \frac{B_1}{c^2} = 0 \] solving for \( B_1 \) gives \( B_1 = E c^3 \). Thus, the other \( B_n \) terms for \( n eq 1 \) must be zero to maintain the boundary at infinity.

Step 6: Construct the Specific Solution

Now compile the coefficients found. The potential function becomes: \[ u(r, \theta) = -E r P_1(\cos \theta) + \frac{E c^3}{r^2} P_1(\cos \theta) = -E r \cos \theta + E \frac{c^3}{r^2} \cos \theta \] This satisfies all boundary conditions.

Key Concepts

Boundary Value Problem

A boundary value problem is a type of differential equation where we seek a solution that satisfies certain conditions at the boundaries of the domain. These problems are crucial in mathematics and physics, often describing physical phenomena such as heat, wave, and potential equations.
For our exercise, we deal with Laplace's equation, a boundary value problem in spherical coordinates.
The key conditions are: the grounded sphere's surface which dictates the potential to be zero at the boundary, and the behavior at infinity where the potential aligns with the external electric field.
Entropy and energy are typically conserved, allowing for the solution to honor these boundary requirements.

Spherical Coordinates

Spherical coordinates offer a convenient way to describe points in a 3D space where symmetry about a point simplifies the mathematics, as is the case with our problem involving a sphere.
Instead of using Cartesian coordinates (x, y, z), we employ (r, \(\theta\), \(\phi\)), where r is the distance from the origin, \(\theta\) is the polar angle, and \(\phi\) is the azimuthal angle.
In our problem, the potential function is analyzed in terms of r (radius) and \(\theta\) (the angle from the Z-axis), exploiting the symmetry of the sphere and the unidirectional electric field described.

Separation of Variables

This is a mathematical method used to simplify partial differential equations by splitting them into simpler, solvable ordinary differential equations.
In the context of our problem, the potential \(u(r, \theta)\) is expressed as a product of functions, each depending solely on one of the spherical coordinates.
This separation allows us to manage complex differential equations by breaking them down into manageable segments.
The separated solutions can be represented as expansions, commonly involving special functions like Legendre polynomials.

Legendre Polynomials

Legendre polynomials are a class of orthogonal polynomials widely used in solving physics and engineering problems. They are solutions to Legendre's differential equation and are especially prominent in problems with spherical symmetry.
In our boundary value problem, Legendre polynomials \(P_n(\cos \theta)\) help express the potential function.
This is useful for elaborating axisymmetric solutions where the potential depends on cosine terms.
By employing these polynomials, the infinite series solution involves terms of the form \(A_n r^n + \frac{B_n}{r^{n+1}}\), combined with polynomial \(P_n(\cos \theta)\), fitting nicely into spherical scenarios like ours.