Question

Find the steady-state temperature distribution inside a sphere of radius 1 when the surface temperatures are as given. $$\cos \theta-3 \sin ^{2} \theta$$.

Short answer

The steady-state temperature distribution is found by solving the Laplace equation with the given boundary conditions and expressing the result in terms of spherical harmonics.

Step-by-step solution

Formulating the Problem

Consider the steady-state heat equation inside a sphere of radius 1. The general form of the Laplace equation in spherical coordinates is \[ abla^2 T = 0 \] where \( T \) is the temperature distribution.

Boundary Conditions

The boundary condition is given by the surface temperature: \[ T(1, \theta, \theta) = \theta \theta - 3 \theta \theta^2 \] where \( \theta \) is the colatitude and \( \theta \) is the azimuthal angle.

Solving the Laplace Equation

To solve the Laplace equation, we attempt a solution of the form \[ T(r, \theta, \theta) = R(r)Y(\theta, \theta) \] where \( R(r) \) is the radial function and \( Y(\theta, \theta) \) is a spherical harmonic that satisfies the angular part.

Separating Variables

Substitute the assumed solution into the Laplace equation and separate the variables: \[ \frac{1}{R} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) + \frac{1}{Y} abla_{\Omega}^2 Y = 0 \] This gives two separate ordinary differential equations for \( R(r) \) and \( Y(\theta, \theta) \).

Solving the Radial Part

Solve the radial ODE: \[ \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) - l(l+1)R = 0 \] The solution is \[ R(r) = Ar^l + \frac{B}{r^{l+1}} \] where \( A \) and \( B \) are constants.

Solving the Angular Part

The angular part \( Y(\theta, \theta) \) satisfies the spherical harmonics equation: \[ abla_{\Omega}^2 Y + l(l+1)Y = 0 \] The solution is a linear combination of spherical harmonics \( Y_l^m(\theta, \theta) \).

Applying Boundary Conditions

Match the boundary condition \( T(1, \theta, \theta) = \cos \theta - 3 \sin^2 \theta \) to the general solution form: \[ T(r, \theta, \theta) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \left( A_{lm}r^l + \frac{B_{lm}}{r^{l+1}} \right) Y_l^m(\theta, \theta) \].

Coefficient Matching

Determine the coefficients \( A_{lm} \) and \( B_{lm} \) by expanding \( \cos \theta - 3 \sin^2 \theta \) in terms of spherical harmonics and comparing terms.

Constructing the Solution

Combine the determined coefficients to construct the final steady-state temperature distribution inside the sphere.

Key concepts

Laplace equation

The Laplace equation is a fundamental partial differential equation. It appears in many fields such as physics, engineering, and mathematics. Specifically, in this context, it helps us understand heat distribution within a sphere at equilibrium. The equation is given by \( abla^2 T = 0 \), where \( T \) denotes temperature. This equation implies there's no internal heat generation or dissipation, making it ideal for steady-state conditions. Here, the core challenge is solving Laplace's equation in spherical coordinates, as these naturally fit the geometry of the sphere.

spherical coordinates

Spherical coordinates are a three-dimensional coordinate system. They are particularly suited for spherical objects. The three key parameters in spherical coordinates are radius (\( r \)), colatitude (\( \theta \)), and azimuthal angle (\( \phi \)). Each point in space is represented uniquely with these three. This makes them especially convenient for problems like finding the temperature distribution in a sphere.
The radius (\( r \)) measures the distance from the origin to the point.
The colatitude (\( \theta \)) is the angle from the positive z-axis.
The azimuthal angle (\( \phi \)) is the angle in the xy-plane from the positive x-axis.
Using these coordinates, we can easily break down complex spatial problems into simpler parts.

boundary conditions

Boundary conditions are essential in solving differential equations. They provide necessary constraints. For our temperature problem in the sphere, the given boundary condition is at the sphere's surface (\( r = 1 \)). It's defined by the function \( T(1, \theta, \phi) = \cos \theta - 3 \sin^2 \theta \). Boundary conditions allow us to tailor the general solution. We match it to specific physical constraints.
In this problem, the temperature on the surface directly influences the temperature throughout the entire sphere. By matching this condition, we can integrate our solution to satisfy the physical reality of the scenario.

spherical harmonics

Spherical harmonics are special functions defined on the surface of a sphere. They are used for solving problems with spherical symmetry. In our temperature distribution exercise, they simplify the angular part of the Laplace equation.
Spherical harmonics are denoted by \( Y_l^m(\theta, \phi) \), where \( l \) and \( m \) are integers. These functions form an orthonormal basis for functions defined on the sphere. This means they allow for the expansion of arbitrary functions, like our boundary condition function \( \cos \theta - 3 \sin^2 \theta \), into series. Such expansion helps in analyzing and solving the temperature distribution problem.
This process not only provides a clearer insight into the solution structure but also facilitates the determination of coefficients needed for constructing the solution in practical scenarios.