Question

Find the steady-state temperature distribution inside a sphere of radius \(a\) when the surface temperature is given by: (a) \(T_{0} \cos ^{2} \theta\), (b) \(T_{0} \cos ^{4} \theta\), (c) \(T_{0}|\cos \theta|\), (d) \(T_{0}\left(\cos \theta-\cos ^{3} \theta\right)\), (e) \(T_{0} \sin ^{2} \theta\), (f) \(\quad T_{0} \sin ^{4} \theta\).

Short answer

The solutions to this heat equation for the steady-state temperature distribution inside a sphere under different surface temperature conditions can be obtained by solving the heat conduction equation, converting it to spherical coordinates and applying appropriate boundary conditions for each specific situation. However, the explicit details of these solutions would depend on the exact interpretations for each given surface temperature function.

Step-by-step solution

Deducing General Solution

First, deduce the general solution. A parts of it would be boundary conditions which involve \( T_{0} \) times some function of \( \theta \). Hence, after solving the general heat conduction equation in Cartesian coordinates, convert the obtained solution to spherical coordinates.

Apply for each surface temperature distribution

Use the boundary conditions given in each part of the exercise: (a) to (f), to solve for the other parts of the general solution.

Evaluate the solutions

Evaluate the solutions for each situation (a) to (f). Check if the solutions fit with the given boundary conditions and ensure they describe a steady-state temperature distribution inside a sphere. If not, the procedure would need to be revised.

Key concepts

Heat Conduction Equation

The heat conduction equation, also known as Fourier's law, is essential to understand when examining how heat spreads within various materials. It's a differential equation that describes the flow of heat in an object over time. For a spherical object, the steady-state heat conduction equation, where the temperature doesn’t change with time, is simplified because there’s no heat being generated or lost; the system is in balance.

Solving the heat conduction equation in steady state cases means we're looking for temperature distribution, which doesn’t vary as time progresses. This is often encountered in mathematical physics, where we can assume that after a long time, the system reaches a condition where temperature at any point in the object stays constant. For the spherical object in your exercise, such as a sphere of radius \(a\), this equation in spherical coordinates revolves around the radial distance from the center of the sphere and the angle, but not time since it's the steady-sate scenario.

Spherical Coordinates

Spherical coordinates are an alternative to Cartesian coordinates for describing locations in three-dimensional space. They are particularly useful for problems involving symmetry around a point, such as the temperature distribution within a sphere. In spherical coordinates, any point in space is described by three values: the radial distance \(r\), the polar angle \(\theta\), also known as colatitude, usually measured from the positive z-axis, and the azimuthal angle \(\phi\), measured in the x-y plane from the positive x-axis.

For your heat conduction problem, we are primarily concerned with the radial distance and the polar angle, since the temperature is symmetrically distributed around the center of the sphere and doesn’t depend on the azimuthal angle. This symmetry drastically simplifies the problem and allows us to find the temperature distribution as a function of \(r\) and \(\theta\) only.

Boundary Conditions

Boundary conditions in mathematical physics are constraints that we apply to differential equations to determine a unique solution. These conditions are critical in problems of heat conduction, as they dictate how heat behaves at the borders of the materials. In the sphere’s case, the boundary conditions specify the temperature distribution at the surface of the sphere, or when \(r = a\).

The exercise provides different surface temperature scenarios, labeled (a) to (f), each presenting a unique boundary condition. In solving for the temperature distribution inside the sphere, the boundary conditions act as the 'clues' that help to lock in the final form of the temperature function within the sphere. You use these conditions to ensure the mathematically derived temperature distribution fits the physical scenario at the sphere's boundary.

Mathematical Physics

Mathematical physics involves making mathematical models to describe physical phenomena. Solving a heat conduction problem in a sphere is a prime example of this. It entails creating a mathematical model (the heat conduction equation) that accounts for the heat transfer inside a spherical object, applying appropriate mathematical tools (spherical coordinates) for the geometry of the problem, and setting up physical constraints (boundary conditions) based on the real-world scenario.

Understanding mathematical physics is crucial because it provides the language to translate physical conditions into mathematical equations that can be solved to predict the behavior of a system. This particular exercise showcases the need to merge theory with practical physical constraints to find a solution that models a real-world scenario.