Question

The upper half of a heat-conducting sphere of radius \(a\) has temperature \(T_{0}\); the lower half is maintained at temperature \(-T_{0}\). The whole sphere is inside an infinitely large mass of heat-conducting material. Find the steady-state temperature distribution inside and outside the sphere.

Short answer

The temperature distribution inside the sphere is given by the solution to Laplace's equation in spherical coordinates, obtained by solving two ordinary differential equations derived from the assumed solution \(T(r, \theta) = R(r) \Theta(\theta)\). The temperature outside the sphere remains constant as the sphere is in an infinitely large heat-conducting material.

Step-by-step solution

- Understand Physical Situation

Realize that, since the sphere is in an infinitely large mass of heat-conducting material, the temperature outside the sphere will not be affected by the sphere and will remain constant. This conclusion is based on the principle of heat conduction which states that heat flows from a region of high temperature to one of low temperature until equilibrium is reached.

- Apply Laplace's Equation

Recall that Laplace's equation in spherical coordinates for a function \(T(r, \theta)\) of the radial and angular coordinates, is given by: \[ \frac{1} {r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial T(r, \theta)}{\partial r} \right) + \frac{1}{r^2 \sin^2(\theta)} \frac{\partial^2 T(r, \theta)}{\partial \theta^2} = 0 \]where \(r\) is the radial distance from the center of the sphere and \(\theta\) is the polar angle. This equation is used to describe the temperature distribution in bodies.

- Assume Solution Form

Assume a solution of the form \(T(r, \theta) = R(r) \Theta(\theta)\), where \(R(r)\) only depends on \(r\) and \(\Theta(\theta)\) only depends on \(\theta\). This assumption helps to simplify the differential equation.

- Solve for \(R(r)\) and \(\Theta(\theta)\)

Substitute the assumed solution form into Laplace's equation to get two ordinary differential equations for \(R(r)\) and \(\Theta(\theta)\). By solving these equations under the boundary conditions of the temperature of the sphere, the temperature distribution inside the sphere can be obtained.

- Construct Temperature Distribution

Combine the solutions for \(R(r)\) and \(\Theta(\theta)\) to obtain \(T(r, \theta)\), the temperature distribution inside the sphere. As noted earlier, the temperature distribution outside the sphere will be a constant value imposed by the infinitely large heat-conducting material.

Key concepts

Laplace's Equation

Understanding Laplace's Equation is fundamental when tackling problems related to heat conduction in spheres. It is a differential equation that describes how a quantity, such as temperature, is distributed in space. When dealing with a steady-state situation, where the temperature distribution does not change over time, we use Laplace's equation to assert this condition. In spherical coordinates, the equation is expressed a bit differently due to the geometry of the sphere.

For a function describing temperature, denoted as \(T(r, \theta)\), where \(r\) represents the radial distance from the center of the sphere and \(\theta\) the polar angle, the equation appears as follows:\[ \frac{1}{r^2} \frac{\text{\textpartial}}{\text{\textpartial} r} \left( r^2 \frac{\text{\textpartial} T(r, \theta)}{\text{\textpartial} r} \right) + \frac{1}{r^2 \sin^2(\theta)} \frac{\text{\textpartial}^2 T(r, \theta)}{\text{\textpartial} \theta^2} = 0 \]When the equation equals zero, it indicates that heat flow has reached equilibrium and there is no net change in temperature. The aim is to find the temperature function, \(T(r, \theta)\), that satisfies this equation and adheres to the physical constraints of the problem.

Steady-State Temperature Distribution

A steady-state temperature distribution implies that the temperature within the material remains constant over time, even though there may be heat transfer occurring within it. This concept is crucial in many engineering applications where the material's response to a thermal environment is under investigation, often over long periods where the initial transient has settled and the system no longer changes.

In the provided problem, we're seeking a steady-state solution inside a sphere that has a peculiar temperature boundary condition: one hemisphere is at temperature \(T_0\) and the other at \(-T_0\). Our goal is to understand how the temperature stabilizes throughout the sphere and in its surrounding material. This involves finding the particular solution to the Laplace's equation that reflects the boundary conditions without changing over time.

Spherical Coordinates

To analyze heat conduction in spheres, it's essential to use spherical coordinates because they align with the symmetry of the problem. This coordinate system breaks down the position of any point within or on the surface of the sphere into three components: the radial distance \(r\) from the center of the sphere, the polar angle \(\theta\) measured from the positive z-axis, and the azimuthal angle \(\phi\) within the x-y plane from the positive x-axis.

In our scenario, since there's symmetry about the z-axis due to the upper and lower hemispheres being maintained at constant and opposite temperatures, we can ignore the \(\phi\) dependency. Therefore, the temperature at any point can be uniquely determined by \(r\) and \(\theta\) which makes our analysis two-dimensional and more manageable. This simplification nicely fits within the separation of variables method applied to solve the Laplace's equation for this problem.

Boundary Conditions

Last but not least, boundary conditions are the crucial piece informing us how the system behaves at its edges. When solving the Laplace's equation for heat conduction, boundary conditions define how the temperature is controlled at the surface or interface of the materials involved. In the problem at hand, the boundary conditions specify the temperatures of different parts of the sphere's surface, with the upper and lower hemispheres each maintained at constant but opposite temperatures \(T_0\) and \(-T_0\) respectively.

This stark contrast at the equator of the sphere sets the stage for how heat must be conducted through the material. For the sphere sitting in an infinitely large volume, outside the sphere, the boundary condition is essentially a constant temperature field imposed by its surrounding. These conditions influence the final form of the solution and ensure that it is physically meaningful and corresponds accurately to the scenario described in the exercise.