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Chapter 19: Problem 5

Denoting the three terms of \(\nabla^{2}\) in spherical polars by \(\nabla_{r}^{2}, \nabla_{\theta}^{2}, \nabla_{\phi}^{2}\) in an obvious way, evaluate \(\nabla_{r}^{2} u\), etc. for the two functions given below and verify that, in each case, although the individual terms are not necessarily zero their sum \(\nabla^{2} u\) is zero. Identify the corresponding values of \(\ell\) and \(m\). (a) \(u(r, \theta, \phi)=\left(A r^{2}+\frac{B}{r^{3}}\right) \frac{3 \cos ^{2} \theta-1}{2}\). (b) \(u(r, \theta, \phi)=\left(A r+\frac{B}{r^{2}}\right) \sin \theta \exp i \phi\).

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01

Understanding the Laplacian in Spherical Coordinates

The Laplacian operator in spherical coordinates is given by: \[ abla^2 u = abla_r^2 u + abla_{\theta}^2 u + abla_{\phi}^2 u \] where: \[ abla_r^2=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial u}{\partial r}\right), \; abla_{\theta}^2=\frac{1}{r^2 \sin \theta}\frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial u}{\partial \theta}\right), \; abla_{\phi}^2=\frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 u}{\partial \phi^2} \]
02

Calculate \( abla_r^2 u \) for Function (a)

For the function \( u(r, \theta, \phi) = \left(A r^2 + \frac{B}{r^3} \right) \frac{3 \cos^2 \theta - 1}{2} \), we have: \[ \frac{\partial u}{\partial r} = \left( 2 A r - 3 B r^{-4} \right) \frac{3 \cos^2 \theta - 1}{2} \] Next, \[ \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right) = \frac{\partial}{\partial r} \left( r^2 \left( 2 A r - 3 B r^{-4} \right) \frac{3 \cos^2 \theta - 1}{2} \right) = \frac{\partial}{\partial r} \left( \left(2 A r^3 - 3 B \right) \frac{3 \cos^2 \theta - 1}{2} \right) \] Simplifying further, \[ \frac{\partial}{\partial r} \left(2 A r^3 - 3 B \right) = 6 A r^2 \] So, \[ abla_r^2 u = \frac{1}{r^2} \cdot 6 A r^2 \frac{3 \cos^2 \theta - 1}{2} = 3 A \left(3 \cos^2 \theta - 1 \right) \]
03

Calculate \( abla_{\theta}^2 u \) for Function (a)

With the same function: \[ \frac{\partial u}{\partial \theta} = \left(A r^2 + \frac{B}{r^3} \right) 3 \cos \theta (- \sin \theta) \] which simplifies to: \[ \Rightarrow \left(A r^2 + \frac{B}{r^3} \right) 3 \cos \theta (- \sin \theta) \] Next, \[ \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial u}{\partial \theta} \right) = \frac{\partial}{\partial \theta} \left( -3\left(A r^2 + \frac{B}{r^3} \right) \sin^2 \theta \cdot \cos \theta \right) \] Simplifying further, \[ \rightarrow \frac{1}{r^2} \frac{1}{\sin \theta} \left(6 \left(A r^2 + \frac{B}{r^3}\right) \cos \theta \sin \theta + \left(3 Ar^2 + \frac{3B}{r^3}\right)\cos \theta \sin^3 \theta \right) \] gives \[ \rightarrow \left( + 3 \left(A r^2 + \frac{B}{r^3} \right) \cos^2 \theta \right) \] divisable by \frac{2}{r^2 =6 A \cos \theta \sin \theta + 12 \left(A r^2 + \frac{B}{r^3} \right)4 abla_{\theta}^2} \left( \right)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Differential Equations
Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. For instance, they can describe how physical quantities such as temperature, pressure, or electromagnetic fields vary over time and space. PDEs are fundamental in the mathematical modeling of many natural phenomena. Unlike ordinary differential equations, which are defined over a single variable, PDEs involve multiple variables and their partial derivatives. An example of a PDE is the Laplace equation: \[ abla^2 u = 0 \], which is significant in fields like electrostatics, fluid dynamics, and potential theory. Solving PDEs often requires specialized techniques, such as separation of variables.
Spherical Polar Coordinates
Spherical polar coordinates are a three-dimensional coordinate system where the position of a point is specified by three numbers: the radial distance (r), the polar angle (胃), and the azimuthal angle (蠁). This system is especially useful in physics and engineering when dealing with problems that exhibit spherical symmetry, such as gravitational fields. The relationships between spherical coordinates \((r, \theta, \theta)\) and Cartesian coordinates \( (x, y, z)\) are:
  • \(x = r \, \text{sin}(\theta) \, \text{cos}(\theta)\)
  • \(y = r \, \text{sin}(\theta) \, \text{sin}(\theta)\)
  • \(z = r \, \text{cos}(\theta)\)
Transforming PDEs into spherical coordinates often simplifies the equations, leveraging their symmetry properties.
Laplacian Operator
The Laplacian operator (鈭嚶�) is a differential operator that appears in various PDEs. In spherical coordinates, it decomposes into three parts corresponding to the radial, polar, and azimuthal directions. The formula for the Laplacian in spherical coordinates is: \[abla^2 u = abla_r^2 u + abla_{\theta}^2 u + abla_{\theta}^2 \], where
  • \(abla_r^2 = \frac{1}{r^2} \frac{\text{d}}{\text{d}r} \bigg( r^2 \frac{\text{d}u}{\text{d}r} \bigg) \)
  • \(abla_{\theta}^2 = \frac{1}{r^2 \text{sin}(\theta) } \frac{\text{d}}{\text{d}\theta} \bigg( \text{sin}(\theta) \frac{\text{d}u}{\text{d}\theta} \bigg) \)
  • \(abla_{\theta}^2 = \frac{1}{r^2 \text{sin}^2(\theta) } \frac{\text{d}^2 u}{\text{d}\theta^2} \)
This operator is central to describing phenomena like heat conduction and wave propagation in spherical systems.
Separation of Variables
Separation of variables is a method used to solve certain PDEs, where the equation is rewritten such that each variable can be solved independently. This technique assumes a solution can be written as the product of functions, each depending only on one variable. For instance, a function \(u(r, \theta, \theta) = R(r) P(\theta) P(\theta)\). By substituting this form into a PDE, we can often rearrange the equation such that each term involves only one of the variables. The resulting ordinary differential equations can be solved separately. In spherical coordinates, this method is particularly useful because it simplifies the equations into forms that are more manageable.

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Most popular questions from this chapter

(a) Show that the gravitational potential due to a uniform disc of radius \(a\) and mass \(M\), centred at the origin, is given for \(ra\) by $$ \frac{G M}{r}\left[1-\frac{1}{4}\left(\frac{a}{r}\right)^{2} P_{2}(\cos \theta)+\frac{1}{8}\left(\frac{a}{r}\right)^{4} P_{4}(\cos \theta)-\cdots\right] $$ where the polar axis is normal to the plane of the disc. (b) Reconcile the presence of a term \(P_{1}(\cos \theta)\), which is odd under \(\theta \rightarrow \pi-\theta\), with the symmetry with respect to the plane of the disc of the physical system. (c) Deduce that the gravitational field near an infinite sheet of matter of constant density \(\rho\) per unit area is \(2 \pi G \rho\).

Find the Green's function \(G\left(\mathbf{r}, \mathbf{r}_{0}\right)\) in the half-space \(z>0\) for the solution of \(\nabla^{2} \Phi=0\) with \(\Phi\) specified in cylindrical polar coordinates \((\rho, \phi, z)\) on the plane \(z=0\) by $$ \Phi(\rho, \phi, z)= \begin{cases}1 & \text { for } \rho \leq 1 \\ 1 / \rho & \text { for } \rho>1\end{cases} $$ Determine the variation of \(\Phi(0,0, z)\) along the \(z\)-axis.

In the region \(-\infty

A conducting cube has as its six faces the planes \(x=\pm a, y=\pm a\) and \(z=\pm a\), and contains no internal heat sources. Verify that the temperature distribution $$ u(x, y, z, t)=A \cos \frac{\pi x}{a} \sin \frac{\pi z}{a} \exp \left(-\frac{2 \kappa \pi^{2} t}{a^{2}}\right) $$ obeys the appropriate diffusion equation. Across which faces is there heat flow? What is the direction and rate of heat flow at the point \((3 a / 4, a / 4, a)\) at time \(t=a^{2} /\left(\kappa \pi^{2}\right) ?\)

Electrostatic charge is distributed in a sphere of radius \(R\) centred on the origin. Determine the form of the resultant potential \(\phi(\mathbf{r})\) at distances much greater than \(R\), as follows. (a) express in the form of an integral over all space the solution of $$ \nabla^{2} \phi=-\frac{\rho(\mathbf{r})}{\epsilon_{0}} $$ (b) show that, for \(r \gg r^{\prime}\), $$ \left|\mathbf{r}-\mathbf{r}^{\prime}\right|=r-\frac{\mathbf{r} \cdot \mathbf{r}^{\prime}}{r}+\mathrm{O}\left(\frac{1}{r}\right) $$ (c) use results (a) and (b) to show that \(\phi(\mathbf{r})\) has the form $$ \phi(\mathbf{r})=\frac{M}{r}+\frac{\mathbf{d} \cdot \mathbf{r}}{r^{3}}+\mathrm{O}\left(\frac{1}{r^{3}}\right) $$ Find expressions for \(M\) and \(\mathbf{d}\), and identify them physically.
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