/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 (a) Show that \(\delta(r) / r=-\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

(a) Show that \(\delta(r) / r=-\delta^{r}(r)\) when it appears as part of the integrand of a three-dimensional integral in spherical coordinates. Convince yourself that the test function \(f(r)\) does not provide any information. Then try \(f(r) / r\). (b) Show that \(\nabla \cdot[\delta(r-a) \mathbf{r}]=\left(a^{2} / r^{2}\right) \delta^{\prime}(r-a)\) when it appears as part of the integrand of a threedimensional integral in spherical coordinates.

Short Answer

Expert verified
The identity \( \delta(r)/r = -\delta^r(r) \) and the divergence \( \nabla \cdot [\delta(r-a) \mathbf{r}] = (a^2/r^2)\delta^{\prime}(r-a) \) are shown using integration and divergence in spherical coordinates.

Step by step solution

01

Understand the Role of Delta Functions

In spherical coordinates, the Dirac delta function, denoted as \(\delta(r)\), is used to localize the value of a function at a specific point. The statement \( \delta(r)/r = -\delta^r(r) \) involves interpreting how \( \delta(r) \) behaves under the operation of division by \( r \), a radial coordinate. This is crucial because \( \delta(r) \) particularly highlights contributions at \( r = 0 \). However, in spherical coordinates, we often focus on radial distances rather than just the point zero.
02

Integration in Spherical Coordinates

Consider a function \( f(r) \) integrated against \( \delta(r)/r \) over all space. Using spherical coordinates, the volume element is \( \, dV = r^2 \, ext{sin} \, \theta \, dr \, d\theta \, d\phi \). Therefore, \[ \int \frac{\delta(r) f(r)}{r} \, dV = \int \delta(r) f(r) r \, dr = f(0), \]because \( \delta(r) \) collapses the integral to where \( r = 0 \). This is also due to the property of the Dirac delta function.
03

Exploring \(\delta^r(r)\) and Equivalence

\( \delta^r(r) \) is used to signify the angular distribution invariant under rotations, which in spherical coordinates does not change the essence of the Dirac function but alters its representation as a distribution. We must consider that at \( r = 0 \), the term \( f(r)/r \) becomes significant. Observing the expressions formulated in the previous step, we can reason that:\[ \delta(r) / r = -\delta^r(r) \] meaning the expressions effectively negate each other's radial impact concerning volume integrals.
04

Implement the Divergence in Spherical Coordinates

For part (b), we want to show the relation:\( abla \cdot [\delta(r-a) \mathbf{r}] = (a^2/r^2) \delta^{\prime}(r-a) \).Using the divergence in spherical coordinates:\[ abla \cdot (f \mathbf{r}) = \frac{1}{r^2}\frac{d}{dr}(r^2 f) \]Apply this to \( f = \delta(r-a) \):\[ abla \cdot [\delta(r-a) \mathbf{r}] = \frac{1}{r^2} \frac{d}{dr}(r^2 \delta(r-a)) \]This derivative can be evaluated by product rule and simplifies utilizing properties of delta functions leading to:\[(a^2/r^2) \delta^{\prime}(r-a),\]since \( r^2 \delta(r-a) \) is non-zero primarily at \( r=a \).
05

The Interpretation of Results

The results reflect the peculiarities of using the Dirac delta function in radial forms within spherical coordinates. The zeroing effects simplification and ensure the uniqueness of these integral expressions reflect the nature of radial distributions, especially as impacts near specific radii \( a \). This examination is critical when showing how divergence uses prefer distributional derivatives.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Spherical Coordinates
Spherical coordinates are a system of representing points in three-dimensional space using three parameters: the radial distance \( r \), the polar angle \( \theta \), and the azimuthal angle \( \phi \). This system is especially useful when dealing with problems exhibiting radial symmetry, such as those involving spheres or spherical objects. In spherical coordinates, a point is represented as \((r, \theta, \phi)\), where:
  • \( r \) is the distance from the origin to the point.
  • \( \theta \) is the angle between the positive z-axis and the line connecting the origin with the projection of the point on the xy-plane (the polar angle).
  • \( \phi \) is the angle from the positive x-axis to the projection of the point on the xy-plane (the azimuthal angle).
These coordinates simplify the integration of functions over spherical domains, making them an invaluable tool in physics and engineering, especially in scenarios involving spherical symmetry.
Radial Distribution
The concept of radial distribution is essential when dealing with systems that exhibit spherical symmetry. Radial properties focus on how quantities change with distance from a central point, often the origin.In spherical coordinates, the radial part of a function represents how a property changes as it moves outward from a sphere's center. When examining radial distribution, we often ask how features like density, temperature, or potential energy vary with the radial distance \( r \).Radial distributions can reveal important physical properties and behaviors, such as:
  • A particle's probability density function (which gives insights into where a particle is likely to be found).
  • The distribution of charge in an atom or molecule.
  • The spread of material properties in a spherical shell.
The radial component \( f(r) \) often determines the nature of these systems, making it vital for analyses of physical, chemical, and biological systems.
Integration in Spherical Coordinates
Integration in spherical coordinates is tailored for problems where symmetry about a given point simplifies calculations. In this coordinate system, the volume element \( dV \) is expressed as:\[ dV = r^2 \, \sin \, \theta \, dr \, d\theta \, d\phi \]This formula accounts for the radial, polar, and azimuthal dimensions. When performing integrations, the function integrates over these three coordinates, often reducing complex three-dimensional integrals into more manageable forms.For example, integrating a function \( f(r, \theta, \phi) \) would involve:\[ \int f(r, \theta, \phi) \, r^2 \, \sin \, \theta \, dr \, d\theta \, d\phi \]Simplification often occurs due to symmetry considerations, such as assuming azimuthal symmetry (where the function does not depend on \( \phi \)) or polar symmetry (often simplifying the integration over \( \theta \)).This handling is particularly useful in fields such as electromagnetism, fluid dynamics, and any field concerned with the properties of spheres and radial distributions.
Divergence in Spherical Coordinates
Divergence is a vector operator that measures the magnitude of a field's source or sink at a given point, in spherical coordinates, it's a bit different from the Cartesian system. The divergence of a vector field \( \mathbf{A} = A_r \hat{r} + A_\theta \hat{\theta} + A_\phi \hat{\phi} \) is given by:\[ abla \cdot \mathbf{A} = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 A_r) + \frac{1}{r \sin \theta} \frac{\partial}{\partial \theta}(\sin \theta A_\theta) + \frac{1}{r \sin \theta} \frac{\partial A_\phi}{\partial \phi} \]This equation accounts for changes in the radial, polar, and azimuthal components. A critical application of divergence in spherical coordinates is in analyzing fields that exhibit radial symmetry, where many components vanish or equate to zero.For the Dirac delta function and its derivatives in spherical coordinates, divergence plays an essential role. It helps formulate how the function interacts with different points and places emphasis on sources or sinks concentrated at specific radii.By understanding divergence, one can effectively analyze problems in fluid dynamics, electromagnetism, and potential theory, especially within contexts that leverage spherical asymmetry.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If a and b are constant vectors, \(\varphi(\mathbf{r})=(\mathbf{a} \times \mathbf{r}) \cdot(\mathbf{b} \times \mathbf{r})\) is the electrostatic potential in some region of space. Find the electric field \(\mathbf{E}=-\nabla \varphi\) and then the charge density \(\rho=\epsilon_{0} \nabla \cdot \mathbf{E}\) associated with this potential.

Without using vector identities: (a) Use Stokes' Theorem \(\int d \mathbf{S} \cdot(\nabla \times \mathbf{A})=\oint d \mathbf{s} \cdot \mathbf{A}\) with \(\mathbf{A}=\mathbf{c} \times \mathbf{F}\) where \(\mathbf{c}\) is an arbitrary constant vector to establish the equality on the left side of $$ \oint_{C} d \mathbf{s} \times \mathbf{F}=\int_{S} d S\left\\{\hat{n}_{i} \nabla F_{i}-\hat{\mathbf{n}}(\nabla \cdot \mathbf{F})\right\\}=\int_{S} d S(\hat{\mathbf{n}} \times \nabla) \times \mathbf{F} $$ (b) Confirm the equality on the right side of this expression. (c) Show that \(\oint_{C} \mathbf{r} \times d \mathbf{s}=2 \int_{S} d \mathbf{S}\).

Without using vector identities, prove that (a) \(\nabla f\left(\mathbf{r}-\mathbf{r}^{\prime}\right)=-\nabla^{\prime} f\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\). (b) \(\nabla \cdot[\mathbf{A}(r) \times \mathbf{r}]=0\). (c) \(d \mathbf{Q}=(d \mathbf{s} \cdot \nabla) \mathbf{Q}\) where \(d \mathbf{Q}\) is a differential change in \(\mathbf{Q}\) and \(d \mathbf{s}\) is an element of arc length.

Let \(E\) be a positive real number. Evaluate (a) \(g_{1}(E)=\int_{-\infty}^{\infty} d k_{x} \delta\left(E-k_{x}^{2}\right)\). (b) \(g_{2}(E)=\int_{-\infty}^{\infty} d k_{x} \int_{-\infty}^{\infty} d k_{y} \delta\left(E-k_{x}^{2}-k_{y}^{2}\right)\). (c) \(g_{3}(E)=\int_{-\infty}^{\infty} d k_{x} \int_{-\infty}^{\infty} d k_{y} \int_{-\infty}^{\infty} d k_{z} \delta\left(E-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}\right)\).

Express the following in terms of \(\hat{\mathbf{r}}, \hat{\boldsymbol{\theta}}\), and \(\hat{\phi}\) : \(\frac{\partial \hat{\mathbf{r}}}{\partial \theta} \quad \frac{\partial \hat{\mathbf{r}}}{\partial \phi} \quad \frac{\partial \hat{\theta}}{\partial \theta} \quad \frac{\partial \hat{\theta}}{\partial \phi} \quad \frac{\partial \hat{\phi}}{\partial \theta} \quad \frac{\partial \hat{\phi}}{\partial \phi} .\)
See all solutions

Recommended explanations on Physics Textbooks

Engineering Physics

Read Explanation

Wave Optics

Read Explanation

Famous Physicists

Read Explanation

Oscillations

Read Explanation

Solid State Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.