Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 18: Problem 23

Consider each of the following situations in a qualitative way and determine the equation type, the nature of the boundary curve and the type of boundary conditions involved. (a) a conducting bar given an initial temperature distribution and then thermally isolated; (b) two long conducting concentric cylinders on each of which the voltage distribution is specified; (c) two long conducting concentric cylinders on each of which the charge distribution is specified; (d) a semi-infinite string the end of which is made to move in a prescribed way.

Short Answer

Expert verified
Heat equation with Neumann condition (a), Laplace equation with Dirichlet conditions (b), Gauss's law with Neumann conditions (c), and wave equation with mixed conditions (d).

Step by step solution

01

- Situation (a)

This situation involves a conducting bar with an initial temperature distribution and thermal isolation. This is a heat conduction problem. The equation type is the heat equation: \ \( \frac{\rightarrow}{\rightarrow t} = \text{thermal diffusivity} \times abla^2 T \) The boundary conditions involve thermal insulation, which means no heat flow across the boundaries. Mathematically, this translates to Neumann boundary conditions: \ \( \frac{\rightarrow T}{\rightarrow n} = 0 \) where n is the normal to the boundary.
02

- Situation (b)

This situation involves two long conducting concentric cylinders with specified voltage distribution. This is an electrostatics problem. The governing equation is Laplace's equation: \ \( abla^2 V = 0 \) The boundary conditions are Dirichlet boundary conditions since the voltage is specified on the cylinders' surfaces: \ \( V = V_{cylinder1} \) and \ \( V = V_{cylinder2} \).
03

- Situation (c)

This situation also involves two long conducting concentric cylinders, but with specified charge distribution. This is another electrostatics problem. Gauss's law for electricity governs this situation: \ \( abla \bullet \textbf{E} = \frac{\rho}{\text{permittivity}} \) The boundary conditions involve specifying the normal component of the electric field, known as Neumann boundary conditions: \ \( \frac{\rightarrow V}{\rightarrow n} = \text{given charge distribution} \).
04

- Situation (d)

This involves a semi-infinite string with one end moving in a prescribed way. It can be modeled by the wave equation: \ \( \frac{\rightarrow^2 y}{\rightarrow t^2} = c^2 \frac{\rightarrow^2 y}{\rightarrow x^2} \) The boundary conditions are mixed; for the end with the prescribed motion, it is a Dirichlet boundary condition: \ \( y(0,t) = f(t) \). At the other end, extending to infinity, typically a boundary condition would involve ensuring no reflected wave or a specified behavior at infinity.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Heat Equation
The heat equation is crucial in understanding how heat diffuses within a material. It describes how the temperature in a given region changes over time due to thermal conduction. The general form of the heat equation in one dimension is: \ \ \[ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} \] where: \ul \ \alpha \ is the thermal diffusivity of the material. \; \ T \ is the temperature. \; \ x \ is the spatial coordinate. By solving the heat equation, one can determine how the temperature evolves.
The boundary conditions define the system's behavior at its boundaries. In the exercise, thermal insulation (Neumann boundary conditions) implies no heat flow at the boundaries: \ \ \[ \frac{\partial T}{\partial n} = 0 \] This equation is crucial in many fields, like climate science, engineering, and physics, to predict thermal behavior.
Laplace's Equation
Laplace's equation is used in electrostatics to describe fields without any free charges. When considering two long conducting concentric cylinders with specified voltage distributions, Laplace's equation is:
\[ abla^2 V = 0 \] where: \ul * \ V \ is the electric potential. There are boundary conditions that need to be met, known as Dirichlet boundary conditions, specify the value of the function on the surface:
Gauss's Law for Electricity
Gauss's law in electrostatics relates the distribution of electric charge to the resulting electric field and is fundamental in understanding electrostatic phenomena. For two long conducting concentric cylinders with a given charge distribution, Gauss's law is: \ \ \[ abla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \] where: \ul * \ \mathbf{E} \ is the electric field. \; * \ \rho \ is the charge density. \; \ \epsilon_0 \ is the permittivity of free space. The boundary conditions in this case involve a specified charge distribution, which translates to Neumann boundary conditions for the potential. Specifically, it involves specifying the normal component of the electric field. This law is applied extensively in fields like electronics and electromagnetics to design and analyze systems.
Wave Equation
The wave equation describes how waves travel through a medium. It is used in various areas, including acoustics, electromagnetism, and string theory. For a semi-infinite string where the end is made to move in a prescribed way, the wave equation is: \ \ \[ \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} \] where: \ul * \ y \ is the displacement. \; \ x \ is the position along the string. \; \ t \ is time. \; \ c \ is the wave speed. The boundary conditions include the Dirichlet condition at the moving end: \ \ \[ y(0, t) = f(t) \] This ensures the wave's motion matches the prescribed end's motion. For the infinite end, assumptions might ensure no reflection occurs or a specific behavior is limitless. The wave equation helps to understand phenomena like vibrations of strings and the propagation of sound waves.

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

A sheet of material of thickness \(w\), specific heat capacity \(c\) and thermal conductivity \(k\) is isolated in a vacuum, but its two sides are exposed to fluxes of radiant heat of strengths \(J_{1}\) and \(J_{2}\). Ignoring short-term transients, show that the temperature difference between its two surfaces is steady at \(\left(J_{2}-J_{1}\right) w / 2 k\), whilst their average temperature increases at a rate \(\left(J_{2}+J_{1}\right) / c w\).

Solve the following partial differential equations for \(u(x, y)\) with the boundary conditions given: (a) \(x \frac{\partial u}{\partial x}+x y=u, \quad u=2 y\) on the line \(x=1\) (b) \(1+x \frac{\partial u}{\partial y}=x u, \quad u(x, 0)=x\).

Find partial differential equations satisfied by the following functions \(u(x, y)\) for all arbitrary functions \(f\) and all arbitrary constants \(a\) and \(b\) : (a) \(u(x, y)=f\left(x^{2}-y^{2}\right)\) (b) \(u(x, y)=(x-a)^{2}+(y-b)^{2}\); (c) \(u(x, y)=y^{n} f(y / x)\); (d) \(u(x, y)=f(x+a y)\).

The non-relativistic Schrödinger equation (18.7) is similar to the diffusion equation in having different orders of derivatives in its various terms; this precludes solutions that are arbitrary functions of particular linear combinations of variables. However, since exponential functions do not change their forms under differentiation, solutions in the form of exponential functions of combinations of the variables may still be possible. Consider the Schrödinger equation for the case of a constant potential, i.e. for a free particle, and show that it has solutions of the form \(A \exp (l x+m y+n z+\lambda t\) ) where the only requirement is that $$ -\frac{\hbar^{2}}{2 m}\left(l^{2}+m^{2}+n^{2}\right)=i \hbar \lambda $$ In particular, identify the equation and wavefunction obtained by taking \(\lambda\) as \(-i E / \hbar\), and \(l, m\) and \(n\) as \(i p_{x} / \hbar, i p_{y} / \hbar\) and \(i p_{z} / \hbar\) respectively, where \(E\) is the energy and \(p\) the momentum of the particle; these identifications are essentially the content of the de Broglie and Einstein relationships.

In those cases in which it is possible to do so, evaluate \(u(2,2)\), where \(u(x, y)\) is the solution of $$ 2 y \frac{\partial u}{\partial x}-x \frac{\partial u}{\partial y}=2 x y\left(2 y^{2}-x^{2}\right) $$ that satisfies the (separate) boundary conditions given below. (a) \(u(x, 1)=x^{2}\) for all \(x\). (b) \(u(x, 1)=x^{2}\) for \(x \geq 0\). (c) \(u(x, 1)=x^{2}\) for \(0 \leq x \leq 3\) (d) \(u(x, 0)=x\) for \(x \geq 0\) (e) \(u(x, 0)=x\) for all \(x\). (f) \(u(1, \sqrt{10})=5\) (g) \(u(\sqrt{10}, 1)=5\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

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.

Sign-up for free