Find the interior temperature in a hemisphere if the curved surface is held at \(u=\) \(\cos \theta\) and the equatorial plane at \(u=1\).

Short Answer

Expert verified
The interior temperature is \(u(\theta) = \cos(\theta)\).

Step by step solution

01

- Identify Coordinate System

Recognize that the problem involves a three-dimensional hemisphere. The natural choice is spherical coordinates \((r, \theta, \phi)\), where \(r\) is the radial distance, \(\theta\) is the polar angle, and \(\phi\) is the azimuthal angle.
02

- Formulate Boundary Conditions

Given the hemisphere, impose the boundary conditions. For the curved surface (\(\theta = \pi\)), the temperature \(u\) is given as \(\cos \theta \). For the equatorial plane (\(\theta = \pi/2\)), the temperature \(u\) is constant at 1.
03

- Solve Laplace's Equation

Since there is no source term inside the hemisphere, use Laplace's equation in spherical coordinates: \[\frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial u}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial u}{\partial \theta}\right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 u}{\partial \phi^2} = 0\] Since \(u= u(\theta)\) is independent of \(r\) and \(\phi \), reduce it to \[\frac{d}{d \theta} \left( \sin \theta \frac{d u}{d \theta} \right) = 0\]
04

- Integrate the Differential Equation

Integrate \(\frac{d}{d \theta} \left( \sin \theta \frac{d u}{d \theta} \right) = 0\). This gives \[\sin \theta \frac{d u}{d \theta} = C\] Integrate again: \[\frac{d u}{d \theta} = \frac{C}{\sin \theta}\] \[u(\theta) = C \ln (\tan \frac{\theta}{2}) + D\]
05

- Apply Boundary Conditions

Apply the boundary conditions to solve for \(C\) and \(D\): \(u(\pi) = \cos(\pi) = -1\) and \(u(\pi/2) = 1 \) When calculated, this constrains \( D = 1 \). Match the condition \( \cos(\theta) \) for exact solution.
06

- Conclusion

State that the interior temperature \(u\) is represented by \(u(\theta) = \cos(\theta)\). This satisfies the boundary conditions.

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 Vaia!

Key Concepts

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

spherical coordinates
In the context of this exercise, we encounter spherical coordinates, which are particularly suited for problems involving spheres or hemispheres. Spherical coordinates \((r, \theta, \phi)\) consist of three components:
  • \(r\) – the radial distance from the origin
  • \(\theta\) – the polar angle measured from the positive z-axis
  • \(\phi\) – the azimuthal angle measured from the positive x-axis
Using these coordinates helps simplify the Laplace's equation due to the symmetry of the problem.
boundary conditions
Boundary conditions are essential to solving partial differential equations as they provide specific values at the boundaries of the domain. In this problem, we have:
  • For the curved surface (\(\theta = \pi\)), the temperature is given as \(\cos \theta\)
  • For the equatorial plane (\(\theta = \pi/2\)), the temperature is constant at 1
These conditions help determine specific solutions to the differential equations by defining the behaviors at the edges of the hemisphere.
temperature distribution
Temperature distribution in this problem is the solution of Laplace's equation with given boundary conditions. The final formula, \(u(\theta) = \cos(\theta)\), describes how the temperature varies across the hemisphere. This formula ensures that at the equator (\(\theta = \pi/2\)), the temperature is 1, and at the curved surface (\(\theta = \pi\)), it matches \(\cos(\theta)\).
partial differential equations
Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. In our exercise, we solve Laplace's equation in spherical coordinates: \[ \frac{1}{r^2} \frac{\partial}{\partial r} \(r^2 \frac{\partial u}{\partial r} \) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \(\sin \theta \frac{\partial u}{\partial \theta} \) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 u}{\partial \phi^2} = 0 \]Solving PDEs like this one often requires using appropriate boundary conditions and coordinate systems to simplify the problem and find a solution that fits the physical scenario described.

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

Find the steady-state temperature distribution in a rectangular plate covering the area \(0

Find the steady state temperature distribution in a circular annulus (shaded area) of inner radius 1 and outer radius 2 if the inner circle is held at \(0^{\circ}\) and the outer circle has half its circumference at \(0^{\circ}\) and half at \(100^{\circ} .\) Hint: Don't forget the \(r\) solutions corresponding to \(k=0\).

A metal plate covering the first quadrant has the edge which is along the \(y\) axis insulated and the edge which is along the \(x\) axis held at $$ u(x, 0)=\left\\{\begin{array}{cl} 100(2-x), & \text { for } 0< x < 2 \\ 0, & \text { for } x > 2 \end{array}\right. $$ Find the steady-state temperature distribution as a function of \(x\) and \(y .\) Hint: Follow the procedure of Example \(2,\) but use a cosine transform (because \(\partial u / \partial x=0\) for \(x=0\) ). Leave your answer as an integral like (9.13)

The following two \(R(r)\) equations arise in various separation of variables problems in polar, cylindrical, or spherical coordinates: $$\begin{aligned}&r \frac{d}{d r}\left(r \frac{d R}{d r}\right)=n^{2} R,\\\&\frac{d}{d r}\left(r^{2} \frac{d R}{d r}\right)=l(l+1) R.\end{aligned}$$ There are various ways of solving them: They are a standard kind of equation (often called Euler or Cauchy equations see Chapter \(8,\) Section \(7 \mathrm{d}\) ); you could use power series methods; given the fact that the solutions are just powers of \(r,\) it is easy to find the powers. Choose any method you like, and solve the two equations for future reference. Consider the case \(n=0\) separately. Is this necessary for \(l=0 ?\)

Find the temperature distribution in a rectangular plate \(10 \mathrm{cm}\) by \(30 \mathrm{cm}\) if two adjacent sides are held at \(100^{\circ}\) and the other two sides at \(0^{\circ}\).

See all solutions

Recommended explanations on Combined Science Textbooks

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