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Consider a uniform rod of material whose temperature

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder ISBN: 9780201380279 40

Solution for problem 62P Chapter 1

An Introduction to Thermal Physics | 1st Edition

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An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

An Introduction to Thermal Physics | 1st Edition

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Problem 62P

Consider a uniform rod of material whose temperature varies only along its length, in the x direction. By considering the heat flowing from both directions into a small segment of length Δx, derive the heat equation,

where K= κt/cρ, c is the specific heat of the material, and ρ is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that K is independent of temperature, show that a solution of the heat equation is

where T0 is a constant background temperature and A is any constant. Sketch (or use a computer to plot) this solution as a function of x, for several values of t. Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.

Step-by-Step Solution:

Solution 62P

To solve this question, we shall have to consider the Fourier heat conduction law. The mathematical equation for Fourier heat conduction law is written as …..(1)

Here,  amount of heat

time taken for the heat to flow

thermal conductivity

 area of cross-section

 change in temperature

 change in length

Step 1</p>

We are given in the question that heat flows from both directions of the rod.

Using equation (1),

For heat coming from one direction,

…..(2)

For heat conduction from the other direction.

…..(3)

Therefore, the net heat flow can be subtracting equation (3) from equation (2),

 …..(2), where …..(4)

Step 2 of 2

Chapter 1, Problem 62P is Solved
Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. This full solution covers the following key subjects: heat, temperature, Energy, solution, rod. This expansive textbook survival guide covers 10 chapters, and 454 solutions. Since the solution to 62P from 1 chapter was answered, more than 262 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1. The full step-by-step solution to problem: 62P from chapter: 1 was answered by , our top Physics solution expert on 07/05/17, 04:29AM. The answer to “Consider a uniform rod of material whose temperature varies only along its length, in the x direction. By considering the heat flowing from both directions into a small segment of length ?x, derive the heat equation, where K= ?t/c?, c is the specific heat of the material, and ? is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that K is independent of temperature, show that a solution of the heat equation is where T0 is a constant background temperature and A is any constant. Sketch (or use a computer to plot) this solution as a function of x, for several values of t. Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.” is broken down into a number of easy to follow steps, and 138 words.

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