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Chapter 13: Problem 9

Stefan's law of radiation gives the rate of change of temperature, \(\frac{\mathrm{d} \theta}{\mathrm{d} t}\), of a body by $$ \frac{\mathrm{d} \theta}{\mathrm{d} t}=k\left(\theta^{4}-T^{4}\right) $$ where \(k\) is a constant, \(\theta(=\theta(t))\) is the temperature of the body at time \(t\) and \(T\) is the constant ambient temperature. Show, by solving the differential equation, that $$ \frac{1}{4 T^{3}}\left[\ln \left(\frac{\theta-T}{\theta+T}\right)-2 \tan ^{-1}\left(\frac{\theta}{T}\right)\right]=k t+C $$ where \(C\) is the constant of integration.

Short Answer

Expert verified
By integrating and substituting variables, it is shown that \(\frac{1}{4 T^{3}}\left[\ln \left(\frac{\theta-T}{\theta+T}\right)-2 \tan^{-1}\left(\frac{\theta}{T}\right)\right]=k t+C\).

Step by step solution

01

Rewrite the given differential equation

The differential equation to solve is \(\frac{\mathrm{d} \theta}{\mathrm{d} t}=k\left(\theta^{4}-T^{4}\right)\).
02

Simplify notation by substituting variables

Let \(u = \frac{\theta}{T}\). Then \(\theta = uT\). Rewriting, \(\theta^4 - T^4 = (uT)^4 - T^4 = T^4(u^4 - 1)\).
03

Rewrite the differential equation using the new variable

With the substitution, the differential equation becomes \(\frac{\mathrm{d} (u T)}{\mathrm{d} t} = k T^4 (u^4 - 1)\). Since \(T\) is a constant, \(\frac{\mathrm{d} u}{\mathrm{d} t}\), the equation simplifies to \(T \frac{\mathrm{d} u}{\mathrm{d} t} = k T^4 (u^4 - 1)\). Dividing both sides by \(T\) gives \(\frac{\mathrm{d} u}{\mathrm{d} t} = k T^3 (u^4 - 1)\).
04

Separate variables and integrate

Rearrange the equation to: \(\frac{\mathrm{d} u}{u^4 - 1} = k T^3 \mathrm{d} t\). Integrate both sides: \(\int \frac{\mathrm{d} u}{u^4 - 1} = \int k T^3 \mathrm{d} t\).
05

Apply partial fraction decomposition and integrate

The partial fraction decomposition of \(\frac{1}{u^4 - 1}\) gives \( \frac{1}{(u-1)(u+1)(u^2+1)} = \frac{A}{u-1} + \frac{B}{u+1} + \frac{Cu + D}{u^2 + 1} \). Solving for coefficients, we get \[ A = \frac{1}{4}, B = \frac{1}{4}, C = -\frac{1}{2}, D = 0 \]. Thus, \(\frac{1}{u^4-1} = \frac{1}{4} \left( \frac{1}{u-1} + \frac{1}{u+1} - \frac{2u}{u^2+1} \right)\).
06

Integrate each component

The integral becomes \(\int \frac{\mathrm{d} u}{u^4 - 1} = \frac{1}{4} \left( \int \frac{\mathrm{d} u}{u-1} + \int \frac{\mathrm{d} u}{u+1} - 2 \int \frac{u \mathrm{d} u}{u^2 + 1} \right)\). Integrating yields: \(\frac{1}{4} \left( \ln | u - 1 | + \ln | u + 1 | - 2 \tan^{-1}(u) \right) + C\).
07

Substitute back for \(u\) and simplify

Resubstitute \(u = \frac{\theta}{T}\): \(\frac{1}{4} \left( \ln | \frac{\theta}{T} - 1 | + \ln | \frac{\theta}{T} + 1 | - 2 \tan^{-1} \left( \frac{\theta}{T} \right) \right) + C\). Which simplifies to: \[ \frac{1}{4} \left( \ln \left( \frac{\theta - T}{\theta + T} \right) - 2 \tan^{-1} \left( \frac{\theta}{T} \right) \right) = kT^3 t + C\]. Lastly multiply both sides by \( \frac{1}{T^3} \) to yield the final form.

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Key Concepts

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

Differential Equations
A differential equation is a mathematical equation that involves functions and their derivatives. It shows the relationship between a function and its rate of change. These equations are essential in describing various physical phenomena, such as motion, heat, and other natural processes.
In this exercise, we are dealing with a first-order differential equation provided by Stefan's law of radiation: \(\frac{\mathrm{d} \theta}{\mathrm{d} t}=k\left(\theta^{4}-T^{4}\right)\). Here, \(\theta\) is the temperature of the body at time \(t\), \(T\) is the ambient temperature, and \(k\) is a constant.
Stefan's Law of Radiation
Stefan's law of radiation, also known as Stefan-Boltzmann law, relates the power radiated by a black body to its temperature. In our context, it states how the temperature of a body changes over time due to radiative heat loss.
The given differential equation is: \(\frac{\mathrm{d} \theta}{\mathrm{d} t}=k\left(\theta^{4}-T^{4}\right)\). This means the rate of change of the temperature (\

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