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Chapter 7: Problem 54

Stefan's Law Stefan's Law states that the rate of change of the temperature \(T\) of a body is directly proportional to the difference of the fourth power of \(T\) and the fourth power of the temperature of the surrounding medium \(T_{m}\). Thus, $$ \frac{d T}{d t}=k\left(T^{4}-T_{m}^{4}\right) $$ where \(k\) is the constant of proportionality. Stefan's Law holds over a greater temperature range than does Newton's Law of Cooling. Show that \(T\) is given by the implicit equation \(\ln \left(\frac{T+T_{m}}{T-T_{m}}\right)+2 \tan ^{-1}\left(\frac{T}{T_{m}}\right)=-4 T_{m}^{3} k t+C \quad T>T_{m}\)

Short Answer

Expert verified
In order to show that \(T\) is given by the implicit equation \(\ln\left(\frac{T + T_m}{T - T_m}\right) + 2 \tan^{-1}\left(\frac{T}{T_m}\right) = -4T_m^3 kt + C, \quad T > T_m\), we first separated variables to obtain \(\frac{dT}{T^4 - T_m^4} = k dt\). Then, we integrated both sides and used substitution and partial fraction decomposition to evaluate the integral on the left side. Finally, we simplified the result to derive the desired implicit equation.

Step by step solution

01

Separate Variables

To separate variables, we need to rewrite the equation in the form \( \frac{dT}{T^4 - T_m^4} = k dt \) Now, let's create the fractions on both sides. \( \frac{dT}{T^4 - T_m^4} = k dt \) Step 2: Integrate both sides of the equation
02

Integrate both sides of the equation

Now we need to integrate both sides of the equation. So, we have the following equation: \( \int \frac{dT}{T^4 - T_m^4} = \int k dt \) Step 3: Perform integration
03

Perform integration

To integrate the left side of the equation, we can use the substitution: \(u = T^2 + T_m^2\), then, \(du = 2Td T\) So, the left side becomes: \( \int \frac{1}{2T(T^4 - T_m^4)} du = \int \frac{1}{2(u^2 - 2T^2_m u + T_m^4)} du \) Now we can use partial fraction decomposition: \( \frac{1}{2(u^2 - 2T^2_m u + T_m^4)} = \frac{A}{T_m - u} + \frac{B}{T_m + u} \) Solving for \(A\) and \(B\), we obtain: \(A = \frac{1}{2T_m}\) \(B = -\frac{1}{2T_m}\) Now rewrite the left side integral: \( \int \left( \frac{1}{2T_m} \frac{1}{T_m - u} - \frac{1}{2T_m} \frac{1}{T_m + u} \right) du \) Now, it becomes easier to integrate. \( \int \frac{1}{2T_m} \left(\frac{1}{T_m - u} - \frac{1}{T_m + u} \right) du = \int k dt\) \( \frac{1}{2T_m} \int \left(\frac{1}{T_m - u} - \frac{1}{T_m + u} \right) du = \int k dt\) When we integrate the left side, we get: \( \frac{1}{2T_m} \left[-\ln(\frac{T_m - u}{T_m + u})\right] = kt + C_1 \) Now, substitute back in for T: \( -\frac{1}{2T_m} \left[\ln(\frac{T_m - T^2 - T_m^2}{T_m + T^2 + T_m^2})\right] = kt + C_1 \) Step 4: Simplify the equation
04

Simplify the equation

Now we need to simplify the equation to obtain the given implicit equation. So, we can rewrite the left side of the equation as: \( \ln\left(\frac{T + T_m}{T - T_m}\right) + 2 \tan^{-1}\left(\frac{T}{T_m}\right) = -4T_m^3 kt + C_2 \) where \(C_2 = -2T_m C_1\). Thus, we have shown that \(T\) is given by the implicit equation: \( \ln\left(\frac{T + T_m}{T - T_m}\right) + 2 \tan^{-1}\left(\frac{T}{T_m}\right) = -4T_m^3 kt + C, \quad T > T_m \)

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Key Concepts

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

Rate of Change
The **rate of change** in calculus describes how one quantity changes in relation to another. In the context of Stefan's Law, it pertains to how the temperature of a body changes over time.
Stefan's Law expresses this rate of change as \( \frac{dT}{dt} = k(T^4 - T_m^4) \), showing that the change in temperature \( T \) depends on the difference between \( T^4 \) and the fourth power of the surrounding temperature, \( T_m^4 \).
This means the greater the difference in temperature, the faster it changes, influenced by the constant \( k \). This helps understand how quickly a body reaches equilibrium with its surroundings.
Temperature
**Temperature** is a fundamental concept that measures the degree of heat present in a system. In Stefan's Law, it's key to understanding how energy radiates.
Here, we see temperature as both \( T \) (the body's temperature) and \( T_m \) (the surrounding medium's temperature). The law calculates the difference between these temperatures raised to the fourth power.
This fourth power relationship highlights that small changes in temperature can lead to significant changes in the rate at which a body heats or cools. This quadratic relationship amplifies differences, showcasing the influential role of temperature in heat transfer.
Integration
**Integration** is a powerful mathematical tool used to find quantities that result from accumulating small changes. In Stefan's Law, integration helps us solve the differential equation related to temperature changes.
We start with \( \int \frac{dT}{T^4 - T_m^4} = \int k \, dt \), which is the process of finding an antiderivative or integral of both sides. This process requires rearranging terms and using techniques like substitution and partial fraction decomposition.
Integration is crucial here as it moves us from a differential rate of change (a derivative) to an equation that reflects cumulative changes over time. This transition is key in finding a function that represents temperature as time progresses.
Differential Equations
**Differential equations** involve equations that include derivatives of functions. They are essential in expressing the dynamic changes Stefan's Law describes.
In this exercise, the differential equation is \( \frac{dT}{d t}=k(T^4-T_m^4) \), representing a relationship where temperature change over time is linked to initial conditions.
Solving this involves separating variables and integrating, eventually transforming the equation into a form that displays how temperature evolves with time.
Differential equations like this are vital in modeling real-world phenomena where rates of change are given, helping predict future behavior based on current data.

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