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Chapter 3: Problem 67

The Stefan-Boltzmann law states that the radiant energy emitted from a unit area of a black surface is given by \(R=k T^{4},\) where \(R\) is the rate of emission per unit area, \(T\) is the temperature (in \(\mathrm{K}\) ), and \(k\) is a constant. If the error in the measurement of \(T\) is \(0.5 \%\), find the resulting percentage error in the calculated value of \(R\).

Short Answer

Expert verified
The percentage error in the calculated value of \(R\) is 2%.

Step by step solution

01

Understanding the Formula

The Stefan-Boltzmann law is given by the formula \(R = k T^4\). Here, \(R\) is the rate of emission per unit area, \(T\) is the temperature, and \(k\) is a constant. Our goal is to find the percentage error in \(R\) when there is a 0.5% error in \(T\).
02

Relate Error in T to Error in R

We know that the percentage error in a power, \(T^n\), can be calculated as \(n\) times the percentage error in \(T\), for a small error. Here, \(n=4\) because \(R = k T^4\). So, the percentage error in \(R\) is \(4\) times the percentage error in \(T\).
03

Calculate the Error in R

Given the percentage error in \(T\) is 0.5%, and using the relationship derived, the percentage error in \(R\) will be \(4 \times 0.5\%\).
04

Compute the Result

Calculate \(4 \times 0.5 = 2\). Thus, the percentage error in \(R\) is 2%.

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Key Concepts

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

Calculus
Calculus is a branch of mathematics that helps us understand changes and motion. In the context of the Stefan-Boltzmann law, calculus allows us to analyze how a small change in temperature (T) affects the radiant energy (R). When using calculus, we often deal with derivatives to find how one variable impacts another.

For the power function T^4, the derivative with respect to T is crucial in error analysis. The derivative \(\frac{d(T^4)}{dT} = 4T^3\) shows us how sensitive R is to changes in T. This is why a small error in measuring T translates into a larger percentage error in R.
Error Analysis
Error analysis is the process used to determine the uncertainty in a measurement or result. When calculating radiant energy using the Stefan-Boltzmann law, it is important to analyze how errors in temperature measurement can affect the overall calculation.

Suppose we measure temperature with a 0.5% error. The derivative with respect to temperature suggests that any error in T is multiplied by the power (in this case, 4). This principle gives us the error propagation rule: for R = kT^4 , the percentage error in R is 4 times the percentage error in T .
  • Understanding potential errors helps us refine measurements.
  • Applying known rules of calculus to quantify how errors propagate through equations.
Radiant Energy
Radiant energy is the energy emitted by a surface in the form of electromagnetic waves. This energy can be conceptualized as light or heat, and it plays a crucial role in numerous scientific applications, from physics to environmental science.

The Stefan-Boltzmann law describes the total radiant energy ( R ) emitted per unit area by a blackbody. This law offers valuable insight into how temperature affects the energy radiated by objects. A key takeaway is that as the object's temperature increases, the radiant energy released escalates dramatically due to its proportional relationship to the fourth power of temperature.
  • Applications include understanding stellar and atmospheric phenomena.
  • Essential for calculating energy budgets in climate science.
Temperature Measurement
Accurate temperature measurement is vital when working with the Stefan-Boltzmann law. Since the radiant energy is highly sensitive to temperature changes, even a small error in measuring temperature can significantly alter the computed value of R .

In practice, errors in temperature measurement typically arise from instrument imperfections or environmental conditions.
  • Precise instruments such as thermocouples and resistance thermometers can minimize these errors.
  • Regular calibration of measuring devices is necessary to improve accuracy.
  • Understanding the relationship between temperature and radiant energy helps emphasize the importance of accurate temperature measurements.

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Most popular questions from this chapter

Find the first and second derivatives. $$ k(r)=(4 r+7)^{5} $$

Show that the equation \(y^{2}=x\) determines an infinite number of implicit functions.

Let \(h=f \circ g\) be a differentiable function. The following tables list some values of \(f\) and \(g\). Use Exercise 51 of Section 3.2 to approximate \(h^{\prime}(-2)\) $$ \begin{array}{|l|rrr} \hline x & -8.48092 & -8.46000 & -8.43908 \\ \hline f(x) & -2.03930 & -2.03762 & -2.03594 \\ \hline \end{array} $$ $$ \begin{array}{|l|rrr|} \hline x & -2.00400 & -2.00000 & -1.99600 \\ \hline g(x) & -8.48092 & -8.46000 & -8.43908 \\ \hline \end{array} $$

If \(k(x)=f(g(x))\) and if \(f(2)=-4, g(2)=2, f^{\prime}(2)=3\), and \(g^{\prime}(2)=5\), find \(k(2)\) and \(k^{\prime}(2)\).

If \(x^{2}+4 x y-y^{2}=8\), find \(y^{\prime \prime}\) by implicit differentiation.
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