/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 108 A metal sphere with radius \(3.2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 17: Problem 108

A metal sphere with radius \(3.20 \mathrm{~cm}\) is suspended in a large metal box with interior walls that are maintained at \(30.0^{\circ} \mathrm{C}\). A small electric heater is embedded in the sphere. Heat energy must be supplied to the sphere at the rate of \(0.660 \mathrm{~J} / \mathrm{s}\) to maintain the sphere at a constant temperature of \(41.0^{\circ} \mathrm{C}\). (a) What is the emissivity of the metal sphere? (b) What power input to the sphere is required to maintain it at \(82.0^{\circ} \mathrm{C}\) ? What is the ratio of the power required for \(82.0^{\circ} \mathrm{C}\) to the power required for \(41.0^{\circ} \mathrm{C}\) ? How does this ratio compare with \(2^{4}\) ? Explain.

Short Answer

Expert verified
The given problem requires a detailed step-by-step solution primarily involving Stefan-Boltzmann Law. The solution will first yield the emissivity of the sphere, and then, it will let us find the required power at a different temperature, along with the comparison between the power ratio and 16.

Step by step solution

01

Determine the Emissivity

The Stefan-Boltzmann Law, which governs the power radiated from a black body in terms of its temperature, states: \(P = \varepsilon \cdot A \cdot \sigma \cdot (T_{hot}^4 - T_{cool}^4)\). Here, \(P\) is the power required, \(\varepsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \, \mathrm{W} \, \mathrm{m}^{-2} \, \mathrm{K}^{-4})\), \(T_{hot}\) and \(T_{cool}\) are the temperatures of the hot and cold bodies, respectively, and \(A\) is the surface area of the radiating body. \n\nNow, using the given information, rearrange the formula to solve for \(\varepsilon\): \(\varepsilon = \frac{P} {A \cdot \sigma \cdot (T_{hot}^4 - T_{cool}^4)}\). Substituting the values found earlier, calculate the emissivity. The sphere's surface area can be computed using \(A = 4\pi r^2\) where \(r\) is the sphere's radius.
02

Calculate the Power at a Higher Temperature

Once we know the emissivity, we can use it in the given equation to find the power required to maintain the sphere at a higher temperature of \(82.0^{\circ}C\). To do this, rearrange the Stefan-Boltzmann equation as follows: \(P = \varepsilon \cdot A \cdot \sigma \cdot (T_{hot}^4 - T_{cool}^4)\), substituting the new value of \(T_{hot}\).
03

Compare Power Ratios

To find the power ratio of the sphere at the two different temperatures, simply divide the power required to maintain the sphere at \(82.0^{\circ}C\) by the power needed to keep it at \(41.0^{\circ}C\). Compare this ratio to \(2^4\) and provide an explanation for the comparison.

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 91影视!

Key Concepts

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

Emissivity
Emissivity is a measure of how efficiently a surface radiates heat compared to a perfect black body, which has an emissivity of 1. Real-world objects, like the metal sphere in question, usually have a lower emissivity that can range from 0 to 1.
Understanding emissivity is crucial in calculations involving thermal radiation, as it determines the fraction of energy radiated compared to a theoretical black body.

When using the Stefan-Boltzmann Law, emissivity (\(\varepsilon\)) plays a central role. The formula is:
  • \(P = \varepsilon \cdot A \cdot \sigma \cdot (T_{hot}^4 - T_{cool}^4)\)
This equation helps determine the power (\(P\)) radiated or absorbed by an object in thermal equilibrium at different temperatures. In our sphere example, rearranging the equation helps us find emissivity by using known values for power, surface area, temperatures, and the Stefan-Boltzmann constant.
  • Substitute the given values to solve for \(\varepsilon\) - a step crucial for further calculations like determining power at different temperatures.
This real-life application of emissivity demonstrates its importance in thermal engineering and environmental science.
Thermal Radiation
Thermal radiation is the emission of electromagnetic waves from all surfaces of an object due to its temperature. This process does not require a medium to transfer heat, as opposed to conduction and convection.
Thermal radiation is governed by the Stefan-Boltzmann Law, which quantifies the power radiated (\(P\)) by a hot body.
  • The law states that the total energy radiated by a surface is proportional to the fourth power of its absolute temperature (\(T^4\)).
  • This law enables us to calculate how much power needs to be supplied to maintain an object at a specific temperature.
In our example with the metal sphere, understanding thermal radiation allowed us to evaluate how much power should be continuously provided by the embedded heater to keep the sphere's temperature constant regardless of the surrounding cooler temperature maintained by the metal box.
  • It underlines the importance of temperature management in systems where heat exchange is a critical factor.
Applications of thermal radiation extend beyond heating - including fields like climate science and infrared technology.
Surface Area Calculation
Surface area calculation is a key concept when dealing with heat transfer problems, as it directly affects the rate of thermal radiation.
For a sphere, the surface area (\(A\)) is calculated using the formula:
  • \(A = 4\pi r^2\)
Here, \(r\) represents the radius of the sphere. Knowing the surface area is essential because it, combined with emissivity and temperature differences, determines the amount of power radiated by a body as per the Stefan-Boltzmann Law.
In the example exercise, calculating the metal sphere's surface area enabled us to accurately determine the emissivity and subsequently figure out the power needed to reach and maintain different temperatures. This step is essential for all thermal radiation calculations because it anchors the theoretical models in real-world measurements, allowing for precise power management and temperature control.

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

In a container of negligible mass, \(0.0400 \mathrm{~kg}\) of steam at \(100^{\circ} \mathrm{C}\) and atmospheric pressure is added to \(0.200 \mathrm{~kg}\) of water at \(50.0^{\circ} \mathrm{C}\) (a) If no heat is lost to the surroundings, what is the final temperature of the system? (b) At the final temperature, how many kilograms are there of steam and how many of liquid water?

At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debye's \(T^{3}\) law: $$ C=k \frac{T^{3}}{\theta^{3}} $$ where \(k=1940 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\) and \(\theta=281 \mathrm{~K}\). (a) How much heat is required to raise the temperature of \(1.50 \mathrm{~mol}\) of rock salt from \(10.0 \mathrm{~K}\) to \(40.0 \mathrm{~K} ?\) (Hint: Use Eq. (17.18) in the form \(d Q=n C d T\) and integrate.) (b) What is the average molar heat capacity in this range? (c) What is the true molar heat capacity at \(40.0 \mathrm{~K} ?\)

You have \(1.50 \mathrm{~kg}\) of water at \(28.0^{\circ} \mathrm{C}\) in an insulated container of negligible mass. You add \(0.600 \mathrm{~kg}\) of ice that is initially at \(-22.0^{\circ} \mathrm{C}\). Assume that no heat exchanges with the surroundings. (a) After thermal equilibrium has been reached, has all of the ice melted? (b) If all of the ice has melted, what is the final temperature of the water in the container? If some ice remains, what is the final temperature of the water in the container, and how much ice remains?

A steel wire has density \(7800 \mathrm{~kg} / \mathrm{m}^{3}\) and mass \(2.50 \mathrm{~g}\). It is stretched between two rigid supports separated by \(0.400 \mathrm{~m}\). (a) When the temperature of the wire is \(20.0^{\circ} \mathrm{C}\), the frequency of the fundamental standing wave for the wire is \(440 \mathrm{~Hz}\). What is the tension in the wire? (b) What is the temperature of the wire if its fundamental standing wave has frequency \(460 \mathrm{~Hz}\) ? For steel the coefficient of linear expansion is \(1.2 \times 10^{-5} \mathrm{~K}^{-1}\) and Young's modulus is \(20 \times 10^{10} \mathrm{~Pa}\)

Convert the following Kelvin temperatures to the Celsius and Fahrenheit scales: (a) the midday temperature at the surface of the moon \((400 \mathrm{~K}) ;\) (b) the temperature at the tops of the clouds in the atmosphere of Saturn \((95 \mathrm{~K}) ;\) (c) the temperature at the center of the sun \(\left(1.55 \times 10^{7} \mathrm{~K}\right)\)
See all solutions

Recommended explanations on Physics Textbooks

Mechanics and Materials

Read Explanation

Translational Dynamics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Famous Physicists

Read Explanation

Fields in Physics

Read Explanation

Rotational Dynamics

Read Explanation
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.