Ch 37: The Foundations of Modern Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 37: The Foundations of Modern Physics

Problem 6

Chapter 37, Problem 6

A 2.0-cm-diameter metal sphere is glowing red, but a spectrum shows that its emission spectrum peaks at an infrared wavelength of 2.0 μm. How much power does the sphere radiate? Assume e=1 .

Verified step by step guidance

Step 1: Identify the relevant formula for power radiated by a blackbody. The power radiated by a blackbody is given by the Stefan-Boltzmann law:

P = eσAT⁴

, where

P

is the power,

e

is the emissivity (given as 1),

σ

is the Stefan-Boltzmann constant (

5.67 × 10⁻⁸ \, \(\text{W/m²K⁴}\)

A

is the surface area of the sphere, and

T

is the temperature in kelvins.

Step 2: Use Wien's displacement law to find the temperature of the sphere. Wien's law is given by

λ_{\(\text{max}\)}T = b

, where

λ_{\(\text{max}\)}

is the peak wavelength (2.0 μm =

2.0 × 10⁻⁶

m),

T

is the temperature, and

b

is Wien's constant (

2.898 × 10⁻³ \, \(\text{m·K}\)

). Solve for

T

T = \(\frac{b}{λ_{\text{max}\)}}

Step 3: Calculate the surface area of the sphere. The surface area of a sphere is given by

A = 4πr²

, where

r

is the radius of the sphere. The diameter is given as 2.0 cm, so the radius is

r = 1.0 \, \(\text{cm}\) = 0.01 \, \(\text{m}\)

. Substitute this value into the formula for

A

Step 4: Substitute the values of

e

σ

A

, and

T

into the Stefan-Boltzmann law. Use the temperature calculated from Wien's law and the surface area calculated in the previous step. The formula becomes

P = (1)(5.67 × 10⁻⁸)(A)(T⁴)

Step 5: Perform the calculations step by step. First, calculate

T

using Wien's law. Then calculate

A

using the surface area formula. Finally, substitute these values into the Stefan-Boltzmann law to find the power

P

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Blackbody Radiation

Blackbody radiation refers to the electromagnetic radiation emitted by an idealized object that absorbs all incident radiation, known as a blackbody. The spectrum of this radiation depends solely on the object's temperature, described by Planck's law. The peak wavelength of emission can be determined using Wien's displacement law, which relates temperature to the wavelength at which the emission is maximized.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the total power radiated per unit area of a blackbody is proportional to the fourth power of its absolute temperature. Mathematically, it is expressed as P = eσAT^4, where P is the power, e is the emissivity, σ is the Stefan-Boltzmann constant, A is the surface area, and T is the temperature in Kelvin. This law is crucial for calculating the total power radiated by the sphere.

Emissivity

Emissivity is a measure of an object's ability to emit thermal radiation compared to a perfect blackbody, with values ranging from 0 to 1. An emissivity of 1 indicates that the object is a perfect blackbody, while lower values indicate less efficient radiation. In this problem, the sphere is assumed to have an emissivity of 1, simplifying the calculations for power radiated.

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Related Practice

Textbook Question

Figure 37.7 identified the wavelengths of four lines in the Balmer series of hydrogen. Predict the wavelength of the fifth line in the spectrum.

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Textbook Question

An electron in a cathode-ray beam passes between 2.5-cm-long parallel-plate electrodes that are 5.0 mm apart. A 2.0 mT, 2.5-cm-wide magnetic field is perpendicular to the electric field between the plates. The electron passes through the electrodes without being deflected if the potential difference between the plates is 600 V. What is the electron's speed?

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Textbook Question

A ceramic cube 3.0 cm on each side radiates heat at 630 W. At what wavelength, in μm, does its emission spectrum peak? Assume e=1.

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Textbook Question

Electrons pass through the parallel electrodes shown in FIGURE EX37.9 with a speed of 5.0×10⁶ m/s. What magnetic field strength and direction will allow the electrons to pass through without being deflected? Assume that the magnetic field is confined to the region between the electrodes.

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