Ch 37: The Foundations of Modern Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 37: The Foundations of Modern Physics

Problema 7

Capitolo 37, Problema 7

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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Step 1: Identify the relevant law for determining the peak wavelength of the emission spectrum. This is Wien's displacement law, which states that the peak wavelength (λ_peak) is inversely proportional to the temperature (T) of the radiating body. The formula is:

λ

_peak=bT, where b is Wien's constant (approximately 2.897 × 10⁻³ m·K).

Step 2: Use the Stefan-Boltzmann law to calculate the temperature of the ceramic cube. The Stefan-Boltzmann law is given by:

P = e σ A T^{4}

, where P is the power radiated (630 W), e is the emissivity (1 in this case), σ is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²·K⁴), A is the surface area of the cube, and T is the temperature.

Step 3: Calculate the surface area of the cube. Since the cube has sides of 3.0 cm, convert this to meters (0.03 m) and calculate the total surface area using the formula for the surface area of a cube:

A = 6 s^{2}

, where s is the side length.

Step 4: Rearrange the Stefan-Boltzmann law to solve for T. Substitute the values for P, e, σ, and A into the equation and isolate T:

T = \sqrt{\sqrt{\frac{P}{e σ A}}}

. Perform the necessary algebraic manipulations to express T in terms of the given quantities.

Step 5: Once T is determined, use Wien's displacement law to calculate the peak wavelength. Substitute the value of T into the formula

λ

_peak=bT. Convert the result from meters to micrometers (μm) by multiplying by 10⁶.

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Blackbody Radiation

Blackbody radiation refers to the electromagnetic radiation emitted by an idealized object that absorbs all incident radiation, known as a blackbody. The intensity and wavelength of this radiation depend solely on the object's temperature, described by Planck's law. This concept is crucial for understanding how objects emit thermal radiation and is foundational for deriving the peak wavelength of emission.

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Radiation Pressure

Wien's Displacement Law

Wien's Displacement Law states that the wavelength at which the emission of a blackbody spectrum peaks is inversely proportional to its absolute temperature. Mathematically, it is expressed as λ_max = b/T, where b is Wien's displacement constant (approximately 2898 μm·K). This law allows us to calculate the peak wavelength of radiation emitted by an object based on its temperature.

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Displacement vs. Distance

Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of its absolute temperature. This law is expressed as P = σAT^4, where P is the power radiated, σ is the Stefan-Boltzmann constant, A is the surface area, and T is the temperature in Kelvin. Understanding this law helps in determining the temperature of the ceramic cube based on its power output.

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Gauss' Law

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