Chapter 5: Q31P (page 245)

Derive the Stefan-Boltzmann formula for the total energy density in blackbody radiation
$\frac{E}{V} = (\frac{π^{2} k_{B}^{4}}{15 ħ^{3} C^{3}}) T^{4} = (7.57 \times 10^{- 16} {Jm}^{- 3} K^{- 4}) T^{4}$

Short Answer

Expert verified

Deriving the Stefan-Boltzmann formula for the total energy density in the blackbody radiation is $= [\frac{π^{2} {(1.3807 \times 10^{23} J / K)}^{4}}{15 {(2.998 \times 10^{8} m / S)}^{3} (1.5046 \times 10^{- 34} J . s)}] T^{4} 7.566 \times 10^{- 16} \frac{J}{m^{3} K^{4}}$

Step by step solution

Definition of Stefan Boltzmann law

According to the Stefan-Boltzmann equation, the amount of radiation emitted by a dark substance per unit area is exactly proportional to the fourth power of the temperature.

Deriving the Stefan-Boltzmann formula for the total energy

From Equation 5.113:

$\frac{E}{V} = \int_{0}^{\infty} ρ (ω) d ω = \frac{h}{π^{2} c^{3}} \int_{0}^{\infty} \frac{ω^{3}}{(e^{h ω / k_{B} T} - 1)} d ω .$

Let $x = \frac{h ω}{k_{B} T} .$ then

$\begin{array}{rcl} \frac{E}{V} & = & \frac{h}{π^{2} c^{3}} {(\frac{k_{B} T}{h})}^{4} \int_{0}^{\infty} \frac{x^{3}}{e^{x} - 1} d x = \frac{{(k_{B} T)}^{4}}{π^{2} c^{3} h^{3}} Γ (4) ς (4) \\ = & \frac{{(k_{B} T)}^{4}}{π^{2} c^{3} h^{3}} . 6 . \frac{π^{4}}{90} \\ = & (\frac{π^{2} {k_{B}}^{4}}{15 c^{3} h^{3}}) T^{4} \end{array}$

$= [\frac{π^{2} {(1.3807 \times 10^{23} J / K)}^{4}}{15 {(2.998 \times 10^{8} m / s)}^{3} {(1.5046 \times 10^{- 34} J . s)}^{3}}] T^{4} \ hfill = [7.566 \times 10^{- 16} \frac{J}{m^{3} K^{4}}] T^{4}$

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Derive the Stefan-Boltzmann formula for the total energy density in blackbody radiation
$\frac{E}{V} = (\frac{π^{2} k_{B}^{4}}{15 ħ^{3} C^{3}}) T^{4} = (7.57 \times 10^{- 16} {Jm}^{- 3} K^{- 4}) T^{4}$

Short Answer

Step by step solution

Definition of Stefan Boltzmann law

Deriving the Stefan-Boltzmann formula for the total energy

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Derive the Stefan-Boltzmann formula for the total energy density in blackbody radiationEV=(π2kB415ħ3C3)T4=7.57×10-16Jm-3K-4T4

Short Answer

Step by step solution

Definition of Stefan Boltzmann law

Deriving the Stefan-Boltzmann formula for the total energy

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Electricity

Quantum Physics

Medical Physics

Modern Physics

Electrostatics

Study anywhere. Anytime. Across all devices.

Derive the Stefan-Boltzmann formula for the total energy density in blackbody radiation
$\frac{E}{V} = (\frac{π^{2} k_{B}^{4}}{15 ħ^{3} C^{3}}) T^{4} = (7.57 \times 10^{- 16} {Jm}^{- 3} K^{- 4}) T^{4}$