Chapter 9, Problem 9.30E

Interpretation Introduction

(a)

Interpretation:

The value and units of the slope of the energy versus wavelength for given Planck’s law equation at given temperatures and wavelengths is to be calculated.

Concept introduction:

Planck’s equation can also be represented in the form of energy density distribution of a black body radiation at a given temperature and wavelength. This equation is known as Planck’s radiation distribution law.

Answer to Problem 9.30E

The value and units of the slope of the energy versus wavelength at given temperature and wavelength is $4.9\times {10}^{-5}$ and ${\text{Jm}}^{-4}$ respectively.

Explanation of Solution

It is given that temperature and wavelength is $1000\text{K}$ and $500\text{nm}$ respectively.

To calculate slope of the energy versus wavelength, the Planck’s law equation used is,

$\frac{d\rho }{d\lambda }=\frac{8\pi hc}{{\lambda }^5}\left(\frac{1}{e^{hc/\lambda kT}-1}\right)$

Where,

• $\rho$ is the energy density.

• $\lambda$ is the wavelength.

• $h$ is the Planck’s constant.

• $k$ is the Boltzmann constant.

• $c$ is the speed of light.

Substitute the values of constants, temperature and wavelength in the given equation.

$\begin{array}{rll}\frac{d\rho }{d\lambda }& =& \frac{8\left(3.14\right)\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)\left(3\times {10}^8{\text{ms}}^{-1}\right)}{{\left(500\times {\text{10}}^{-9}\text{m}\right)}^5}\left(\frac{1}{e^{\frac{6.626\times {10}^{-34}\text{J}\cdot \text{s}\times 3\times {10}^8{\text{ms}}^{-1}}{500\times {10}^{-9}\text{m}\times 1.38\times {10}^{-23}{\text{JK}}^{-\text{1}}\times 1000\text{K}}}-1}\right)\\ & =& 4.9\times {10}^{-5}{\text{Jm}}^{-4}\end{array}$

Thus, the slope of the energy versus wavelength is $4.9\times {10}^{-5}{\text{Jm}}^{-4}$.

Conclusion

The value and units of the slope of the energy versus wavelength at given temperature and wavelength is $4.9\times {10}^{-5}$ and ${\text{Jm}}^{-4}$ respectively.

Interpretation Introduction

(b)

Interpretation:

The value and units of the slope of the energy versus wavelength for given Planck’s law equation at given temperatures and wavelengths is to be calculated.

Concept introduction:

Planck’s equation can also be represented in the form of energy density distribution of a black body radiation at a given temperature and wavelength. This equation is known as Planck’s radiation distribution law.

Answer to Problem 9.30E

The value and units of the slope of the energy versus wavelength at given temperature and wavelength is $89.3$ and ${\text{Jm}}^{-4}$ respectively.

Explanation of Solution

It is given that temperature and wavelength is $2000\text{K}$ and $500\text{nm}$.

To calculate slope of the energy versus wavelength, the Planck’s law equation used is,

$\frac{d\rho }{d\lambda }=\frac{8\pi hc}{{\lambda }^5}\left(\frac{1}{e^{hc/\lambda kT}-1}\right)$

Where,

• $\rho$ is the energy density.

• $\lambda$ is the wavelength.

• $h$ is the Planck’s constant.

• $k$ is the Boltzmann constant.

• $c$ is the speed of light.

Substitute the values of constants, temperature and wavelength in the given equation.

$\begin{array}{rll}\frac{d\rho }{d\lambda }& =& \frac{8\left(3.14\right)\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)\left(3\times {10}^8{\text{ms}}^{-1}\right)}{{\left(500\times {\text{10}}^{-9}\text{m}\right)}^5}\left(\frac{1}{e^{\frac{6.626\times {10}^{-34}\text{J}\cdot \text{s}\times 3\times {10}^8{\text{ms}}^{-1}}{500\times {10}^{-9}\text{m}\times 1.38\times {10}^{-23}{\text{JK}}^{-\text{1}}\times 2000\text{K}}}-1}\right)\\ & =& 89.3{\text{Jm}}^{-4}\end{array}$

Thus, the slope of the energy versus wavelength is $89.3{\text{Jm}}^{-4}$.

Conclusion

The value and units of the slope of the energy versus wavelength at given temperature and wavelength is $89.3$ and ${\text{Jm}}^{-4}$ respectively.

Interpretation Introduction

(c)

Interpretation:

The value and units of the slope of the energy versus wavelength for given Planck’s law equation at given temperatures and wavelengths is to be calculated.

Concept introduction:

Planck’s equation can also be represented in the form of energy density distribution of a black body radiation at a given temperature and wavelength. This equation is known as Planck’s radiation distribution law.

Answer to Problem 9.30E

The value and units of the slope of the energy versus wavelength at given temperature and wavelength is $496.1$ and ${\text{Jm}}^{-4}$ respectively.

Explanation of Solution

It is given that temperature and wavelength is $2000\text{K}$ and $5000\text{nm}$.

To calculate slope of the energy versus wavelength, the Planck’s law equation used is,

$\frac{d\rho }{d\lambda }=\frac{8\pi hc}{{\lambda }^5}\left(\frac{1}{e^{hc/\lambda kT}-1}\right)$

Where,

• $\rho$ is the energy density.

• $\lambda$ is the wavelength.

• $h$ is the Planck’s constant.

• $k$ is the Boltzmann constant.

• $c$ is the speed of light.

Substitute the values of constants, temperature and wavelength in the given equation.

$\begin{array}{rll}\frac{d\rho }{d\lambda }& =& \frac{8\left(3.14\right)\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)\left(3\times {10}^8{\text{ms}}^{-1}\right)}{{\left(5000\times {\text{10}}^{-9}\text{m}\right)}^5}\left(\frac{1}{e^{\frac{6.626\times {10}^{-34}\text{J}\cdot \text{s}\times 3\times {10}^8{\text{ms}}^{-1}}{5000\times {10}^{-9}\text{m}\times 1.38\times {10}^{-23}{\text{JK}}^{-\text{1}}\times 2000\text{K}}}-1}\right)\\ & =& 496.1{\text{Jm}}^{-4}\end{array}$

Thus, the slope of the energy versus wavelength is $496.1{\text{Jm}}^{-4}$.

Conclusion

The value and units of the slope of the energy versus wavelength at given temperature and wavelength is $496.1$ and ${\text{Jm}}^{-4}$ respectively.

Interpretation Introduction

(d)

Interpretation:

The value and units of the slope of the energy versus wavelength for given Planck’s law equation at given temperatures and wavelengths is to be calculated.

Concept introduction:

Planck’s equation can also be represented in the form of energy density distribution of a black body radiation at a given temperature and wavelength. This equation is known as Planck’s radiation distribution law.

Answer to Problem 9.30E

The value and units of the slope of the energy versus wavelength at given temperature and wavelength is $47.4$ and ${\text{Jm}}^{-4}$ respectively.

Explanation of Solution

It is given that temperature and wavelength is $2000\text{K}$ and $10000\text{nm}$.

To calculate slope of the energy versus wavelength, the Planck’s law equation used is,

$\frac{d\rho }{d\lambda }=\frac{8\pi hc}{{\lambda }^5}\left(\frac{1}{e^{hc/\lambda kT}-1}\right)$

Where,

• $\rho$ is the energy density.

• $\lambda$ is the wavelength.

• $h$ is the Planck’s constant.

• $k$ is the Boltzmann constant.

• $c$ is the speed of light.

Substitute the values of constants, temperature and wavelength in the given equation.

$\begin{array}{rll}\frac{d\rho }{d\lambda }& =& \frac{8\left(3.14\right)\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)\left(3\times {10}^8{\text{ms}}^{-1}\right)}{{\left(10000\times {\text{10}}^{-9}\text{m}\right)}^5}\left(\frac{1}{e^{\frac{6.626\times {10}^{-34}\text{J}\cdot \text{s}\times 3\times {10}^8{\text{ms}}^{-1}}{10000\times {10}^{-9}\text{m}\times 1.38\times {10}^{-23}{\text{JK}}^{\text{-1}}\times 2000\text{K}}}-1}\right)\\ & =& 47.4{\text{Jm}}^{-4}\end{array}$

Thus, the slope of the energy versus wavelength is $47.4{\text{Jm}}^{-4}$.

Conclusion

The value and units of the slope of the energy versus wavelength at given temperature and wavelength is $47.4$ and ${\text{Jm}}^{-4}$ respectively.

Interpretation Introduction

(e)

Interpretation:

The results are to be compared with those of exercise $9.25$.

Concept introduction:

Planck’s equation can also be represented in the form of energy density distribution of a black body radiation at a given temperature and wavelength. This equation is known as Planck’s radiation distribution law.

The slope of the plot of energy versus wavelength for the Rayleigh-Jeans law is given by a rearrangement of equation $9.20$ which is given below.

$\frac{d\rho }{d\lambda }=\frac{8\pi kT}{{\lambda }^4}$

Answer to Problem 9.30E

The values of slope of the plot of energy versus wavelength from Rayleigh-Jeans law is similar to that of Planck’s radiation distribution law at temperature and wavelength $2000\text{K}$ and $10000\text{nm}$ respectively and most dissimilar at temperature and wavelength $1000\text{K}$ and $500\text{nm}$ respectively.

Explanation of Solution

On comparing the results from exercise $9.25$, it is found that the values of slope of the plot of energy versus wavelength from Rayleigh-Jeans law is similar to that of Planck’s radiation distribution law at temperature and wavelength $2000\text{K}$ and $10000\text{nm}$ respectively and most dissimilar at temperature and wavelength $1000\text{K}$ and $500\text{nm}$ respectively.

Conclusion

The values of slope of the plot of energy versus wavelength from Rayleigh-Jeans law is similar to that of Planck’s radiation distribution law at temperature and wavelength $2000\text{K}$ and $10000\text{nm}$ respectively and most dissimilar at temperature and wavelength $1000\text{K}$ and $500\text{nm}$ respectively.

Interpretation Introduction

(f)

Interpretation:

The temperatures and spectral regions at which the Rayleigh-Jeans law is close to Planck’s law are to be identified.

Concept introduction:

Planck’s equation can also be represented in the form of energy density distribution of a black body radiation at a given temperature and wavelength. This equation is known as Planck’s radiation distribution law.

The slope of the plot of energy versus wavelength for the Rayleigh-Jeans law is given by,

$\frac{d\rho }{d\lambda }=\frac{8\pi kT}{{\lambda }^4}$

Answer to Problem 9.30E

At higher temperatures and longer wavelengths, the Rayleigh-Jeans law is close to Planck’s law.

Explanation of Solution

On comparing the results from exercise $9.25$, it is found that the values of slope of the plot of energy versus wavelength from Rayleigh-Jeans law is similar to that of Planck’s radiation distribution law at temperature and wavelength $2000\text{K}$ and $10000\text{nm}$ respectively and most dissimilar at temperature and wavelength $1000\text{K}$ and $500\text{nm}$ respectively. Thus, at higher temperatures and longer wavelengths the Rayleigh-Jeans law is close to Planck’s law.

Conclusion

At higher temperatures and longer wavelengths the Rayleigh-Jeans law is close to Planck’s law.