2. The Sun can be modeled as a sphere of gas in thermal equilibrium. This is justified by that the spectrum of sunlight is well approximated by the blackbody spectrum. (a) The peak frequency of the spectrum is f = 3.4× 104 Hz. What is the temperature of the Sun? (b) The radius of the Sun is estimated to be 700,000 km. Find the total energy radiated by the Sun in a day (24 hours) in units of Joules and of eV. Hint: the Stefan-Boltzmann law.
2. The Sun can be modeled as a sphere of gas in thermal equilibrium. This is justified by that the spectrum of sunlight is well approximated by the blackbody spectrum. (a) The peak frequency of the spectrum is f = 3.4× 104 Hz. What is the temperature of the Sun? (b) The radius of the Sun is estimated to be 700,000 km. Find the total energy radiated by the Sun in a day (24 hours) in units of Joules and of eV. Hint: the Stefan-Boltzmann law.
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![**Exercise #5 (updated)**
In class, we found the number of standing wave modes (with polarizations) contained in between \( \omega \) and \( \omega + d\omega \),
\[
N(\omega) \, d\omega = \frac{V \omega^2}{\pi^2 c^3} \, d\omega \tag{1}
\]
where \( V \) is the volume of the cavity, \( V = L^3 \). With this number density \( N(\omega) \), the total mean energy of the blackbody radiation inside the cavity is obtained as
\[
E = \int_0^\infty \overline{E}(\omega, T) N(\omega) \, d\omega = \int_0^\infty \rho(\omega) \, d\omega \tag{2}
\]
where \( \overline{E}(\omega, T) = \frac{\hbar \omega}{e^{\hbar \omega / k_B T} - 1} \), and \( \rho(\omega) = E(\omega, T) N(\omega) \) is called the blackbody spectrum.
a. What is the SI unit for the blackbody spectrum \( \rho(\omega) \)?
b. Do the integral of Eq.(2) to find the energy density \( u = E/V \) of the blackbody radiation inside the cavity. Use the following formula:
\[
\int_0^\infty \frac{x^3}{e^x - 1} \, dx = \frac{\pi^4}{15} \tag{3}
\]
c. Borrowing a formula from electromagnetic theory, the radiation flux \( J \) from the surface of the cavity can be written in terms of the energy density as \( J = cu/4 \). For the blackbody radiation, this can be written as \( J = \sigma T^4 \) with \( \sigma \) the Stefan-Boltzmann constant. Find the expression of \( \sigma \) in terms of \( k_B \), \( \hbar \), and \( c \), then compute the value. (If your calculator cannot handle it, you will need to work out powers by hand.)
d. What is the SI unit for \( \sigma \) supposed to be? Does your \( \sigma](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa4426dc-92c3-4ac7-bd5a-9ba1dd10b9af%2F57a4454f-4838-41fc-b076-05d5ae04523b%2Fdak8ru_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise #5 (updated)**
In class, we found the number of standing wave modes (with polarizations) contained in between \( \omega \) and \( \omega + d\omega \),
\[
N(\omega) \, d\omega = \frac{V \omega^2}{\pi^2 c^3} \, d\omega \tag{1}
\]
where \( V \) is the volume of the cavity, \( V = L^3 \). With this number density \( N(\omega) \), the total mean energy of the blackbody radiation inside the cavity is obtained as
\[
E = \int_0^\infty \overline{E}(\omega, T) N(\omega) \, d\omega = \int_0^\infty \rho(\omega) \, d\omega \tag{2}
\]
where \( \overline{E}(\omega, T) = \frac{\hbar \omega}{e^{\hbar \omega / k_B T} - 1} \), and \( \rho(\omega) = E(\omega, T) N(\omega) \) is called the blackbody spectrum.
a. What is the SI unit for the blackbody spectrum \( \rho(\omega) \)?
b. Do the integral of Eq.(2) to find the energy density \( u = E/V \) of the blackbody radiation inside the cavity. Use the following formula:
\[
\int_0^\infty \frac{x^3}{e^x - 1} \, dx = \frac{\pi^4}{15} \tag{3}
\]
c. Borrowing a formula from electromagnetic theory, the radiation flux \( J \) from the surface of the cavity can be written in terms of the energy density as \( J = cu/4 \). For the blackbody radiation, this can be written as \( J = \sigma T^4 \) with \( \sigma \) the Stefan-Boltzmann constant. Find the expression of \( \sigma \) in terms of \( k_B \), \( \hbar \), and \( c \), then compute the value. (If your calculator cannot handle it, you will need to work out powers by hand.)
d. What is the SI unit for \( \sigma \) supposed to be? Does your \( \sigma
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