In class we found the number of standing wave modes (with polarizations) contained in between w and w + dw, Vw? N(w) dw = dw (1) where V is the volume of the cabity, v = L³. With this number density N(w), the total mean energy of the blackbody radiation inside the cavity is obtained as = [ E(w,T') N(w)dw = plw) duw (2) where E(w, T') : , and p(w) = E(w,T) N (w) is called the blackbody spectrum. a. What is the SI unit for the blackbody spectrum p(w)? 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: xp- 15 (3) c. Borrowing a formula from electro-magnetic 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 = oT with o the Stefan-Boltzmann constant. Find the expression of o in terms of kg, h, and c, then compute the value. (If your calculator cannot handle it, you will need to work out powers by hand.)
In class we found the number of standing wave modes (with polarizations) contained in between w and w + dw, Vw? N(w) dw = dw (1) where V is the volume of the cabity, v = L³. With this number density N(w), the total mean energy of the blackbody radiation inside the cavity is obtained as = [ E(w,T') N(w)dw = plw) duw (2) where E(w, T') : , and p(w) = E(w,T) N (w) is called the blackbody spectrum. a. What is the SI unit for the blackbody spectrum p(w)? 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: xp- 15 (3) c. Borrowing a formula from electro-magnetic 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 = oT with o the Stefan-Boltzmann constant. Find the expression of o in terms of kg, h, and c, then compute the value. (If your calculator cannot handle it, you will need to work out powers by hand.)
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Please do A, B, and C
Transcribed Image Text:Exercise #5 (updated)
In class we found the number of standing wave modes (with polarizations) contained in
between w and w + dw,
Vw?
N(w) dw =
(1)
where V is the volume of the cabity, V = L³. With this number density N(w), the total
mean energy of the blackbody radiation inside the cavity is obtained as
|E(w, T) N (w)dw = | plw) dw
E =
(2)
where E(w, T) =
ehw /kpT
and p(w) = E(w,T') N (w) is called the blackbody spectrum.
a. What is the SI unit for the blackbody spectrum p(w)?
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:
73
(3)
c. Borrowing a formula from electro-magnetic 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 = oT with o the Stefan-Boltzmann
constant. Find the expression of o in terms of kg, h, 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 o supposed to be? Does your o have the correct unit?
e. Find the peak frequency of p(w) as a function of T by solving
dp
= 0
dw
(4)
for w. Use the solution for 3+ e"(x – 3) = 0, x = 2.824.
f. What is the total power of the black body radiation from an object whose surface area
is A and temperature is T? Use the Stefan-Boltzmann law, J = oTª.
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× 1014 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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