Question

Construct plots that show the wavelength-dependent energy spectrum of a blackbody at a temperature of 5800 K (approx. temperature of the Sun) using both the Planck distribution and the Raleigh-Jeans distribution. Confirm agreement between the two at long wavelength. a. What is the maximum emission wavelength at this temperature? b. What is the total power output (W/m^2) ? c. Using the Planck distribution, estimate what fraction of the Sun's total power output is emitted in visible wavelengths (400-750 nm)

Answer 1

(a)

A should be weawned in nK 2.3 へ2/ 녀 Planka laud h 664x 13 -34 2x 10 2S x 10 2 0 LC

The plot is

Energy Density v/s Wavelength 12 Rayleigh Plancks_ 10 5000 0000 15000 20000 25000 30000 Wavelength(nm)

Gnuplot is used for plotting

The text document is

set zeroaxis
set xlabel 'Wavelength(nm)'
set ylabel 'energy density(J/kg m^3)'
set title 'Energy Density v/s Wavelength'
set term png
set output 'rp.png'
r(x)=(8*pi*8*10**(15))/x**4
p(x)=((8*pi*2*10**(20))/x**5)*(1/(exp(2.5*10**(4)/x)-1))
plot [1000:32000][0.1:12] r(x) title 'Rayleigh',p(x) title 'Plancks'

Write all this in a txt file and in gnuplot write

load 'filename.txt'

From graph one can see:

At higher wavelength i.e. (2500 nm- 3000 nm) there is agreement between Plancks and Rayleigh

maximum wavelength at this tempertaure is

$\lambda_{max}=5000\text{nm}$

(b) Total power emitted

$Q=\sigma T^4$

where $\sigma$ is stefans constant

$Q=5.6\times 10^{-8} \times 5800^4$

$Q=63.37\times 10^6\mathrm{J/m^2 s}$