Planck's law gives the spectral radiance of an ideal blackbody cavity as L_ (T) =…

PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Photodetection, photonic sources and quantum optics · Photodetection, photonic sources and quantum optics

Optics and Photonics, Ch The photon

Problem

Planck's law gives the spectral radiance of an ideal blackbody cavity as

L_\lambda(T) = \frac{2hc^2}{\lambda^5}\,\frac{1}{e^{hc/(\lambda k_BT)}-1}.

The total radiance is

L(T) = \int_0^\infty L_\lambda(T)\,d\lambda

, and Chapter 3's Lambertian relation gives the radiant exitance of the cavity's aperture as

M_e = \pi L

. (a) Substitute

x = hc/(\lambda k_BT)

into the integral for

L(T)

and show that it becomes

L(T) = \frac{2(k_BT)^4}{h^3c^2}\int_0^\infty \frac{x^3}{e^x-1}\,dx.

(b) Using the supplied result

\int_0^\infty x^3/(e^x-1)\,dx = \pi^4/15

, show that

M_e = \sigma T^4

with

\sigma = \frac{2\pi^5k_B^4}{15h^3c^2},

and evaluate

\sigma

numerically from

h = 6.626\times10^{-34}

J s,

k_B = 1.381\times10^{-23}

J K

^{-1}

and

c = 3.00\times10^8

m s

^{-1}

. (c) Sirius A radiates as an ideal blackbody at surface temperature

9940

K. Use your value of

\sigma

to find its radiant exitance

M_e

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