Question

2.2. Instead of ρν, a spectral energy density PA can be defined, pa being such that ρλdλ gives the nergy density for the e.m. waves of wavelength lying between λ and λ +d. Find the relationship between ρλ and ρν.

Please solve this question fron the book Principles of Laser, written by Orazio Sevelto, 5th edition

Chapter no 2
Quesrion no 2.2

Answer 1

solution ;

$\rho_{\nu}$ is the spectral energy density in terms of frequency and $\rho_{\lambda}$ is in terms of wavelength

$\rho_{\nu} = \frac{8\pi h\nu^3}{c^3 } \frac{1}{e^{h \nu/kT}-1}$    (1)and

$\rho_{\lambda} = \frac{8\pi hc}{\lambda^5 } \frac{1}{e^{h /\lambda kT}-1}$    (2)  

where we just use the fact frequency of light = speed of light / wavelength of light

on dividing (1) by (2) we get

$\frac{\rho_{\nu}}{\rho_{\lambda}} = \frac{8\pi h \nu^3}{c^3}/\frac{8\pi hc}{\lambda^5 } \\ \frac{\rho_{\nu}}{\rho_{\lambda}} = \frac{\lambda^5 \nu^3}{c^3} = \lambda^2$ where

LV