# Energy Density of Blackbody Radiation

1. Jan 14, 2014

### stefan10

1. The problem statement, all variables and given/known data

$$\mbox{Let} \ p(< \nu_{0}) \mbox{be the total energy density of blackbody radiation in all frequencies less than} \ \nu_{0}, \mbox{where} \ h \nu_{0} << kT. \mbox{Derive an expression for} \ p (< \nu_{0})$$

2. Relevant equations

$$p(v) dv = \dfrac{8 \pi h} {c^3} \dfrac {\nu^3}{e^\frac{h\nu}{kT} -1} dv$$

3. The attempt at a solution

We want to find the total energy density, so that means we'll have to integrate the Planck's Law. The limits of integration will be from 0 to v-knot.

$$h \nu < h \nu_{0} << KT \Rightarrow h \nu << KT$$

Which if we simplify for this limit gives:

$$p(v) dv = \dfrac{8 \pi h} {c^3} \dfrac {\nu^3}{e^\frac{h\nu}{kT} -1} dv$$

$$p(v) dv = \dfrac{8 \pi h} {c^3} \dfrac {\nu^3}{\dfrac{h\nu}{kT} +1 -1} dv$$

$$p(v) dv = \dfrac{8 \pi h} {c^3} \dfrac {\nu^3}{\frac{h\nu}{kT} } dv$$

$$p(v) dv = \dfrac{8 \pi KT}{c^3} \nu^2 dv \ \mbox{Rayleigh-Jeans Formula}$$

$p(<v_{0}) = \dfrac{8 \pi KT}{c^3}\int _ {0} ^{\nu_{0}} \nu^2 dv$

$p(<v_{0}) = \dfrac{8 \pi KT}{3 c^3}\nu_{0}^3$

Is there anything wrong? Thank you very much!

Last edited: Jan 14, 2014
2. Jan 15, 2014

### TSny

Looks good to me.