Planck's radiation law assumes a body radiating into a vacuum. If the body is radiating within another medium the wavelength of the emitted radiation changes because the speed of light in the medium changes even if the frequency of the radiation remains the same. This leads to changes in Wien's displacement law and the power output of a blackbody of the form:

\[E_{b b}^{r}=\int_{0}^{\infty} \frac{2 \pi c_{m}^{2} \hbar}{\lambda_{m}^{5}\left[\exp \left(\frac{f_{1} c_{m}}{\lambda_{m} k_{b} T}\right)-1\right]} d \lambda_{m} \quad \lambda_{\max } T=\frac{2.884 \times 10^{-3}}{\eta}\]

a. Integrate the power density function above to show that in the medium of refractive index, \(\eta\), the blackbody emissive power is:

\[E_{b b}^{r}=\eta^{2} \sigma^{r} T^{4}\]

Here we know that in the medium the wavelength and velocity are related to their vacuum wavelength and velocity via:

\[\lambda_{m}=\frac{\lambda}{\eta} \quad c_{m}=\frac{c}{\eta}\]

b. If we have an object in a dense medium of refractive index 1.5 that is radiating at a temperature of \(3000 \mathrm{~K}\), determine the blackbody emissive power and the wavelength of maximum emission.