Use Wien’s law (see Problem 37) to answer the following questions: (a) The cosmic background radiation peaks in intensity at a wavelength of.$\mathbf{1}\mathbf{.1}\mathbf{\text{mm}}$To what temperature does this correspond? (b) About$\mathbf{379000}\mathbf{\text{y}}$ after the big bang, the universe became transparent to electromagnetic radiation. Its temperature then was$\mathbf{2970}\mathbf{\text{K}}$ . What was the wavelength at which the background radiation was then most intense?

By Wien’s law, the wavelength at which a thermal radiator at temperature $T$radiates electromagnetic waves most intensely is.${\mathit{\lambda }}_{\mathbf{m}\mathbf{a}\mathbf{x}}\mathbf{=}\mathbf{\left(}\mathbf{\text{2898μm.K}}\mathbf{\right)}\mathbf{\text{/T}}$

Consider the formulas required:

a) $\mathit{T}\mathbf{=}\frac{\mathbf{\lambda }}{{\mathbf{\lambda }}_{\mathbf{m}\mathbf{a}\mathbf{x}}}$ …… (1)

b) ${\mathit{\lambda }}_{\mathbf{m}\mathbf{a}\mathbf{x}}\mathbf{=}\frac{\mathbf{\lambda }}{\mathbf{T}}$ ……. (2)

(a)

Consider that the cosmic background radiation peaks in intensity at a wavelength of .

For the universal microwave background, Wien’s law leads to,

$\begin{array}{l}T=\frac{2898\text{\hspace{0.17em}μm}\cdot \text{K}}{{\lambda }_{\max }}\\ T=\frac{2898\text{\hspace{0.17em}mm}\cdot \text{K}}{1.1\text{\hspace{0.17em}mm}}\\ T=2.6\text{\hspace{0.17em}K}\end{array}$

Therefore, the given wavelength correspondsto the temperature.$T=2.6\text{\hspace{0.17em}K}$

b)

Consider that$379000\text{y}$after the big bang, the universe became transparent to electromagnetic radiation. Its temperature then was.$2970\text{K}$

At, decoupling, the wavelength can be calculate as follows:

.$\begin{array}{l}{\lambda }_{\max }=\frac{2898\text{\hspace{0.17em}μm}\cdot \text{K}}{T}\\ {\lambda }_{\max }=\frac{2898\text{\hspace{0.17em}μm}\cdot \text{K}}{2970\text{\hspace{0.17em}K}}\\ {\lambda }_{\max }=0.976\text{\hspace{0.17em}μm}\end{array}$

${\lambda }_{\max }=976\text{\hspace{0.17em}nm}$

Therefore,the wavelength at which the background radiation was then most intenseis ${\lambda }_{\max }=976\text{\hspace{0.17em}nm}$.