Question

The cosmic microwave background peaks at a wavelength of about 1 mm, and the universe has a temperature of about 3 K. If the microwave background peaked at a wavelength of 10 micrometers, what would its temperature be?

Answer 1

Using Wien's law, the temperature at which the cosmic microwave background would peak at 10 micrometers is calculated to be 289.8 K.

Step-by-step explanation:

The cosmic microwave background (CMB) radiation is a relic from the early universe and acts as a perfect blackbody radiation with a measured temperature of approximately 2.73 K. The peak wavelength of the CMB is about 1 mm (or 1000 micrometers). Using Wien's law, we can find the temperature corresponding to a different peak wavelength. Wien's law states that the product of the peak wavelength (λmax) and the temperature (T) is constant, and for blackbody radiation, this constant is approximately 2.898 x 10-3 m·K. Therefore, if the peak wavelength were 10 micrometers, we can use the formula:

λmax T = 2.898 x 10-3 m·K

to find the new temperature (T') by solving for T' when λmax = 10 x 10-6 meters. The calculation would be:

T' = (2.898 x 10-3) / (10 x 10-6)

T' = 289.8 K

Thus, if the cosmic microwave background peaked at a wavelength of 10 micrometers, its temperature would be 289.8 K.