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? (Hint: see Cosmic Calculations 5.1 and 17.1 in the textbook)

Answer 1

1. If the cosmic microwave background peaked at a wavelength of 10 micrometers instead of 1 mm, its temperature would be approximately 300 K.

Step-by-step explanation:

The temperature of the cosmic microwave background (CMB) is related to the peak wavelength through Wien's Law, which states that the product of the peak wavelength (λ) and the temperature (T) is a constant (b). Mathematically, this relationship is represented as λ * T = b. Given that the CMB peaks at 1 mm with a temperature of 3 K, we can use this information to find the temperature for a peak wavelength of 10 micrometers.

For the initial scenario with a peak wavelength of 1 mm:

`\[1 \, mm * 3 \, K = b.\]`

Now, for the modified scenario with a peak wavelength of 10 micrometers (converting micrometers to millimeters):

`\[10 \, \mu m = 0.01 \, mm.\]`

`\[0.01 \, mm * T = b.\]`

Setting the two expressions for 'b' equal to each other:

`\[1 \, mm * 3 \, K = 0.01 \, mm * T.\]`

Solving for T (temperature):

`\[T = (1 \, mm * 3 \, K)/(0.01 \, mm) = 300 \, K.\]`

Therefore, if the cosmic microwave background peaked at a wavelength of 10 micrometers, its temperature would be approximately 300 K. This calculation demonstrates the application of Wien's Law in determining the temperature of the CMB for different peak wavelengths.