The cosmic microwave background radiation is light left over from the Big Bang that has been Doppler-shifted to microwave frequencies by the expansion of the universe. It now fills the universe with 450 photons/cm³ at an average frequency of 160 GHz. How much energy from the cosmic microwave background, in MeV, fills a small apartment that has 95 m² of floor space and 2.5-m-high ceilings?

Verified step by step guidance

1. Convert the volume of the apartment from meters cubed to centimeters cubed. The volume is calculated as the product of the floor area and the ceiling height: \( V = \text{floor area} \times \text{ceiling height} \). Since 1 m = 100 cm, the conversion factor is \( 1 \text{ m}^3 = 10^6 \text{ cm}^3 \).
2. Determine the total number of photons in the apartment. Multiply the photon density (450 photons/cm³) by the volume of the apartment in cm³: \( N = \text{photon density} \times V \).
3. Calculate the energy of a single photon using the formula \( E = h \nu \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J·s} \)) and \( \nu \) is the frequency (160 GHz, or \( 160 \times 10^9 \, \text{Hz} \)). Convert the energy from joules to MeV using the conversion factor \( 1 \text{ J} = 6.242 \times 10^{12} \text{ MeV} \).
4. Calculate the total energy of the photons in the apartment by multiplying the energy of a single photon by the total number of photons: \( E_{\text{total}} = E_{\text{single photon}} \times N \).
5. Express the final result in MeV, ensuring all units are consistent and properly converted throughout the calculation.

Key Concepts

Cosmic Microwave Background Radiation (CMB)

The Cosmic Microwave Background Radiation is the afterglow of the Big Bang, consisting of low-energy photons that fill the universe. It is a remnant from the early universe, providing critical evidence for the Big Bang theory. The CMB is isotropic, meaning it has a uniform temperature across the sky, and its study helps us understand the universe's evolution and structure.

Doppler Shift

Doppler Shift refers to the change in frequency or wavelength of waves in relation to an observer moving relative to the source of the waves. In the context of the CMB, the expansion of the universe causes the wavelengths of the original high-energy photons to stretch, shifting them into the microwave region of the electromagnetic spectrum. This phenomenon is crucial for understanding how the universe has expanded over time.

Energy Density of Photons

Energy density of photons refers to the amount of energy contained in a given volume of space due to the presence of photons. In the case of the CMB, with a density of 450 photons/cm^3, this energy can be calculated using the relationship between photon energy and frequency, given by E = hf, where h is Planck's constant and f is the frequency. This concept is essential for determining the total energy present in a specified volume, such as an apartment.