(a)(i) To calculate the change in wavelength of light received by an observer on the Earth, we can use the formula for redshift:
z = ∆λ/λ = v/c
where z is the redshift, ∆λ is the change in wavelength, λ is the original wavelength, v is the velocity of the galaxy, and c is the speed of light.
Substituting the given values, we get:
z = ∆λ/6.2 × 10−7 m = 3.9 × 104 km/s / 3.0 × 105 km/s
Solving for ∆λ, we get:
∆λ = λz = 6.2 × 10−7 m × 3.9 × 104 km/s / 3.0 × 105 km/s
∆λ = 8.06 × 10−11 m
Therefore, the change in the wavelength of the light received by an observer on Earth is 8.06 × 10−11 m.
(ii) The wavelength of the light that is received by the observer on Earth can be calculated using the formula:
λ' = λ + ∆λ
where λ' is the new wavelength and λ is the original wavelength.
Substituting the given values, we get:
λ' = 6.2 × 10−7 m + 8.06 × 10−11 m
λ' = 6.2008 × 10−7 m
Therefore, the wavelength of the light received by the observer on Earth is 6.2008 × 10−7 m.
(b) The redshift of galaxies supports the Big Bang theory in two ways:
1. According to the Big Bang theory, the universe is expanding. As the universe expands, galaxies move away from each other, and their light is redshifted. The greater the redshift, the faster the galaxy is moving away from us. The observation of redshift in distant galaxies provides evidence that the universe is indeed expanding.
2. The Big Bang theory predicts that the early universe was much denser and hotter than it is now. This high density and temperature would have caused the universe to emit a lot of radiation, including light. As the universe expanded, this radiation would have cooled and stretched, leading to a cosmic microwave background radiation that fills the universe. The observed spectrum of this radiation is consistent with the predictions of the Big Bang theory. The redshift of distant galaxies provides further evidence for the Big Bang theory, as it is consistent with the idea that the universe was much denser and hotter in the past.