In a laboratory experiment, light from a hydrogen discharge lamp is used to produce a line emission spectrum - Scottish Highers Physics - Question 10 - 2018

To find the frequency of the photon emitted during the transition from E1 to E3:

1. Calculate the change in energy (ΔE): $$ext{ΔE} = E3 - E1 = -2.42 \times 10^{-19} \, \text{J} - (-0.871 \times 10^{-19} \, \text{J}) = -1.55 \times 10^{-19} \, \text{J}$$
2. Use the equation relating energy and frequency: $$E = hf \rightarrow f = \frac{E}{h}$$ where h = Planck's constant, approximately $6.63 \times 10^{-34} \, \text{Js}$.
3. Substitute values: $$f = \frac{1.55 \times 10^{-19}}{6.63 \times 10^{-34}} \approx 2.33 \times 10^{14} \, \text{Hz}$$

To find the recessional velocity (v) of the distant galaxy, we use the formula:

1. First, determine the shift in wavelength: $$z = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = \frac{661 \text{ nm} - 656 \text{ nm}}{656 \text{ nm}} \approx 0.007628$$
2. Relate the redshift (z) to velocity using the formula: $$v = zc$$ where c ≈ $3.00 \times 10^8 \, \text{m/s}$.
3. Substituting the values: $$v = 0.007628 \times (3.00 \times 10^8) \approx 2.29 \times 10^6 \, \text{m/s}$$