To determine the frequency shift of the hydrogen line in the spectrum of galaxy NGC 4414, we'll use the Doppler effect formula for electromagnetic waves. The formula for the frequency shift ([tex]\(\Delta f\)[/tex]) in the context of a source moving away from the observer is:
[tex]\[ \Delta f = \left( \frac{v}{c} \right) f \][/tex]
Where:
- [tex]\(v\)[/tex] is the velocity of the source moving away from the observer.
- [tex]\(c\)[/tex] is the speed of light in vacuum.
- [tex]\(f\)[/tex] is the original frequency of the emitted light.
Given values:
- Velocity, [tex]\(v = 7.16 \times 10^5 \, \text{m/s}\)[/tex]
- Frequency, [tex]\(f = 6.91 \times 10^{14} \, \text{Hz}\)[/tex]
- Speed of light, [tex]\(c = 3 \times 10^8 \, \text{m/s}\)[/tex]
Let's substitute these values into the formula to find the frequency shift:
[tex]\[ \Delta f = \left( \frac{7.16 \times 10^5 \, \text{m/s}}{3 \times 10^8 \, \text{m/s}} \right) \times 6.91 \times 10^{14} \, \text{Hz} \][/tex]
First, calculate the ratio [tex]\(\frac{v}{c}\)[/tex]:
[tex]\[ \frac{7.16 \times 10^5}{3 \times 10^8} = 2.3867 \times 10^{-3} \][/tex]
Then, multiply this ratio by the original frequency:
[tex]\[ \Delta f = 2.3867 \times 10^{-3} \times 6.91 \times 10^{14} \][/tex]
[tex]\[ \Delta f = 1.6491866666666668 \times 10^{12} \, \text{Hz} \][/tex]
Therefore, the frequency of the hydrogen line will be shifted by approximately:
[tex]\[ \boxed{1.6491866666666668} \][/tex]
(Note: The units are [tex]\( \times 10^{12} \, \text{Hz} \)[/tex] as required by the question.)
