The Doppler effect is a phenomenon that describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. This effect is not limited to sound waves; it also applies to electromagnetic waves, such as light. When an observer moves towards a light source, the observed frequency increases, while it decreases when moving away. The fundamental equation for the Doppler effect in light is given by:
f_{\text{observed}} = f_{\text{source}} \left(1 \pm \frac{v_{\text{relative}}}{c}\right)
In this equation, fobserved is the frequency observed, fsource is the frequency emitted by the source, vrelative is the relative velocity between the observer and the source, and c is the speed of light (approximately \(3 \times 10^8\) m/s). The sign used in the equation depends on the direction of motion: a plus sign is used when the observer and source are moving closer together, while a minus sign is used when they are moving apart.
To illustrate this, consider a distant star emitting light with a wavelength of 630 nanometers. If the star is receding from Earth at a speed of \(3 \times 10^6\) m/s, we can calculate the observed frequency. First, we convert the wavelength to frequency using the relationship:
f_{\text{source}} = \frac{c}{\lambda_{\text{source}}}
Substituting the values, we find:
f_{\text{source}} = \frac{3 \times 10^8 \text{ m/s}}{630 \times 10^{-9} \text{ m}} \approx 4.76 \times 10^{14} \text{ Hz}
Next, we apply the Doppler effect equation to find the observed frequency:
f_{\text{observed}} = 4.76 \times 10^{14} \left(1 - \frac{3 \times 10^6}{3 \times 10^8}\right)
Calculating this gives:
f_{\text{observed}} \approx 4.71 \times 10^{14} \text{ Hz}
Finally, to find the observed wavelength, we use the frequency we just calculated:
\lambda_{\text{observed}} = \frac{c}{f_{\text{observed}}} = \frac{3 \times 10^8 \text{ m/s}}{4.71 \times 10^{14} \text{ Hz}} \approx 636 \text{ nm}
This example demonstrates that even at significant velocities, the change in wavelength due to the Doppler effect can be relatively small unless the speeds approach a significant fraction of the speed of light. Understanding the Doppler effect is crucial in fields such as astronomy, where it helps in determining the movement of stars and galaxies.
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