Wave Motion 3 Question 3
3. An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency $10 GHz$. What is the frequency of the microwave measured by the observer? (speed of light $=3 \times 10^{8} ms^{-1}$ )
(2017 Main)
(a) $12.1 GHz$
(b) $17.3 GHz$
(c) $15.3 GHz$
(d) $10.1 GHz$
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Answer:
Correct Answer: 3. (b)
Solution:
- As observer is moving with relativistic speed; formula $\frac{\Delta f}{f}=\frac{v _{\text {radial }}}{c}$, does not apply here.
Relativistic doppler’s formula is
$$ f _{\text {observed }}=f _{\text {actual }} \cdot\left(\frac{1+v / c}{1-v / c}\right)^{1 / 2} $$
Here, $\quad \frac{v}{c}=\frac{1}{2}$
So, $f _{\text {observed }}=f _{\text {actual }}\left(\frac{3 / 2}{1 / 2}\right)^{1 / 2}$
$\therefore \quad f _{\text {observed }}=10 \times \sqrt{3}=17.3 GHz$
