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:

  1. 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, $f _{\text {observed }}=f _{\text {actual }}\left(\frac{3 / 2}{1 / 2}\right)^{1 / 2}$

$\therefore \quad f _{\text {observed }}=10 \times \sqrt{3}=17.3 GHz$



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