Question

An observer is moving with one-third of the speed of light towards a stationary microwave source emitting waves at frequency $8\text{GHz}$. What is the frequency of the microwave measured by the observer?

• A. $8\text{GHz}$
• B. $8\sqrt{3}\text{GHz}$
• C. $8\sqrt{2}\text{GHz}$
• D. $\frac{8}{3}\text{GHz}$

Answer 1

The correct option is C $8\sqrt{2}\text{GHz}$

As the observer is moving towards a stationary source with relativistic velocity, distance between the light source and observer is decreasing, then the apparent frequency of the source will be given by
$f^{\prime }=f_0\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$

From the data given in the question,

$v=\frac{c}{3}$ and $f_0=8\text{GHz}$
$f^{\prime }\to$ observer frequency
So
$f^{\prime }=8{\left[\frac{1+\frac{1}{3}}{1-\frac{1}{3}}\right]}^{\frac{1}{2}}=8{\left(\frac{4}{2}\right)}^{\frac{1}{2}}=8\sqrt{2}\text{GHz}$