Question

How long would it take a radio wave with a frequency of $7.6\times {{10}^{5}}Hz$ to travel from Mars to Earth if the distance between the two planets is approximately $8.00\times {{10}^{7}}km$?

Answer 1

Hint: radio wave is an electromagnetic wave whose wavelength is in the range of 1 mm to 10,000 km. Electromagnetic waves are transverse waves that are created because of synchronized oscillations or vibrations between a magnetic field and the electric field.

Complete step-by-step answer:First. we need to know about wave frequency. The frequency of a wave can be defined as the number of wavelengths that pass through a certain point in a unit length of time. Its SI unit is Hertz (Hz).
Now, to determine the time taken by an electromagnetic wave to travel a certain distance, we need to know about the factors affecting the speed of an electromagnetic wave.
The speed of an electromagnetic wave is completely dependent on the medium through which the wave is passing.
When passing through a medium, the speed of an electromagnetic wave is given by
\[v=\dfrac{c}{n({{\lambda }_{free}})}\]
Where c is the speed of light and \[n({{\lambda }_{free}})\] is the medium's index of refraction.
From this formula, we can see that the value of frequency does not affect the speed of an electromagnetic wave.
Now, we know that the atmosphere is absent in space, and hence is a vacuum. In a vacuum, the speed of the electromagnetic wave is the same as the speed of light as the index of refraction \[n({{\lambda }_{free}})\] of the vacuum medium is 1.
So, time taken by radio wave to travel the distance $8.00\times {{10}^{7}}km$ between mars and earth at the speed of light is
\[t=\dfrac{D}{s}=\dfrac{8.00\times {{10}^{7}}\times 1000\text{ m}}{3.00\times {{10}^{8}}\text{ m}\text{.}{{\text{s}}^{-1}}}\cong 266.67\text{ s}\]

Additional Information: The relation between wavelength and frequency, in a vacuum, can be defined by the
\[v=\dfrac{c}{\lambda }\]
Where v is the frequency, $\lambda $ is the wavelength and c is the speed of light.
From this formula, we can see that frequency and wavelength are inversely proportional. That means that when there is an increase in wavelength, the frequency decreases.

Note:We know that the index of refraction \[n({{\lambda }_{free}})\] of a medium is the ratio of the speed of an electromagnetic wave in a vacuum to the speed of an electromagnetic wave in a medium (of greater density).
Since the index of refraction \[n({{\lambda }_{free}})\] is always greater than 1 in any medium, hence the speed of an electromagnetic wave in any medium will always be less than the speed of an electromagnetic wave in a vacuum.