Question

A light beam travels at a speed \[1.94 \times {10^8}\,{\text{m/s}}\] in quartz. The wavelength found in quartz is \[355\,{\text{nm}}\]. What would be the wavelength in air?
A. \[179\,{\text{nm}}\]
B. \[549\,{\text{nm}}\]
C. \[355\,{\text{nm}}\]
D. \[707\,{\text{nm}}\]

Answer 1

Hint: Use the formula for the velocity of a wave. This formula gives the relation between the wavelength of the light wave, frequency of the light wave and the frequency of the light wave.

Complete step by step answer:
The equation for the velocity of a wave is given by
\[v = n\lambda \] …… (1)

Here, \[\lambda \] is the wavelength of the wave and \[n\] is the frequency of the wave.

Complete step by step answer:

The speed \[v\] of the light beam in quartz is \[1.94 \times {10^8}\,{\text{m/s}}\] and its wavelength \[\lambda \] in quartz is \[355\,{\text{nm}}\].

The frequency of any wave remains the same in every medium.

Rewrite equation (1) for the speed of the light beam in the quartz medium.
\[v = n\lambda \] …… (2)

Here, \[n\] is the frequency of the light beam.

The speed \[c\] of the light beam in air is \[3 \times {10^8}\,{\text{m/s}}\].

Rewrite equation (1) for the speed of the light beam in the air medium.
\[c = n{\lambda _a}\] …… (3)

Here, \[{\lambda _a}\] is the wavelength of the light beam in air.

Divide equation (3) by equation (2).
\[\dfrac{c}{v} = \dfrac{{n{\lambda _a}}}{{n\lambda }}\]
\[ \Rightarrow \dfrac{c}{v} = \dfrac{{{\lambda _a}}}{\lambda }\]

Rearrange the above equation for the wavelength \[{\lambda _a}\] of the light in the air medium.
\[{\lambda _a} = \dfrac{{c\lambda }}{v}\]

Substitute \[3 \times {10^8}\,{\text{m/s}}\] for \[c\], \[355\,{\text{nm}}\] for \[\lambda \] and \[1.94 \times {10^8}\,{\text{m/s}}\] for \[v\] in the above equation.
\[{\lambda _a} = \dfrac{{\left( {3 \times {{10}^8}\,{\text{m/s}}} \right)\left( {355\,{\text{nm}}} \right)}}{{1.94 \times {{10}^8}\,{\text{m/s}}}}\]
\[ \Rightarrow {\lambda _a} = 548.96\,{\text{nm}}\]
\[ \Rightarrow {\lambda _a} \approx 549\,{\text{nm}}\]

Therefore, the wavelength of the light beam in air is \[549\,{\text{nm}}\].

Hence, the correct option is B.

Note: There is no need to convert the unit of the wavelength from nanometer to meter in the SI system of units as the ultimate unit of the wavelength is to be in nanometer.