Question

For a certain light there are \[2\times {{10}^{3}}\] waves in \[1.5mm\] in air. What will be the wavelength of light?
\[\begin{align}
  & A.750nm \\
 & B.75\overset{0}{\mathop{A}}\, \\
 & C.750\overset{0}{\mathop{A}}\, \\
 & D.75\times {{10}^{7}}m \\
\end{align}\]

Answer 1

Hint: The wavelength of the light used can be found by taking the ratio of the distance of the waves in the air to the number of waves in air. Substitute the values in it. The answer obtained will be in the micrometer. This has to be converted in terms of Nanometer. This will help you in answering this question.

Complete step by step solution:
It has been given in the question that the distance of the waves in air be,
\[d=1.5mm=1.5\times {{10}^{-3}}m\]
The number of waves can be shown as,
\[n=2\times {{10}^{3}}\]
The wavelength of the light used can be found by taking the ratio of the distance of the waves in air to the number of waves in air. This can be written as,
\[\lambda =\dfrac{\text{distance}}{n}\]
Substituting the values in it will give,
\[\lambda =\dfrac{\text{distance}}{n}=\dfrac{1.5\times {{10}^{-3}}}{2\times {{10}^{3}}}=0.75\mu m\]
The micrometer is to be converted into Nanometer. For that let us first of all convert the micro meter into metre. That is,
\[\lambda =0.75\mu m=0.75\times {{10}^{-6}}m\]
Now the wavelength is in meters. This is to be converted into a Nanometer. That is,
\[\lambda =0.75\times {{10}^{6}}\times {{10}^{9}}nm=750nm\]
Therefore the wavelength of the used light will be obtained as \[750nm\]. This has been mentioned as option A.

Note: Wavelength can be explained as the distance between two consecutive crests or troughs of a wave. This can be measured in the direction of the wave. In detail, we can say that as long as the wavelength, lower will be the frequency of the wave. In the same sense, as shorter as the wavelength, higher will be the frequency of the wave. the product of the frequency and the wavelength of the wave will be equivalent to the velocity of the wave.