Question

The dimensional formula of wave number is:
$A\left[ {{M^0}{L^0}{T^0}} \right] \\\ B\left[ {{M^0}{L^{ - 1}}{T^0}} \right] \\\ C\left[ {{M^0}{L^0}{T^{ - 1}}} \right] \\\ D\left[ {{M^{ - 1}}{L^{ - 1}}{T^0}} \right] \\\$

Answer 1

- Hint: In order to solve this question, firstly we must know the concept of wavenumber i.e. it is the number of waves that occurs across a fixed distance. Then we will compare all the given options to get the required result.

Complete step by step answer:
The number of waves per unit can be matched to the number of waves. In other words, it is the number of waves that occurs across a fixed distance. It is also known as propagation constant or angular wave number. It is a scalar quantity represented by k and the mathematical representation is given by-
$k = \dfrac{1}{\lambda }$
Where k is the wavenumber
$\lambda$is the wavelength.
Wavenumber is reciprocal of wavelength. Units of wavenumber are cycles per centimeter or inverse centimeters.
Wavenumber is mathematically expressed as the number of the complete cycle of a wavelength which is given by-
Wavenumber (k)$= \dfrac{{2\pi }}{\lambda }$ (measured using radian per meter)
Where $\lambda$is the wavelength.
As we know that the wavelength is the distance between identical points (adjacent crests) in the adjacent cycles of a waveform signal propagated in space or along a wire.
And the dimensional formula of wavelength i.e.${M^0}{L^1}{T^0}$
So putting the value of wavelength in the formula of wavenumber,
We get-
$k = \dfrac{1}{{{M^0}{L^1}{T^0}}}$
$\therefore k = {M^0}{L^{ - 1}}{T^0}$
Therefore, the dimensional formula of wave number,$k = {M^0}{L^{ - 1}}{T^0}$.
Hence, option B is correct.

Note- While solving this question, we must know that the SI unit of wave number is ${m^{ - 1}}$. Also one can get confused, so it should be clear that wavenumber is also known as propagation constant or angular wave number.