Question

Which of the following do not have the same dimensional formula as the velocity?
(Given that ${\mu _o} = $ permeability of free space, ${\varepsilon _o} = $ permittivity of free space, $\upsilon = $ frequency, $\lambda = $ wavelength, $P = $ pressure, $\rho = $ density, $\omega = $ angular frequency, $k = $ wave number}
A) $\dfrac{1}{{\sqrt {{\mu _o}{\varepsilon _o}} }}$
B) $\upsilon \lambda $
C) $\sqrt {P/\rho } $​
D) $\omega k$

Answer 1

Hint: In order to solve this question, one should have the knowledge of dimensions of the Physical Quantities. First, determine the dimensions of velocity and compare it to the dimension of the given four options and then find out which one is different from the dimensions of velocity.

Complete step by step answer:
The dimensions of velocity is given as $[L{T^{ - 1}}]$
Here in option A we have, $\dfrac{1}{{\sqrt {{\mu _o}{\varepsilon _o}} }}$
As we know, ${F_c} = \dfrac{1}{{4\pi {\varepsilon _o}}} \times \dfrac{{{q_1}{q_2}}}{{{r^2}}}$
Here, ${\varepsilon _o}$ is permittivity of vacuum, its dimensions would be given by $[M{L^{ - 3}}{T^4}{I^2}]$
Now, ${\mu _o} = 4\pi \times {10^{ - 7}}N{A^{ - 2}}$
Now, its dimensions would be given by $[M{L^{ - 3}}{T^{ - 2}}{I^{ - 2}}]$
So, the dimensions of $\dfrac{1}{{\sqrt {{\mu _o}{\varepsilon _o}} }}$ is given by $[L{T^{ - 1}}]$ which is same as the dimension of velocity.
In option B we have, $\upsilon \lambda $

The dimensions of frequency $\upsilon $is $[{T^{ - 1}}]$
And the dimension of $\lambda $ is $[L]$
So, the dimensions of $\upsilon \lambda $ is $[L{T^{ - 1}}]$ which is the same as the dimension of velocity.
In option C we have, $\sqrt {P/\rho } $

The dimensions of momentum $P$ is given by $[ML{T^{ - 1}}]$
And the dimensions of $\rho $ is given by $[M{L^3}{T^{}}]$
So, the dimensions of $\rho $ is $[L{T^{ - 1}}]$ which is the same as the dimension of velocity.
Now, option D we have, $[\omega k]$

Dimensions of $[\omega k] = \dfrac{1}{T} \times \dfrac{1}{L} = [{T^{ - 1}}{L^{ - 1}}]$ but the dimension for velocity is $[L{T^{ - 1}}]$ .
So, the dimensions of $[\omega k]$ is different from the dimensions of velocity.

Hence, $[\omega k]$ do not have the same dimensional formula as the velocity.
Therefore, option (D) is the correct option.

Note: Dimensions of Physical Quantities describes the nature of the physical quantity. It is described as the powers to which fundamental quantities are raised in order to represent a specific quantity. Dimension of a physical quantity is enclosed in []. There are seven fundamental quantities which are enclosed in [] to give the dimension of a physical quantity.