Question

The dimensions of Planck’s constant is same as that of
A. Angular momentum
B. Linear momentum
C. Work
D. Coefficient of viscosity

Answer 1

Hint: Convert the derived physical quantities into fundamental physical quantities. First try to find the dimensional formula for energy and frequency by breaking them into components of fundamental quantities. So, we can find the dimensional formula of h from the dimension of these two quantities. Then compare it with the dimensions of the given quantities in the answer.

Complete step-by-step answer:
All the derived physical quantities can be expressed in terms of the fundamental quantities. The derived units are dependent on the 7 fundamental quantities. Fundamental units are mutually independent of each other.
Dimension of a physical quantity is the power to which the fundamental quantities are raised to express that physical quantity.
Now, Energy can be expressed as,
$E=m{{c}^{2}}$
Where m is the mass and c is the velocity of light.
Now. dimension of mass= $\left[ {{M}^{1}}{{L}^{0}}{{T}^{0}} \right]$
And, dimension of velocity= $\left[ {{M}^{0}}{{L}^{1}}{{T}^{-1}} \right]$
So, dimension of energy= $\left[ {{M}^{1}}{{L}^{0}}{{T}^{0}} \right]\times {{\left[ {{M}^{0}}{{L}^{1}}{{T}^{-1}} \right]}^{2}}=\left[ {{M}^{1}}{{L}^{0}}{{T}^{0}} \right]\times \left[ {{M}^{0}}{{L}^{2}}{{T}^{-2}} \right]=\left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}} \right]$
Now, frequency is the number of complete wave cycles formed in a unit time. given by,
$\nu =\dfrac{1}{T}$
Where, $\nu $ is the frequency of radiation and T is the time period.
Now, dimension of frequency $\nu $ = $\left[ {{M}^{0}}{{L}^{0}}{{T}^{-1}} \right]$

Dimension of h = $\dfrac{[{{M}^{1}}{{L}^{2}}{{T}^{-2}}]}{[{{M}^{0}}{{L}^{0}}{{T}^{-1}}]}=[{{M}^{1}}{{L}^{2}}{{T}^{-1}}]$
Linear momentum can be expressed as, p=mv
Where m is mass and v is the velocity
Dimension of linear momentum = $\left[ {{M}^{1}}{{L}^{1}}{{T}^{-1}} \right]$
Angular momentum can be expressed as, L=rp
Dimension of angular momentum = $\left[ {{M}^{1}}{{L}^{2}}{{T}^{-1}} \right]$
Dimension of work = $\left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}} \right]$
Dimension of coefficient of viscosity = $\left[ {{M}^{1}}{{L}^{-1}}{{T}^{-1}} \right]$
So, the dimension of Planck’s constant is the same as the dimension of angular momentum.
The correct answer is (A)

Note: The quantity h is called the Planck’s constant. It is a universal constant with its value $h=6.626\times {{10}^{34}}\text{ Joule second}$.
The unit for energy is Joule.
The unit for frequency is hertz (Hz).