Question

The energy of a photon is given by $E=h\nu $, where $\nu $ is the frequency of radiation. Use this equation to get the dimensional formula of ‘h’.

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.

Complete Step-by-Step solution:
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 = $\dfrac{[{{M}^{1}}{{L}^{2}}{{T}^{-2}}]}{[{{M}^{0}}{{L}^{0}}{{T}^{-1}}]}=[{{M}^{1}}{{L}^{2}}{{T}^{-1}}]$

Additional information:
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).

Note: Don’t try to remember the dimensional formula. You may get confused. Always express the derived quantities in terms of the fundamental quantities and you will get the dimension of quantities.