Question

Dual nature of matter was proposed by de Broglie in 1923, it was experimentally verified by Davisson and Germer by diffraction experiment. Wave character of matter has significance only for microscopic particles. The de Broglie wavelength or wavelength of matter-wave can be calculated using the following relation: \[\lambda = \dfrac{h}{{mv}}\;\] where' and 'v' is the mass and velocity of the particle. De Broglie hypothesis suggested that electron waves were being diffracted by the target, much as X-rays are diffracted by planes of atoms in the crystals.
Planck’s constant has the same dimension as that of:
(A) Work
(B) Energy
(C) Power
(D) Angular momentum

Answer 1

Hint: First find out the dimensional formula of the Planck’s constant using the formula: $E = h\nu $. The dimensional formula of energy is ${M^1}{L^2}{T^{ - 2}}$ and that of frequency is ${T^{ - 1}}$. Then find the dimensional formulas of various terms given in the options and check which one has this formula similar to that of the Planck’s constant.

Complete step by step solution:
-First of all, we will talk about dual nature by taking the example of light.
If we are taking the case of light we should know that the wave character of light explains some phenomenon like diffraction and interference. While certain other phenomena such as black body radiation and photoelectric effect can be explained only on the basis of its particle nature. Hence, we can say that light has dual nature as both a wave and a particle. Studies to prove the above-given conclusions were done by Einstein in 1905.
- We will now see what Planck’s constant is.
We all know that radiation, such as light, is emitted, transmitted, and absorbed in discrete energy packets, or quanta, determined by the frequency of the radiation and the value of Planck’s constant. The energy ‘E’ of each quantum, or each photon, equals Planck’s constant ‘h’ times the radiation frequency and is symbolized by the Greek letter ‘ν’. Hence we can write energy in mathematical forms to be:
$E = h\nu $ (1)
Now, we should know that the value of Planck’s constant in metre-kilogram-second units is defined as exactly $6.626 \times {10^{ - 34}}$ Joule second.
-From equation (1) we can say that the Planck’s constant (h) can be defined as a proportionality constant that relates the energy (E) of a photon to the frequency (ν) of its associated electromagnetic wave.
Mathematically, using eqn. (1) we can write ‘h’ to be:
Planck’s constant (h) = \[\dfrac{{Energy\left( E \right)}}{{Frequency\left( \nu \right).}}{\text{ }}\]\[\] (2)
-Now using equation (2) and the dimensional formulas of energy and frequency we can find the dimensional formula of Planck’s constant.
-First we will find out the dimensional formula of energy: The unit of energy is Joule (J) which is also equal to $kg{\left( {\dfrac{m}{{\sec }}} \right)^2}$.
So from this, the dimensional formula of energy can be written as: \[{{\text{M}}^{\text{1}}}{{\text{L}}^{\text{2}}}{{\text{T}}^{ - {\text{2}}}}\]
-Now we will write down the dimensional Formula of frequency: Its unit is ${\sec ^{ - 1}}$ and so its dimensional formula would be: \[{{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^{ - {\text{1}}}}\]
-We will now put these dimensional values of energy and frequency in equation (2):
$h = \dfrac{E}{\nu }$
= $\dfrac{{{M^1}{L^2}{T^{ - 2}}}}{{{T^{ - 1}}}}$ = ${M^1}{L^2}{T^{ - 1}}$
Hence we now know that the dimensional formula of Planck’s constant is: ${M^1}{L^2}{T^{ - 1}}$
-We will now check the dimensional formulas of all the options to see which one has a dimensional formula similar to the Planck’s constant.
-For (A) Work: We know that work is the product of force and displacement. And the dimensional formula of force is ${M^1}{L^1}{T^{ - 2}}$ and that of displacement is ${L^1}$. So, for work:
Work = Force\[ \times \]Displacement
= ${M^1}{L^1}{T^{ - 2}} \times {L^1}$
= ${M^1}{L^2}{T^{ - 2}}$
Here we have seen that the dimensional formula of work is not the same as that of Planck’s constant. So, this option is not correct.
-For (B) Energy: We know that energy is the product of force and displacement and also that work is also a form of energy. Hence both work and energy have the same dimensional formula which is ${M^1}{L^2}{T^{ - 2}}$. Even the dimensional formula of energy is not the same as that of Planck’s constant so this option is also not correct.
-For (C) Power: We know that power is defined as work per unit time and the dimensional formula of work is ${M^1}{L^2}{T^{ - 2}}$ and that of time is ${T^1}$. So, for power:
\[Power = \dfrac{{Work}}{{time}}\]
$Power = \dfrac{{{M^1}{L^2}{T^{ - 2}}}}{{{T^1}}}$
= ${M^1}{L^2}{T^{ - 3}}$
Here also we have seen that the dimensional formula of power is not the same as that of Planck’s constant. So, this option is also not correct.
-For (D) Angular momentum: Angular momentum is basically the rotational equivalent of linear momentum and mathematically written as: $mvr = \dfrac{{nh}}{{2\pi }}$. For its dimensional formula:
Angular momentum = mvr
= $M \times L{T^{ - 1}} \times L$
= ${M^1}{L^2}{T^{ - 1}}$
Here we can see that the dimensional formula of angular momentum is the same as that of the Planck’s constant and that is ${M^1}{L^2}{T^{ - 1}}$.

So, the correct option will be: (D) Angular momentum.

Note: We should know that momentum is the product of mass, velocity and distance of the object from the central point. It should be noted that any object moving with mass possesses momentum. The only difference between linear and angular momentum is that angular momentum deals with rotating or spinning objects.