Question

The speed of the particle that can take discrete values is proportional to
$A.{{n}^{\dfrac{-3}{2}}}$
$B.{{n}^{-1}}$
$C.{{n}^{\dfrac{1}{2}}}$
$D.n$

Answer 1

We have to apply the concept that wavelength of the standing wave is related to the linear momentum of the particle with de Broglie relation. We have to use the fact that during propagation of waves, it oscillates up and down. During this oscillation, each segment sweeps out the loop. We have to relate the concept of loop with de Broglie relation.

Formula used:
We will use the following relations to solve this problem:-
$\lambda =\dfrac{h}{mv}$ and $n\lambda =2\pi a$.

Complete step by step answer:
We have to relate the speed of the particle with the number of loops. To find the relation we are going to use the following relation:-
$\lambda =\dfrac{h}{mv}$……………….. $(i)$
Where, $\lambda$ is de Broglie wavelength, $h$ is Planck’s constant, $m$ is the mass of the particle and $v$ is the speed of the particle during the motion.
We also know that $n\lambda =2\pi a$………………… $(ii)$
Where $a$ is the acceleration of the particle
From the equation $(ii)$ we get
$\lambda =\dfrac{2\pi a}{n}$…………………. $(iii)$
From equations $(i)$ and $(iii)$ we get,
$\dfrac{h}{mv}=\dfrac{2\pi a}{n}$
$v=\dfrac{nh}{2\pi am}$…………………….. $(iv)$
From the equation $(iv)$ we get,
$v\propto n$
That is, velocity is directly proportional to the number of loops.

So, the correct answer is “Option D”.

Additional Information:
The above problem is solved with the use of de Broglie relation. Therefore, we must have some basic concept of de Broglie relation. de Broglie relation of dual nature states that a matter particle moving with a velocity v can be treated as a wave of wavelength λ. This λ is called de-Broglie wavelength & it is defined as:
$\lambda =\dfrac{h}{p}=\dfrac{h}{mv}$.

Note:
In solving these types of problems we should apply the concept of loops in relation with wave motion. We should always begin our solution with de Broglie relation. We should also consider the fact that light has dual characteristics. Light wave is an electromagnetic wave which is produced by vibrating electric charge.