Question

If the de-Broglie wavelength of a particle of mass m is 100 times its velocity then its value in terms of its mass (m) and Planck’s constant (h) is
A) $\dfrac{1}{{10}}\sqrt {\dfrac{{\text{m}}}{{\text{h}}}}$
B) $10\sqrt {\dfrac{{\text{h}}}{{\text{m}}}}$
C) $\dfrac{1}{{10}}\sqrt {\dfrac{{\text{h}}}{{\text{m}}}}$
D) $10\sqrt {\dfrac{{\text{m}}}{{\text{h}}}}$

Answer 1

Using the de-Broglie equation we will first get the relation between velocity and wavelength. We will put the given condition to calculate the value of velocity and hence calculate the final relation between velocity and wavelength for the given condition.

Formula used:
$\lambda = \dfrac{{\text{h}}}{{{\text{mv}}}}$
Here $\lambda$ is wavelength, m is the mass, v is the velocity and h is the planck's constant.

Complete step by step solution:
According to de-Broglie equation the relation between velocity and wavelength will be given as:
${\text{ }}\lambda = \dfrac{{\text{h}}}{{{\text{mv}}}}{\text{ }}$
We have been given that the wavelength is 100 times than that of velocity. So we will get:
$\lambda = 100{\text{v}}$
Substituting the value in the de-Broglie formula we will get:
$100{\text{v}} = \dfrac{{\text{h}}}{{{\text{mv}}}}$
Rearranging the above equation we will get:
${{\text{v}}^2} = \dfrac{1}{{100}} \times \dfrac{{\text{h}}}{{\text{m}}}$
Taking the under root we will get:
${\text{v}} = \sqrt {\dfrac{1}{{100}} \times \dfrac{{\text{h}}}{{\text{m}}}}$
We can write it as:
${\text{v}} = \dfrac{1}{{10}}\sqrt {\dfrac{{\text{h}}}{{\text{m}}}}$
Again using the condition given in question in the formula:
$\lambda = 100 \times \dfrac{1}{{10}}\sqrt {\dfrac{{\text{h}}}{{\text{m}}}}$
On simplification we will get:
$\lambda = 10\sqrt {\dfrac{{\text{h}}}{{\text{m}}}}$

Hence, the correct option is B.

Note:
De Broglie states that the matter has both wave and particle-like properties. That is, any matter has wavelength just like a wave and momentum just like a particle. De Broglie has given the relation between wavelength and momentum which we have used above. Momentum is the product of mass and velocity of a body. The mass of the body is lesser will be its wavelength and the reverse is also true. That is why we cannot observe the wavelength of a moving ball because of its higher mass; its wavelength is almost negligible but this does not mean that it does not have wavelength.