Question

The ratio of the energies of a moving particle and a photon is 1/100. Their velocities are in the ratio 1/10, then the ratio of their de – Broglie wavelengths is
A) 5:1
B) 1:5
C) 10:1
D) 1:10

Answer 1

Hint: De Broglie stated that just like light matter also shows wave-particle duality, since light behaves both as a wave and as a particle, which means that the matter would also follow the same equation for wavelength as light.

Formula used:
$\lambda = \dfrac{h}{p}$

Complete answer:
It is known that matter has a dual nature of wave – particles. De – Broglie waves is known as the property of a material object which varies with time or space while behaving similar to the waves.
Mathematically de – Broglie wavelength is given as,
$\lambda = \dfrac{h}{p} = \dfrac{h}{{mv}}$
Where ‘m’ is the mass, ‘v’ is the velocity of the particle and ‘h’ is the Planck’s constant. Planck's constant is known as the quantum of electromagnetic action which relates a photon's energy to its frequency.
Now, $\lambda = \dfrac{h}{p} = \dfrac{h}{{mv}}$
$ \Rightarrow \lambda = \dfrac{k}{{\sqrt {2me} }}$
Now, the ratio of the their de – Broglie wavelengths is given as,
$\dfrac{{{\lambda _1}}}{{{\lambda _2}}} = \sqrt {\dfrac{{e{}_2}}{{{e_1}}}} $
$ \Rightarrow \dfrac{{{\lambda _1}}}{{{\lambda _2}}} = \sqrt {\dfrac{1}{{\dfrac{1}{{100}}}}} = 10:1$

So, the correct answer is “Option C”.

Additional Information:
The dual nature of light explains that, in some of the experiments, light behaves as a wave and as a particle. The particle nature of light explains the path of the light in which it travels i.e. light travels in straight lines. The wave nature of the light explains the diffraction or bending of light around the object.

Note:
De – Broglie wavelength is the chief concept for studying quantum mechanics. The wavelength (λ) which is associated with an object in relation to its momentum ‘p’ or ‘mv’ and mass ‘m’ is known as the de Broglie wavelength. A particle’s de Broglie wavelength is generally inversely proportional to its force.