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Dual Nature of Radiation and Matter — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsDual Nature of Radiation and Matter1 Mark
Question
Relativistic corrections become neccssary when the expression for the kinetic energy $\frac{1}{2}\text{mv}^2$, becomes comparable with $mc^2,$ where m is the mass of the particle. At what de Broglie wavelength will relativistic corrections become important for an electron?
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Answer

  1. $\lambda =10^{-4}nm.$
  2. $\lambda =10^{-6}nm.$
Key concept : De $-$ Brogile or matter wave is independent of die charge on the material particle. It means, matter wave of de $-$ Broglie wave is associated with every moving particle $($whether charged or uncharged$)$.
The de $-$ Broglie wavelength at which relativistic corrections become important that the phase velocity of the matter waves can be greater than the speed of the light $(3 \times 10^8m/s).$
The wavelength of de $-$ Broglie wave is given by
$\lambda=\frac{\text{h}}{\text{p}}=\frac{\text{h}}{\text{mv}}$
Here $, h = 6.6 \times 10^{-34}Js$
and for electron $, m = 9 \times 10^{-31}\ kg$
To approach these types of problem we use hit and trial method by picking up each option one by one.
In option $(a), \lambda_1=10\text{nm}=10\times10^{-9}\text{m}=10^{-8}\text{m}$
$\Rightarrow\ \text{v}_1=\frac{6.6\times10^{-34}}{(9\times10^{-31})\times10^{-8}}$
$=\frac{2.2}{3}\times10^5\approx10^{5}\frac{\text{m}}{\text{s}}$
In option $(b), \lambda_2=10^{-1}\text{nm}=10^{-1}\times10^{-9}\text{m}=10^{-10}\text{m}$
$\Rightarrow\ \text{v}_2=\frac{6.6\times10^{-34}}{(9\times10^{-31})\times10^{-10}}\approx10^7\frac{\text{m}}{\text{s}}$
In option $(c), \lambda_3=10^{-4}\text{nm}=10^{-4}\times10^{-9}\text{m}=10^{-13}\text{m}$
$\Rightarrow\ \text{v}_3=\frac{6.6\times10^{-34}}{(9\times10^{-31})\times10^{-13}}\approx10^{10}\frac{\text{m}}{\text{s}}$
In option $(d), \lambda_4=10^{-6}\text{nm}=10^{-6}\times10^{-9}\text{m}=10^{-15}\text{m}$
$\Rightarrow\ \text{v}_4=\frac{6.6\times10^{-34}}{(9\times10^{-31})\times10^{-15}}\approx10^{12}\frac{\text{m}}{\text{s}}$
Thus, options $(c)$ and $(d)$ are correct as $v_3$ and $v_4$ is greater than $3\times10^{8}\text{m/s}.$

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