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JEE Main 201815 April
Exam
JEE Main
Year
2018
Date
15 April
Subject
Physics
Chapter
Dual Nature of Matter

JEE Main 2018 · 15 April · Physics · Dual Nature of Matter

Two electrons are moving with non-relativistic speeds perpendicular...

Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de brogile wavelengths are λ1{\lambda}_{1} and λ2{\lambda}_{2}, their de brogile wavelength in the frame of reference attached to their centre of mass is:

Options

  1. A1λCM=1λ1+1λ2\dfrac{1}{{\lambda}_{CM}} = \dfrac{1}{{\lambda}_{1}} + \dfrac{1}{{\lambda}_{2}}
  2. BλCM=2λ1λ2λ12+λ22{\lambda}_{CM} = \dfrac{2{\lambda}_{1}{\lambda}_{2}}{\sqrt{{{\lambda}_{1}}^{2} + {{\lambda}_{2}}^{2}}}
  3. CλCM=λ1=λ2{\lambda}_{CM} = {\lambda}_{1} = {\lambda}_{2}
  4. DλCM=λ1+λ22{\lambda}_{CM} = \dfrac{{\lambda}_{1} + {\lambda}_{2}}{2}

Correct answer

B. λCM=2λ1λ2λ12+λ22{\lambda}_{CM} = \dfrac{2{\lambda}_{1}{\lambda}_{2}}{\sqrt{{{\lambda}_{1}}^{2} + {{\lambda}_{2}}^{2}}}

Step-by-step solution

Momentum of each electron \dfrac{h}{{\lambda}_{1}}\overset{^}{i}and\dfrac{h}{{\lambda}_{2}}\overset{^}{j}

Velocity of centre of mass

{V}_{cm} = \dfrac{h}{2m{\lambda}_{1}}\overset{^}{i} + \dfrac{h}{2m{\lambda}_{2}}\overset{^}{j}

Velocity of 1st{1}^{st} particle about centre of mass

{V}_{cm} = \dfrac{h}{2m{\lambda}_{1}}\overset{^}{i} - \dfrac{h}{2m{\lambda}_{2}}\overset{^}{j}

λcm=hh24λ12+h24λ22=2λ1λ2λ12+λ22{\lambda}_{cm} = \dfrac{h}{\sqrt{\dfrac{{h}^{2}}{4{\lambda}_{1}^{2}} + \dfrac{{h}^{2}}{4{\lambda}_{2}^{2}}}} = \dfrac{2{\lambda}_{1}{\lambda}_{2}}{\sqrt{{\lambda}_{1}^{2} + {\lambda}_{2}^{2}}}
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