Two electrons are moving with non-relativisti | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the momenta of the two electrons be $\vec{p}_1 = \frac{h}{\lambda_1} \hat{i}$ and $\vec{p}_2 = \frac{h}{\lambda_2} \hat{j}$.Since both are ele

Vedclass · Accepted Answer

$\lambda_{CM} = \frac{2\lambda_1\lambda_2}{\sqrt{\lambda_1^2 + \lambda_2^2}}$

Vedclass · Answer

(C) Let the momenta of the two electrons be $\vec{p}_1 = \frac{h}{\lambda_1} \hat{i}$ and $\vec{p}_2 = \frac{h}{\lambda_2} \hat{j}$.Since both are electrons,they have the same mass $m$.The velocity of the centre of mass is $\vec{V}_{CM} = \frac{\vec{p}_1 + \vec{p}_2}{2m} = \frac{h}{2m\lambda_1} \hat{i} + \frac{h}{2m\lambda_2} \hat{j}$.The velocity of the first electron relative to the centre of mass is $\vec{v}_{1,CM} = \vec{v}_1 - \vec{V}_{CM} = \frac{\vec{p}_1}{m} - \frac{\vec{p}_1 + \vec{p}_2}{2m} = \frac{\vec{p}_1 - \vec{p}_2}{2m} = \frac{h}{2m\lambda_1} \hat{i} - \frac{h}{2m\lambda_2} \hat{j}$.The momentum of the electron in the $CM$ frame is $\vec{p}_{CM} = m \vec{v}_{1,CM} = \frac{h}{2\lambda_1} \hat{i} - \frac{h}{2\lambda_2} \hat{j}$.The magnitude of this momentum is $p_{CM} = \sqrt{(\frac{h}{2\lambda_1})^2 + (-\frac{h}{2\lambda_2})^2} = \frac{h}{2} \sqrt{\frac{1}{\lambda_1^2} + \frac{1}{\lambda_2^2}} = \frac{h}{2} \frac{\sqrt{\lambda_1^2 + \lambda_2^2}}{\lambda_1 \lambda_2}$.The de Broglie wavelength in the $CM$ frame is $\lambda_{CM} = \frac{h}{p_{CM}} = \frac{2\lambda_1\lambda_2}{\sqrt{\lambda_1^2 + \lambda_2^2}}$.

Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de Broglie wavelengths are $\lambda_1$ and $\lambda_2$,their de Broglie wavelength in the frame of reference attached to their centre of mass is

Similar Questions

What is the de-Broglie wavelength associated with an electron moving with a speed of $6.4 \times 10^{6} \ m/s$ (in $nm$)? $(m_{e} = 9.11 \times 10^{-31} \ kg, h = 6.63 \times 10^{-34} \ J \cdot s)$

At room temperature $(27^{\circ}C)$ and $1$ atmospheric pressure,the de Broglie wavelength associated with a helium atom is compared with the average distance between two atoms. Which of the following relations is correct?

When the kinetic energy of an electron is increased,the wavelength of the associated wave will:

The speed of electrons in a scanning electron microscope is $1 \times 10^{7} \, m/s$. If protons having the same speed are used instead of electrons,then the resolving power of the scanning proton microscope will be changed by a factor of:

Vedclass Test Series

Exam Paper Generator

Online Exam Module