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  • An electron (mass m ) with an initial velocity <img alt="v=v_0 \hat{\mathrm{i}}\left(v_0>0\right)" src="https://entrancecorner.oncodecogs.com/gif.latex?v%3Dv_0%20%5Chat%7B%5Cmathrm%7Bi%7D%7D%5Cl

An electron (mass m ) with an initial velocity v=v_0 \hat{\mathrm{i}}\left(v_0>0\right) is in an electric field \mathrm{E}=-\mathrm{E}_0 \hat{\mathrm{i}}\left(\mathrm{E}_0=\right. \text{constant }\left.>0\right). Its de-Broglie wavelength at time t is given by:

 

Option: 1

\frac{\lambda_0}{\left(1+\frac{e E_0}{m} \frac{t}{v_0}\right)}


Option: 2

\lambda_0\left(1+\frac{e E_0 t}{m v_0}\right)


Option: 3

\quad \lambda_0


Option: 4

\quad \lambda_0 \mathrm{t}


Answers (1)

best_answer

Initial de-Broglie wavelength of electron, \lambda_0=\frac{\mathrm{h}}{\mathrm{mv}_0}

Force on electron in electric field, \mathrm{F}=-\mathrm{eE}=-\mathrm{e}\left[-\mathrm{E}_0 \hat{\mathrm{i}}\right]=\mathrm{eE}_0 \hat{\mathrm{i}}

Acceleration of electron, a=\frac{F}{m}=\frac{e E_0 \hat{\mathrm{i}}}{\mathrm{m}}a=\frac{F}{m}=\frac{e E_0 \hat{\mathrm{i}}}{\mathrm{m}}

Velocity of electron after time t, v=v_0 \hat{i}+\left(\frac{e E_0 \hat{i}}{m}\right) t=\left(v_0+\frac{e E_0}{m} t\right) \hat{i}=v_0\left(1+\frac{e E_0}{m_0} t\right) \hat{i}

de-Broglie wavelength associated with electron at time t is \lambda=\frac{\mathrm{h}}{\mathrm{mv}}
\Rightarrow=\frac{h}{m\left[v_0\left(1+\frac{e E_0}{m_0} t\right)\right]}=\frac{\lambda_0}{\left[1+\frac{e E_0}{m v_0} t\right]} \quad \quad \quad \quad \left[\because \lambda_0=\frac{\mathrm{h}}{\mathrm{mv}_0}\right]

 

Posted by

shivangi.bhatnagar

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