Question

An electron of mass m and charge e initially at rest gets accelerated by a constant electric Field E. Find the rate of change of de-Broglie wavelength of this electron at time t ?(ignore relativistic effects)

Answer 1

The force on the electron due to the electric field in,

$F-eE$. [Electrons have charge $q=-e$]

now, as F = ma, acceleration of electron :-

$a=\frac{F}{m}=\frac{-eE}{m}$

initially electron was at rest (u = 0) so at a time 't' its velocity would be :-

$v=u+at\Rightarrow v=\frac{-eE}{m}t$

Thus, at a time 't' its momentum would be,

$p=mv=-eEt$

and its de-Broglie wavelength :

$\lambda =\frac{h}{p}=\frac{-h}{eEt}$

Thus rate of change of de-Broglie wavelength,

$\frac{d\lambda }{dt}=\frac{h}{eE{t}^2}$ $[\frac{d}{dt}\left(\frac{1}{t}\right)=\frac{-1}{t^2}]$