Question 266: Dual Nature of Matter and Radiation

Question

An electron of mass ' $m$ ' and charge ' $e$ ' initially at rest gets accelerated by a constant electric field ' E '. The rate of change of de-Broglie wavelength of the electron at time ' $t$ ' is
(Ignore relativistic effect)( $\mathrm{h}=$ Planck's constant)

Accepted Answer

Force on electron : $\mathrm{F}=\mathrm{eE}$
Since it's initially at rest, using Newton's $2^{	ext {nd }}$
Law: $\mathrm{a}=\frac{\mathrm{F}}{\mathrm{m}}=\frac{\mathrm{eE}}{\mathrm{m}}$
Velocity after time t :
$$ \begin{aligned} & v=u+a t \Rightarrow v=a t \ & v=\frac{e E}{m} t\,\,\,\,\,\,\,\, \ldots . .(\mathrm{u}=0)\end{aligned} $$
By de- Broglie wavelength formula.
$$ \lambda=\frac{\mathrm{h}}{\mathrm{p}}=\frac{\mathrm{h}}{\mathrm{mv}} \Rightarrow \lambda=\frac{\mathrm{h}}{\mathrm{~m}\left(\frac{\mathrm{eE}}{\mathrm{~m}} \cdot \mathrm{t}ight)} $$
$$ \begin{aligned} &	herefore \quad \lambda=\frac{\mathrm{h}}{\mathrm{eEt}}\ &	ext { Now differentiate } \lambda 	ext { w.r.t. } \mathrm{t}\ &\begin{array}{ll}  & \frac{\mathrm{d} \lambda}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{~h}}{\mathrm{eEt}}ight)=\frac{-\mathrm{h}}{\mathrm{eEt}^2} \ 	herefore \quad & \frac{\mathrm{~d} \lambda}{\mathrm{dt}}=-\frac{\mathrm{h}}{\mathrm{eEt}^2} \end{array} \end{aligned} $$

Answer

$-\frac{\mathrm{h}}{\mathrm{eEt}^2}$

Answer

$-\frac{\mathrm{eEt}}{\mathrm{h}}$

Answer

$\frac{-\mathrm{mh}}{\mathrm{eEt}^2}$

Answer

$-\frac{\mathrm{h}}{\mathrm{eE}}$

Dual Nature of Matter and Radiation

An electron of mass ' m ' and charge ' e ' initially at rest gets accelerated by a constant electric field ' E '. The rate of change of de-Broglie wavelength of the electron at time ' t ' is (Ignore relativistic effect)( \mathrm{h}= Planck's constant)

Choose the correct answer: