Single Choice

An electron of mass $$m$$ and magnitude of charge $$|e|$$ initially at rest gets accelerated by a constant electric field $$E$$. The rate of change of de-Broglie wavelength of this electron at time $$t$$ ignoring relativistic effects is:

A$$\dfrac {-h}{|e| Et^2}$$
Correct Answer
B$$-\dfrac {h}{|e| Et}$$
C$$-\dfrac {h}{|e| Et \sqrt t}$$
D$$\dfrac {|e| Et}{h}$$

Solution

$$P=\dfrac{h}{\lambda_D}$$

$$\lambda_D=\dfrac{h}{p}=\dfrac{h}{mv}$$

$$\because \dfrac{v}{t}=a\Rightarrow v=at$$

$$v=at$$

$$\because F=qE$$

$$ma=qE$$

$$\boxed{a=\dfrac{qE}{m}=\dfrac{eE}{m}}$$

So, $$v=\dfrac{|e|E}{m}t$$

$$\therefore \lambda_D=\dfrac{h}{m\left(\dfrac{|e|E}{m}\right)t}=\dfrac{h}{|e|Et}$$

Rate of change of de-Broglie wavelength is given by differentiating the wavelength w.r.t. time.
$$\boxed{\dfrac{d\lambda_D}{dt}=-\dfrac{h}{|e|Et^2}}$$

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