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) ($h$ = Planck's constant) ______.

• A. $-\frac{h}{e E t^2}$
• B. $-\frac{e E t}{h}$
• C. $-\frac{m h}{e E t^2}$
• D. $-\frac{h}{e E}$

Hint:

It's fascinating to note that the rate of change of the de-Broglie wavelength in a uniform electric field is completely independent of the mass of the particle! Only its charge matters.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The de-Broglie wavelength is $\lambda = \frac{h}{p}$. We need to find the rate of change $\frac{d\lambda}{dt}$.

Step 2: Formula Application:
Force $F = eE = ma$, so acceleration $a = \frac{eE}{m}$. Velocity at time $t$ (starting from rest) is $v = at = \frac{eEt}{m}$. Momentum $p = mv = m \left( \frac{eEt}{m} \right) = eEt$.

Step 3: Explanation:
Substitute $p$ into the wavelength formula: $\lambda = \frac{h}{eEt}$. To find the rate of change, differentiate with respect to $t$: $\frac{d\lambda}{dt} = \frac{d}{dt} \left( \frac{h}{eE} \cdot t^{-1} \right) = \frac{h}{eE} \cdot (-1) t^{-2} = -\frac{h}{e E t^2}$.

Step 4: Final Answer:
The rate of change is $-\frac{h}{e E t^2}$.