Question 3: Dual Nature of Matter and Radiation

Question

An electron of mass $$m$$ and charge $$e$$ initially at rest gets accelerated by a constant field $$2 E$$. The rate of change of de-Broglie wavelength of this electron at time $$t$$ ignoring relativistic effects is

Accepted Answer

To determine the rate of change of the de-Broglie wavelength of the electron at time $$t$$, we need to start with the expression for the de-Broglie wavelength, which is given by
$$\lambda = \frac{h}{p}$$
where $$h$$ is Planck's constant and $$p$$ is the momentum of the electron.
Since the electron experiences a constant electric field of $$2E$$, it undergoes a uniform acceleration. The force $ F $ on the electron is given by
$$F = e \cdot 2E = 2eE$$
This force causes the electron to accelerate, so the acceleration $a$ is given by
$$a = \frac{F}{m} = \frac{2eE}{m}$$
Initially, the electron is at rest, so its velocity $v$ at time $t$ can be obtained using the equation of motion:
$$v = u + at = 0 + \left(\frac{2eE}{m}\right) t = \frac{2eE t}{m}$$
The momentum $$p$$ of the electron at time $$t$$ is given by
$$p = mv = m \left( \frac{2eE t}{m} \right) = 2eE t$$
Now substituting the expression for momentum into the de-Broglie wavelength equation, we get
$$\lambda = \frac{h}{p} = \frac{h}{2eE t}$$
To find the rate of change of de-Broglie wavelength with respect to time, we take the derivative of $$\lambda$$ with respect to $$t$$:
$$\frac{d\lambda}{dt} = \frac{d}{dt} \left( \frac{h}{2eE t} \right) = h \cdot \frac{d}{dt} \left( \frac{1}{2eE t} \right) = h \cdot \left( \frac{-1}{(2eE t)^2} \cdot 2eE \right) = \frac{-h}{2eE} \cdot \frac{1}{t^2} \cdot 2eE = \frac{-h}{t^2}$$
Thus, the rate of change of de-Broglie wavelength of the electron at time $$t$$ is
$$\frac{d\lambda}{dt} = \frac{-h}{2eE t^2}$$
So the correct answer is:
Option A: $$-\frac{h}{\left(2eEt^2\right)}$$

Answer

$$-\frac{h}{\left(2 e E t^2\right)}$$

Answer

$$\frac{-e h t}{2 E}$$

Answer

$$\left(\frac{-2 m h}{e E t^2}\right)$$

Answer

$$\left(\frac{-h}{e E}\right)$$

Dual Nature of Matter and Radiation

An electron of mass $m and charge e initially at rest gets accelerated by a constant field 2 E. The rate of change of de-Broglie wavelength of this electron at time t$ ignoring relativistic effects is

Choose the correct answer: