An electron is released from rest near an infinite non-conducting sheet of uniform charge density $-\sigma$. The rate of change of de-Broglie wave length associated with the electron varies inversely as $n^{th}$ power of time. The numerical value of $n$ is ____.
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Solution Steps
1
Identify the Electric Field
The electric field $E$ near an infinite non-conducting sheet is uniform and given by:
$$E = \frac{\sigma}{2\epsilon_0}$$
2
Calculate Acceleration and Velocity
The force on the electron is $F = eE$. Since the field is uniform, the acceleration $a$ is constant:
$$a = \frac{eE}{m} = \frac{e\sigma}{2m\epsilon_0}$$
Since it starts from rest ($u=0$), the velocity at time $t$ is $v = at$.
3
Express de-Broglie Wavelength
The de-Broglie wavelength $\lambda$ is defined as:
$$\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{m(at)}$$
This shows $\lambda \propto \frac{1}{t}$ or $\lambda = k t^{-1}$, where $k$ is a constant.
4
Find the Rate of Change
The “rate of change of wavelength” is the derivative with respect to time:
$$\frac{d\lambda}{dt} = \frac{d}{dt}(k t^{-1}) = -k t^{-2}$$
The magnitude of the rate of change is proportional to $\frac{1}{t^2}$.
5
Determine the value of n
The problem states the rate varies inversely as the $n^{th}$ power of time ($\propto \frac{1}{t^n}$):
Comparing $\frac{1}{t^2}$ with $\frac{1}{t^n}$, we get $n = 2$.
Final Answer: 2
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Theory
1. de-Broglie Hypothesis
The de-Broglie hypothesis proposes that every moving particle exhibits wave-like properties. The wavelength $\lambda$ associated with a particle is inversely proportional to its linear momentum $p$, expressed as $\lambda = h/p$. This duality is fundamental to quantum mechanics. For a particle accelerated by a constant force, its momentum increases linearly with time ($p = Ft$), causing the de-Broglie wavelength to decrease over time. This relationship is crucial in electron microscopy and particle physics experiments.
2. Electric Field of Infinite Sheets
According to Gauss’s Law, the electric field generated by an infinite non-conducting sheet with surface charge density $\sigma$ is $E = \sigma/2\epsilon_0$. Key characteristics include that the field is uniform, meaning its magnitude and direction do not change with distance from the sheet. This uniformity ensures that any charged particle placed in this field experiences a constant force $F = qE$, leading to uniform acceleration $a = qE/m$ as per Newton’s second law.
3. Kinematics of Charged Particles
When a charged particle is released from rest in a uniform electric field, it follows the laws of uniformly accelerated motion. The velocity $v$ at any time $t$ is $v = at$, and the displacement is $s = \frac{1}{2}at^2$. In this specific problem, since the electron is released from rest, its instantaneous momentum is $p(t) = m \cdot (a \cdot t)$. The fact that velocity increases linearly with time directly dictates the power-law decay of the associated de-Broglie wavelength.
4. Rates of Change in Physics
In calculus-based physics, the “rate of change” of a quantity $y$ refers to its derivative $dy/dt$. For power-law relationships where $y \propto t^m$, the rate of change $dy/dt$ will be proportional to $t^{m-1}$. Here, $\lambda \propto t^{-1}$, so its rate of change involves $t^{-2}$. Understanding these power relationships allows for quick analysis of how physical observables evolve, which is a common theme in JEE Main “Numerical Value” type questions.
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FAQs
1
Why does $n=2$ and not $n=1$?
Because the question asks for the rate of change (derivative) of wavelength, not the wavelength itself. Wavelength $\propto t^{-1}$, but its derivative $\propto t^{-2}$.
2
Does the distance from the sheet matter?
No, for an infinite sheet, the electric field is uniform and does not depend on distance.
3
What is $\sigma$?
$\sigma$ is the surface charge density, representing charge per unit area on the sheet.
4
Will the answer change for a proton?
No, while the magnitude of acceleration would be different due to mass, the power relationship with time ($n=2$) would remain identical.
5
Is the electron moving towards or away from the sheet?
Since the sheet is negative ($-\sigma$) and the electron is negative, it will be repelled and move away.
6
What if the field was not uniform?
Then $n$ would not be a simple integer, as acceleration would vary with position/time.
7
What does ‘non-conducting’ imply?
It means the charge is distributed throughout the sheet (or on the surface) and won’t move even if an external field is applied.
8
How is $\lambda$ related to kinetic energy?
$\lambda = h/\sqrt{2mK}$. Since $K = \frac{1}{2}mv^2 \propto t^2$, we find $\lambda \propto 1/t$.
9
What is the significance of $n$ in the inverse variation?
$n$ is the exponent in the denominator. $\propto 1/t^n$ is the same as $\propto t^{-n}$.
10
Is this a relativistic calculation?
No, JEE Main assumes non-relativistic speeds ($v \ll c$) unless stated otherwise.
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