An electron is released from rest near an infinite non-conducting sheet of uniform charge density '−σ'. The rate of change of de-Broglie wavelength associated with the electron varies inversely as nᵗʰ power of time. The numerical value of n is:

 

⚙️ de-Broglie Wavelength vs Time – Charged Sheet Problem | Doubtify JEE

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📌 Question:

An electron is released from rest near an infinite non-conducting sheet of uniform negative charge density (−σ).
The rate of change of de-Broglie wavelength associated with the electron varies inversely as the nᵗʰ power of time.

👉 Find the numerical value of n.

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📸 Question Image:

🧠 Concept & Approach:

We combine Electrostatics + Modern Physics concepts:

• Electric field near infinite sheet:
  $E = \frac{σ}{2ε₀} \quad (\text{constant})$
  

• Force on electron:
  $F = -eE = \text{constant} \Rightarrow \text{uniform acceleration}$
  

• If acceleration is constant:
  $v \propto t \Rightarrow p = mv \propto t$
  

• de-Broglie wavelength:
  $\lambda = \frac{h}{p} \propto \frac{1}{t}$

• Therefore:
  $\frac{d\lambda }{dt}\propto -\frac{1}{t^{\mathrm{2}}}$

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🔢 Step-by-Step Solution:

✅ Step 1: Use field of sheet
Electric field = $\frac{σ}{2ε₀}$ → constant

✅ Step 2: Constant force → constant acceleration
From rest:
$v = at \Rightarrow p = mv \propto t$

✅ Step 3: de-Broglie wavelength
$\lambda = \frac{h}{p} \Rightarrow \lambda \propto \frac{1}{t}$

✅ Step 4: Rate of change
$\frac{d\lambda}{dt} \propto \frac{d}{dt}\left(\frac{1}{t}\right) = -\frac{1}{t^2}$

So,
$\frac{d\lambda}{dt} \propto \frac{1}{t^n}$⇒ n = 2

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✅ Final Answer:

n = 2

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🖼️ Solution Image:

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🎥 Watch the Step-by-Step Video Solution:

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🧠 Pro Tip:

In de-Broglie + Electrostatics combo questions:

• Electric fields from symmetric charge distributions are often constant

• Constant force → Uniform acceleration → Linear momentum-time relationship

• de-Broglie formulas help translate mechanics into wave behavior

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