Rate of Change of de Broglie Wavelength Problem

❓ Question

An electron is released from rest near an infinite non-conducting sheet of uniform charge density:

$$-\sigma$$

The rate of change of de-Broglie wavelength associated with the electron varies inversely as:

$$n^{th}$$

power of time.

The numerical value of:

$$n$$

is ______.

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🖼 Question Image

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✍️ Short Explanation

This problem is based on:

👉 de-Broglie wavelength
👉 Motion under constant electric field
👉 Momentum-wavelength relation.

Main idea:

$$\lambda=\frac{h}{p}$$

and electric field due to infinite sheet is constant.

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🔷 Step 1 — Electric Field Due to Infinite Sheet 💯

For an infinite non-conducting sheet:

$$E=\frac{\sigma}{2\varepsilon_0}$$

Since charge density is uniform:

$$E=\text{constant}$$

Hence force on electron:

$$F=eE$$

is also constant.

Therefore acceleration:

$$a=\frac{F}{m}$$

is constant.

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🔷 Step 2 — Momentum Variation with Time

Electron starts from rest.

For constant acceleration:

$$v\propto t$$

Momentum:

$$p=mv$$

Hence:

$$p\propto t$$

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🔷 Step 3 — Use de-Broglie Relation

de-Broglie wavelength:

$$\lambda=\frac{h}{p}$$

Since:

$$p\propto t$$

therefore:

$$\lambda\propto\frac1t$$

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🔷 Step 4 — Differentiate with Respect to Time

Given:

$$\lambda\propto t^{-1}$$

Differentiate:

$$\frac{d\lambda}{dt}\propto -t^{-2}$$

Thus:

$$\left|\frac{d\lambda}{dt}\right| \propto\frac1{t^2}$$

Hence:

$$n=2$$

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🔷 Step 5 — JEE Trap Alert 🚨

❌ Electric field ko variable maan lena

❌ Momentum instead of wavelength differentiate kar dena

❌ $\lambda \propto \frac1{t}$ ke baad derivative na lena

Remember:

$$\boxed{ \lambda=\frac{h}{p} }$$

and for constant acceleration:

$$p\propto t$$

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✅ Final Answer

$$\boxed{2}$$

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📚 Related Topics

• RC Circuit Phase 90° Condition
• Work Done in Cyclic Process from PV Diagram
• Instantaneous Power for Time-Dependent Force