Q.33 Physics ⭐ 360° Option Elimination Active

NEET 2021 Physics Question 33 Solution

Topic: Dual Nature of Radiation and Matter · Official NTA Booklet Code M1

Question

An electromagnetic wave of wavelength 'λ' is incident on a photosensitive surface of negligible work function. If 'm' mass is of photoelectron emitted from the surface has de-Broglie wavelength λd, then:

✓

Verified Against NTA Final Answer Key

Option (3) : λ = (2mc / h) λ_d^2

📚 Dual Nature of Radiation and Matter Easy

📋 Given Parameters & Problem Setup

Incident EM Wave Wavelength λ

Work Function (ϕ_0) Negligible (≈ 0)

Photoelectron Mass m

Photoelectron de-Broglie Wavelength λ_d

⚡

The 30-Second Exam Hall Shortcut: ⚡ 3-Step Quantum Coupling: E = hc/λ = p²/2m, p = h/λ_d

⏱️ Save ~20 seconds

$K = \frac{hc}{\lambda} = \frac{p^2}{2m} = \frac{h^2}{2m \lambda_d^2}$ . Cross-multiply: $\lambda = \left(\frac{2mc}{h}\right) \lambda_d^2$ ! Instant 15-second algebraic relation matching Option (3)!

1 Kinetic energy of emitted photoelectron:

K = E_{\text{photon}} - \phi_0 = \frac{hc}{\lambda} - 0 = \frac{hc}{\lambda}

2 De-Broglie wavelength of electron:

\lambda_d = \frac{h}{p} = \frac{h}{\sqrt{2mK}}

3 Square both sides:

\lambda_d^2 = \frac{h^2}{2mK} = \frac{h^2}{2m(hc/\lambda)} = \frac{h\lambda}{2mc}

4 Rearrange for

\lambda

\lambda = \left(\frac{2mc}{h}\right) \lambda_d^2

5 Match Option (3).

🛡️

Why You Can 100% Trust This Shortcut: Canonical de-Broglie hypothesis and Einstein's photoelectric equation.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Applying Einstein's Photoelectric Equation

For incident photons of wavelength $\lambda$ incident on a surface of negligible work function ( $\phi_0 \approx 0$ ), the maximum kinetic energy $K$ of emitted photoelectrons is:

K = \frac{hc}{\lambda} - \phi_0 = \frac{hc}{\lambda}

Step 2

Expressing Kinetic Energy via de-Broglie Wavelength

The de-Broglie wavelength $\lambda_d$ of a particle of mass $m$ and kinetic energy $K$ is given by:

\lambda_d = \frac{h}{p} = \frac{h}{\sqrt{2mK}} \implies \lambda_d^2 = \frac{h^2}{2mK}

Step 3

Connecting Incident Wavelength λ with Matter Wavelength λ_d

Substitute $K = \frac{hc}{\lambda}$ into the expression for $\lambda_d^2$ :

\lambda_d^2 = \frac{h^2}{2m \left(\frac{hc}{\lambda}\right)} = \frac{h^2 \lambda}{2mhc} = \frac{h}{2mc} \lambda

Step 4

Solving for λ

Multiply both sides by $\frac{2mc}{h}$ :

\lambda = \left(\frac{2mc}{h}\right) \lambda_d^2

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 1

✗ Incorrect

Verdict & Scientific Reason:

Inverts the speed of light $c$ to the denominator (dimensionally inconsistent).

⚠️

Student Fallacy (Why candidates pick this trap): Algebraic manipulation error.

2 2

✗ Incorrect

Verdict & Scientific Reason:

Swaps $\lambda$ and $\lambda_d$ exponents ( $lambda$ is linear with $lambda_d^2$ , not vice versa).

⚠️

Student Fallacy (Why candidates pick this trap): Inverting which wavelength carries the square power.

3 3

✓ 100% Correct

Verdict & Scientific Reason:

Exact mathematical relationship between electromagnetic wavelength $lambda$ and de-Broglie wavelength $lambda_d$ .

💡

Why this is airtight: None; rigorous quantum mechanics derivation.

4 4

✗ Incorrect

Verdict & Scientific Reason:

Inverts Planck constant $h$ and mass $m$ .

⚠️

Student Fallacy (Why candidates pick this trap): Reciprocal error in constant factor.

⚠️

NTA Trap Alert: Exponent Placement Trap (Option 2 vs. Option 3)

Students frequently remember the constant factor $\frac{2mc}{h}$ but place the square on the wrong wavelength, confusing $\lambda = \left(\frac{2mc}{h}\right) \lambda_d^2$ with $\lambda_d = \left(\frac{2mc}{h}\right) \lambda^2$ .

🛡️ How to avoid the -1 mark: Dimensional check:

\lambda_d \propto \frac{1}{\sqrt{K}} \propto \sqrt{\lambda} \implies \lambda_d^2 \propto \lambda

. Thus

\lambda

is proportional to

\lambda_d^2

📖

Core NEET Syllabus Standard

NCERT Physics · Dual Nature of Radiation and Matter Standard Benchmark

Core Scientific Principle Tested:

"The de Broglie wavelength of an electron is related to its kinetic energy by $\lambda_d = \frac{h}{\sqrt{2mK}}$ ."

Syllabus Focus: Wave Nature of Matter (de Broglie Wavelength)

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2021 Physics Question 33 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ 3-Step Quantum Coupling: E = hc/λ = p²/2m, p = h/λ_d

Step-by-Step Rigorous Workflow & Mathematical Proof

Applying Einstein's Photoelectric Equation

Expressing Kinetic Energy via de-Broglie Wavelength

Connecting Incident Wavelength λ with Matter Wavelength λ_d

Solving for λ

360° Option Elimination Engine Key Feature

NTA Trap Alert: Exponent Placement Trap (Option 2 vs. Option 3)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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