Physics Modern and Semiconductor - How to Solve: Dual Nature of Matter and Radiation (NEET UG Physics)

How to Solve: Dual Nature of Matter and Radiation (NEET UG Physics)

Complete Guide

Introduction

"Mastering the photoelectric effect and de Broglie wavelength can get you 4–6 marks in NEET—enough to push you into the top 10%! These concepts explain how solar panels work, why X-rays detect fractures, and even how electron microscopes see atoms. Let’s break them down so you never lose marks again."

WHAT YOU NEED TO KNOW FIRST

1. Energy of a photon: ( E = h\nu ) (Planck’s constant ( h = 6.63 \times 10^{-34} \, \text{Js} ))
2. Kinetic energy: ( KE = \frac{1}{2}mv^2 )
3. Momentum: ( p = mv )

(If you’re shaky on these, pause and review them first.)

KEY TERMS & FORMULAS

1. Photoelectric Effect

Key Terms: - Threshold frequency (( \nu_0 )): Minimum frequency of light needed to eject electrons. - Work function (( \phi )): Minimum energy required to remove an electron from a metal (( \phi = h\nu_0 )). - Stopping potential (( V_0 )): Minimum voltage needed to stop the fastest photoelectrons.

Formulas:
1. Einstein’s photoelectric equation: [ h\nu = \phi + KE_{\text{max}} ] - ( h\nu ): Energy of incident photon (MEMORISE) - ( \phi ): Work function of metal (MEMORISE) - ( KE_{\text{max}} ): Maximum kinetic energy of ejected electrons (MEMORISE)

1. Maximum kinetic energy in terms of stopping potential: [ KE_{\text{max}} = eV_0 ]
2. ( e ): Charge of electron (( 1.6 \times 10^{-19} \, \text{C} )) (given on exam sheet)

3. ( V_0 ): Stopping potential (MEMORISE)

4. Threshold frequency: [ \nu_0 = \frac{\phi}{h} ] (MEMORISE)

2. de Broglie Wavelength

Key Term: - de Broglie wavelength (( \lambda )): Wavelength associated with a moving particle.

Formula: [ \lambda = \frac{h}{p} = \frac{h}{mv} ] - ( h ): Planck’s constant (given on exam sheet) - ( p ): Momentum of particle (MEMORISE) - ( m ): Mass of particle (MEMORISE) - ( v ): Velocity of particle (MEMORISE)

(For electrons, use ( m_e = 9.1 \times 10^{-31} \, \text{kg} ).)

3. Davisson-Germer Experiment

Key Term: - Bragg’s law: Confirms wave nature of electrons via diffraction. [ 2d \sin \theta = n\lambda ] - ( d ): Spacing between atomic planes (given in question) - ( \theta ): Angle of diffraction (MEMORISE) - ( n ): Order of diffraction (usually ( n = 1 )) (MEMORISE) - ( \lambda ): de Broglie wavelength of electron (MEMORISE)

STEP-BY-STEP METHOD

Photoelectric Effect Problems

Step 1: Identify what’s given and what’s asked. - Given: Frequency (( \nu )), work function (( \phi )), stopping potential (( V_0 )), etc. - Asked: ( KE_{\text{max}} ), ( \nu_0 ), ( V_0 ), etc.

Step 2: Write down Einstein’s equation: [ h\nu = \phi + KE_{\text{max}} ]

Step 3: Plug in known values. Solve for the unknown. - If ( KE_{\text{max}} ) is asked, rearrange: [ KE_{\text{max}} = h\nu - \phi ] - If ( V_0 ) is asked, use: [ KE_{\text{max}} = eV_0 \implies V_0 = \frac{KE_{\text{max}}}{e} ]

Step 4: Check units. - ( h\nu ) and ( \phi ) must be in joules (J) or electron volts (eV). - ( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} ).

Step 5: Calculate and box the answer.

de Broglie Wavelength Problems

Step 1: Identify the particle (electron, proton, etc.) and its given properties (mass, velocity, momentum, or kinetic energy).

Step 2: Write the de Broglie formula: [ \lambda = \frac{h}{p} = \frac{h}{mv} ]

Step 3: If velocity (( v )) is not given but kinetic energy (( KE )) is: - Use ( KE = \frac{1}{2}mv^2 ) to find ( v ): [ v = \sqrt{\frac{2KE}{m}} ] - Then plug ( v ) into ( \lambda = \frac{h}{mv} ).

Step 4: Plug in values. Use ( h = 6.63 \times 10^{-34} \, \text{Js} ).

Step 5: Calculate and box the answer in meters (m).

Davisson-Germer Experiment Problems

Step 1: Identify given values: - Accelerating voltage (( V )) of electron. - Angle of diffraction (( \theta )). - Spacing between planes (( d )).

Step 2: Find the de Broglie wavelength of the electron. - First, find the velocity (( v )) of the electron using: [ eV = \frac{1}{2}mv^2 \implies v = \sqrt{\frac{2eV}{m}} ] - Then, find ( \lambda ): [ \lambda = \frac{h}{mv} ]

Step 3: Use Bragg’s law to find the unknown (usually ( \theta ) or ( d )): [ 2d \sin \theta = n\lambda ] - For first-order diffraction, ( n = 1 ).

Step 4: Solve for the unknown and box the answer.

WORKED EXAMPLES

Example 1 – Basic (Photoelectric Effect)

Question: Light of frequency ( 8 \times 10^{14} \, \text{Hz} ) falls on a metal with work function ( 3.2 \times 10^{-19} \, \text{J} ). Find the maximum kinetic energy of the ejected electrons.

Solution: Step 1: Given: - ( \nu = 8 \times 10^{14} \, \text{Hz} ) - ( \phi = 3.2 \times 10^{-19} \, \text{J} ) - Asked: ( KE_{\text{max}} )

Step 2: Write Einstein’s equation: [ h\nu = \phi + KE_{\text{max}} ]

Step 3: Plug in values: [ (6.63 \times 10^{-34})(8 \times 10^{14}) = 3.2 \times 10^{-19} + KE_{\text{max}} ] [
5.304 \times 10^{-19} = 3.2 \times 10^{-19} + KE_{\text{max}} ]

Step 4: Solve for ( KE_{\text{max}} ): [ KE_{\text{max}} = 5.304 \times 10^{-19} - 3.2 \times 10^{-19} = 2.104 \times 10^{-19} \, \text{J} ]

Step 5: Box the answer: [ \boxed{2.10 \times 10^{-19} \, \text{J}} ]

What we did and why: We used Einstein’s photoelectric equation to relate photon energy to the work function and kinetic energy of ejected electrons. The key was ensuring all units were in joules.

Example 2 – Medium (de Broglie Wavelength)

Question: An electron is accelerated through a potential difference of ( 100 \, \text{V} ). Find its de Broglie wavelength.

Solution: Step 1: Given: - ( V = 100 \, \text{V} ) - ( m_e = 9.1 \times 10^{-31} \, \text{kg} ) - ( e = 1.6 \times 10^{-19} \, \text{C} ) - Asked: ( \lambda )

Step 2: Find the velocity (( v )) of the electron: [ eV = \frac{1}{2}mv^2 \implies v = \sqrt{\frac{2eV}{m}} ] [ v = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 100}{9.1 \times 10^{-31}}} = \sqrt{3.516 \times 10^{13}} = 5.93 \times 10^6 \, \text{m/s} ]

Step 3: Use de Broglie formula: [ \lambda = \frac{h}{mv} = \frac{6.63 \times 10^{-34}}{(9.1 \times 10^{-31})(5.93 \times 10^6)} ] [ \lambda = \frac{6.63 \times 10^{-34}}{5.396 \times 10^{-24}} = 1.23 \times 10^{-10} \, \text{m} ]

Step 4: Box the answer: [ \boxed{1.23 \times 10^{-10} \, \text{m}} ]

What we did and why: We first found the electron’s velocity using energy conservation, then applied the de Broglie formula. The key was converting potential energy to kinetic energy correctly.

Example 3 – Exam-Style (Davisson-Germer)

Question: In a Davisson-Germer experiment, electrons accelerated through ( 54 \, \text{V} ) are diffracted by a nickel crystal with atomic spacing ( 0.091 \, \text{nm} ). Find the angle of first-order diffraction.

Solution: Step 1: Given: - ( V = 54 \, \text{V} ) - ( d = 0.091 \, \text{nm} = 0.091 \times 10^{-9} \, \text{m} ) - ( n = 1 ) (first-order) - Asked: ( \theta )

Step 2: Find the de Broglie wavelength (( \lambda )): - First, find ( v ): [ v = \sqrt{\frac{2eV}{m}} = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 54}{9.1 \times 10^{-31}}} = 4.36 \times 10^6 \, \text{m/s} ] - Then, find ( \lambda ): [ \lambda = \frac{h}{mv} = \frac{6.63 \times 10^{-34}}{(9.1 \times 10^{-31})(4.36 \times 10^6)} = 1.67 \times 10^{-10} \, \text{m} ]

Step 3: Use Bragg’s law: [ 2d \sin \theta = n\lambda ] [ 2 \times 0.091 \times 10^{-9} \times \sin \theta = 1 \times 1.67 \times 10^{-10} ] [ \sin \theta = \frac{1.67 \times 10^{-10}}{2 \times 0.091 \times 10^{-9}} = 0.9176 ] [ \theta = \sin^{-1}(0.9176) = 66.5^\circ ]

Step 4: Box the answer: [ \boxed{66.5^\circ} ]

What we did and why: We combined de Broglie wavelength with Bragg’s law to find the diffraction angle. The key was ensuring all units were consistent (meters, not nanometers).

COMMON MISTAKES

1. MISTAKE: Using ( KE = h\nu ) instead of ( KE = h\nu - \phi ). WHY IT HAPPENS: Forgetting that the work function must be subtracted. CORRECT APPROACH: Always write ( h\nu = \phi + KE_{\text{max}} ) first.

2. MISTAKE: Confusing ( \nu ) (frequency) with ( \lambda ) (wavelength). WHY IT HAPPENS: Not distinguishing between ( c = \nu \lambda ) and ( E = h\nu ). CORRECT APPROACH: Label variables clearly. Frequency (( \nu )) is in Hz; wavelength (( \lambda )) is in meters.

3. MISTAKE: Forgetting to convert eV to joules. WHY IT HAPPENS: Mixing units in calculations. CORRECT APPROACH: Use ( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} ) for all energy terms.

4. MISTAKE: Using ( \lambda = \frac{h}{p} ) for photons (which is correct) but forgetting photons have no mass. WHY IT HAPPENS: Overgeneralizing the de Broglie formula. CORRECT APPROACH: For photons, use ( p = \frac{h}{\lambda} ) or ( E = \frac{hc}{\lambda} ).

5. MISTAKE: Ignoring the order (( n )) in Bragg’s law. WHY IT HAPPENS: Assuming ( n = 1 ) without checking. CORRECT APPROACH: Read the question carefully. If not specified, assume ( n = 1 ).

EXAM TRAPS

1. TRAP: Giving work function in eV but asking for ( KE_{\text{max}} ) in joules. HOW TO SPOT IT: The question mixes eV and joules. HOW TO AVOID IT: Convert all energies to joules (or all to eV) before calculating.

2. TRAP: Asking for the de Broglie wavelength of a photon (which is just its regular wavelength). HOW TO SPOT IT: The question says "photon" but uses the de Broglie formula. HOW TO AVOID IT: For photons, use ( \lambda = \frac{hc}{E} ), not ( \lambda = \frac{h}{mv} ).

3. TRAP: Davisson-Germer questions where the angle is given, but you must find the voltage. HOW TO SPOT IT: The question asks for ( V ) but gives ( \theta ) and ( d ). HOW TO AVOID IT: Work backward: Use Bragg’s law to find ( \lambda ), then find ( v ), then find ( V ).

1-MINUTE RECAP

"Listen up—this is your last-minute checklist for NEET:

1. Photoelectric effect: Write ( h\nu = \phi + KE_{\text{max}} ). If they give stopping potential, use ( KE_{\text{max}} = eV_0 ). Always subtract the work function!
2. de Broglie wavelength: ( \lambda = \frac{h}{mv} ). If they give voltage, find ( v ) first using ( eV = \frac{1}{2}mv^2 ).
3. Davisson-Germer: Use Bragg’s law ( 2d \sin \theta = n\lambda ). Find ( \lambda ) from the electron’s velocity.
4. Units: Joules for energy, meters for wavelength, volts for potential. Convert eV to joules if needed.
5. Common traps: Don’t mix photons and electrons. Don’t forget the work function. Don’t ignore the order in Bragg’s law.

Now go crush those 6 marks!