Photoelectric effect and wave-particle Duality

The energy of a photon is: $E = h\nu = \frac{hc}{\lambda}$

Where:

• $h$ is Planck's constant (6.626 × 10^-34 Js or 4.136 × 10^-15 eVs)
• $\nu$ is the frequency of light
• $c$ is the speed of light (3.0 × 10^8 m/s)
• $\lambda$ is the wavelength of light

This equation shows that photon energy is directly proportional to frequency and inversely proportional to wavelength.

Effects of light intensity according to photon theory:

On photoelectric current:

• Higher intensity = more photons per unit area per unit time
• More photons = more electron-photon interactions
• More interactions = more ejected electrons
• More ejected electrons = increased saturation photocurrent
• Relationship is directly proportional

On kinetic energy of photoelectrons:

• Intensity has NO effect on maximum kinetic energy
• Each photon has fixed energy $E = h\nu$ for a given frequency
• Each electron interacts with at most one photon
• Maximum kinetic energy depends only on frequency: $K_{max} = h\nu - \phi$

This behavior demonstrates light's particle nature: increasing intensity increases the number of photons, not the energy per photon.

Step 1: Calculate energy of one photon $E_{photon} = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34} \text{ J·s})(3 \times 10^8 \text{ m/s})}{500 \times 10^{-9} \text{ m}} = 3.98 \times 10^{-19} \text{ J}$

Step 2: Calculate total energy crossing the area in one second $E_{total} = \text{Intensity} \times \text{Area} \times \text{Time} = (200 \text{ W/m²})(2 \times 10^{-4} \text{ m²})(1 \text{ s}) = 4 \times 10^{-2} \text{ J}$

Step 3: Calculate number of photons $n = \frac{E_{total}}{E_{photon}} = \frac{4 \times 10^{-2} \text{ J}}{3.98 \times 10^{-19} \text{ J}} = 1.01 \times 10^{17} \text{ photons}$

Note: Increasing intensity would increase this number proportionally, while changing wavelength would change the energy per photon.

Classical Wave Theory:

• Light energy distributed continuously over wavefront
• Electron energy should gradually accumulate
• Predicts higher intensity = higher electron energy
• No threshold wavelength should exist
• All wavelengths should work with sufficient intensity
• Predicts continuous spectrum of electron energies
• Equation: Not applicable (incorrect model)

Photon Theory:

• Light energy delivered in discrete quanta (photons)
• Each photon transfers all energy to one electron
• Predicts maximum kinetic energy depends only on frequency
• Clear threshold wavelength exists
• Wavelength must be below threshold for any emission
• Predicts discrete maximum kinetic energy
• Equation: $K_{max} = \frac{hc}{\lambda} - \phi$

The photon theory correctly predicts the linear relationship between $K_{max}$ and $\frac{1}{\lambda}$ with slope $hc$, while wave theory fails to explain this relationship.

Quantum mechanics must be used for particles smaller than ~10^-4 cm because:

1. Wave-particle duality becomes significant:

   • De Broglie wavelength $\lambda = \frac{h}{p}$ becomes measurable
   • Wave-like behavior (diffraction, interference) becomes observable

2. Quantization effects dominate:

   • Energy exists in discrete levels rather than continuous values
   • Transitions occur in quantum jumps rather than smooth changes

3. Uncertainty principle becomes limiting:

   • Position and momentum cannot be simultaneously known: $\Delta x \Delta p \geq \frac{h}{4\pi}$
   • Creates fundamental limits on measurement precision

4. Probabilistic nature emerges:

   • Deterministic trajectories give way to probability distributions
   • Measurement outcomes have inherent randomness

5. Superposition states become possible:

   • Particles can exist in multiple states simultaneously
   • Classical either/or logic fails to describe quantum both/and states

These quantum effects are negligible for larger objects but fundamental to understanding atomic and subatomic behavior.

The photocurrent-voltage characteristic curve has three distinct regions:

1. Initial region (negative voltage): Photocurrent decreases as negative voltage increases due to electrons being repelled back to the cathode.

2. Transition region: Photocurrent rises rapidly as voltage becomes less negative or slightly positive, as more photoelectrons can reach the anode.

3. Saturation region: Photocurrent reaches a maximum constant value regardless of further voltage increase, indicating all emitted photoelectrons are being collected.

The point where photocurrent becomes zero corresponds to the stopping potential, which equals the maximum kinetic energy of photoelectrons divided by the elementary charge.

The stopping potential (V₀) relates to light and metal properties by:

$V_0 = \frac{hc}{e\lambda} - \frac{\phi}{e}$

Where:

• h = Planck's constant
• c = speed of light
• e = elementary charge
• λ = wavelength of incident light
• φ = work function of the metal

Key insights:

• V₀ is directly proportional to 1/λ with slope hc/e
• The y-intercept equals -φ/e
• V₀ is independent of light intensity
• For λ > λ₀ (threshold wavelength), V₀ is negative
• When V₀ = 0, λ = λ₀ = hc/φ

This relationship allows experimental determination of Planck's constant and work functions by measuring stopping potentials at different wavelengths.

In wave theory:

• Energy is continuously distributed over a surface
• All free electrons receive energy gradually and simultaneously
• Energy accumulates slowly over time in each electron
• Requires long exposure time before any electron could acquire enough energy to overcome work function

In particle theory (photon theory):

• Energy arrives in discrete packets (photons)
• Each photon delivers its entire energy to a single electron in an instantaneous interaction
• No gradual accumulation - either an electron receives sufficient energy or it doesn't
• Explains immediate emission of photoelectrons regardless of light intensity

This fundamental difference explains why photoelectric emission occurs instantaneously even with very low light intensities, contradicting wave theory predictions.

Wave behavior (continuous distribution):

• Like watering plants with a sprinkler system or hose spray
• Water (energy) spreads evenly across all plants (electrons) simultaneously
• Each plant receives a small, continuous amount of water
• Plants would need long exposure before any single plant receives enough water
• All plants get similar amounts depending on their position

Particle behavior (discrete packets):

• Like throwing water balloons at plants randomly
• Each balloon (photon) delivers its entire content to just one plant
• A plant either gets an entire balloon's worth of water or nothing
• Plants receive water immediately upon being hit
• The number of plants getting water depends on how many balloons are thrown

This analogy helps explain why:

1. Photoelectric emission occurs immediately (like plants getting wet as soon as hit by balloons)
2. Emission depends on photon energy/frequency, not intensity (like needing balloons with enough water)
3. More intense light increases electron count but not energy (like throwing more balloons, not bigger ones)

The saturation current in the photoelectric effect is the maximum photocurrent reached when all photoelectrons emitted from the cathode are collected by the anode.

Key characteristics:

• Appears as a plateau (flat region) on the current vs. voltage curve
• Occurs once the anode potential is sufficient to overcome space charge effects
• Independent of further increases in anode potential
• Directly proportional to incident light intensity

Physical explanation:

1. The number of emitted photoelectrons is determined solely by:

   • Incident photon flux (light intensity)
   • Material's quantum efficiency
   • NOT by the electric field strength

2. Saturation mechanism:

   • At low anode potentials: some electrons fail to reach the anode
   • At sufficient positive potential: all emitted electrons are collected
   • Beyond this point: additional voltage cannot increase electron production

Fundamental principles demonstrated:

• Confirms the quantum nature of light (photon theory)
• Shows that each photon can eject at most one electron
• Provides evidence that photoemission depends on photon flux, not field strength
• Supports Einstein's photoelectric equation where electron emission is governed by photon energy (hf), not intensity

Note: While increasing voltage beyond saturation doesn't increase current, increasing light intensity will raise the saturation current level proportionally.