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.

The stopping potential ($V_0$) relates to light and material properties by: $V_0 = \frac{hc}{e\lambda} - \frac{\phi}{e}$

Where:

• $e$ is the elementary charge
• $\lambda$ is the wavelength of incident light
• $\phi$ is the work function of the emitter metal
• $h$ is Planck's constant
• $c$ is the speed of light

Key relationships:

• $V_0$ is proportional to $\frac{1}{\lambda}$ with slope $\frac{hc}{e}$ (same for all metals)
• $V_0 = 0$ when $\lambda = \lambda_0$ (threshold wavelength)
• $V_0$ is independent of light intensity
• $V_0$ increases as wavelength decreases

The stopping potential directly measures the maximum kinetic energy of ejected electrons: $eV_0 = K_{max}$

Key contradictions between wave theory predictions and photoelectric effect observations:

1. Time delay:

   • Wave theory: Electrons should require significant time to accumulate sufficient energy
   • Observation: Electron emission occurs almost instantaneously

2. Intensity dependence:

   • Wave theory: Higher intensity should increase electron kinetic energy
   • Observation: Intensity only affects the number of electrons (current), not their energy

3. Threshold frequency:

   • Wave theory: Sufficiently intense light of any wavelength should eject electrons
   • Observation: No emission occurs below threshold frequency regardless of intensity

4. Energy distribution:

   • Wave theory: Predicts continuous distribution of electron energies
   • Observation: Discrete energy distribution with clear maximum value

These contradictions arise because light energy is quantized in photons rather than continuously distributed across wavefronts.

The threshold wavelength ($\lambda_0$) is the maximum wavelength of light that can cause photoelectric emission from a given metal.

Mathematical relationship with work function: $\lambda_0 = \frac{hc}{\phi}$

Where:

• $h$ is Planck's constant
• $c$ is the speed of light
• $\phi$ is the work function of the metal

When $\lambda > \lambda_0$:

• Photon energy ($\frac{hc}{\lambda}$) < work function ($\phi$)
• No electrons can be emitted regardless of intensity

When $\lambda = \lambda_0$:

• Photon energy equals work function
• Electrons can be ejected with zero kinetic energy

When $\lambda < \lambda_0$:

• Photon energy exceeds work function
• Electrons are ejected with positive kinetic energy

For metals with higher work functions, the threshold wavelength is shorter (higher frequency is required).

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.

Essential properties of photons:

Unique to photons:

• Always travel at speed $c = 3.0 \times 10^8$ m/s in vacuum (in all reference frames)
• Zero rest mass
• Energy and momentum related by $E = pc$ (unlike classical particles where $E = \frac{1}{2}mv^2$)
• Can be created or destroyed in interactions (not conserved in number)
• Cannot be localized to a specific position while having a definite momentum

Particle-like properties:

• Carry discrete quanta of energy $E = h\nu$
• Carry definite momentum $p = \frac{h}{\lambda}$
• Can collide with material particles
• Energy and momentum are conserved in collisions

Wave-like properties:

• Associated with a wavelength $\lambda$ and frequency $\nu$
• Can exhibit interference and diffraction
• Distributed according to probability waves

These properties reflect the fundamental quantum nature of light that cannot be fully described by either classical particle or wave models alone.

The stopping potential (V₀) varies linearly with 1/λ (inverse of wavelength) according to:

$V_0 = \frac{hc}{e}\left(\frac{1}{\lambda}\right) - \frac{\phi}{e}$

Where:

• h is Planck's constant
• c is speed of light
• e is elementary charge
• φ is the work function of the metal

Properties of this relationship:

• The slope (hc/e) is identical for all metals
• Each metal has a unique threshold wavelength (λ₀) where V₀ = 0
• When λ > λ₀, no photoelectric effect occurs
• The x-intercept occurs at 1/λ₀ = φ/hc
• Different metals have different work functions, resulting in different threshold wavelengths

This relationship directly confirms Einstein's photoelectric equation: Kₘₐₓ = hν - φ

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)

Photocurrent Response to Experimental Variables

1. Response to Light Intensity:

   • Increasing intensity → proportionally increases saturation photocurrent
   • Physical reason: More photons per second → more electrons ejected per second
   • Critical insight: Intensity affects only the rate of electron emission (current magnitude), NOT their kinetic energy
   • This directly supports the photon theory of light (each electron interacts with a single photon)

2. Response to Anode Potential:

   • Negative potential region:
     • Current decreases as negative potential increases
     • At stopping potential (V₀), current becomes zero (all electrons repelled)
     • V₀ remains constant regardless of light intensity changes
   • Positive potential region:
     • Current increases until reaching saturation
     • At saturation, all emitted electrons reach the anode

3. Combined Effects (Intensity × Potential):

   • Higher intensities (I₃ > I₂ > I₁) produce higher saturation currents
   • All intensity curves share the same stopping potential
   • Curves differ only in height, not horizontal position

4. Quantum Significance:

   • Contradicts classical wave theory predictions (which expected intensity to affect electron energy)
   • Confirms Einstein's photoelectric equation: E = hf - φ
   • Demonstrates light's particle-like behavior in electron interactions
   • Shows energy transfer occurs in discrete quanta, not continuously

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.