7.3 Einstein's Photoelectric Theory
In 1905, Albert Einstein successfully explained the photoelectric effect by applying Max Planck's quantum theory, earning him the Nobel Prize in Physics in 1921. Einstein proposed that light is made up of discrete packets of energy called photons. When light strikes a metal surface, each photon completely transfers its energy to an electron.
Learning Objectives
- State and explain Einstein's Photoelectric Theory and the photon energy equation, \(E = hf\).
- Formulate Einstein's photoelectric equation based on the Principle of Conservation of Energy: \(hf = W + \frac{1}{2}mv_{\text{max}}^2\) or \(hf = W + K_{\text{max}}\).
- Define work function (\(W\)), threshold frequency (\(f_0\)), and threshold wavelength (\(\lambda_0\)).
- Relate work function to threshold frequency using \(W = hf_0\) and threshold wavelength using \(W = \frac{hc}{\lambda_0}\).
- Interpret the graph of maximum kinetic energy (\(K_{\text{max}}\)) against frequency (\(f\)) and against \(\frac{1}{\lambda}\).
- Solve numerical problems using Einstein's photoelectric equation and related formulae.
- Explain the generation of photoelectric current in a photocell circuit.
- Identify and describe real-world applications of the photoelectric effect.
Graph of Maximum Kinetic Energy (\(K_{\text{max}}\)) against Frequency (\(f\)) for Different Metals
The gradient equals Planck's constant (\(h\)) for all metals, while the horizontal intercept represents the threshold frequency (\(f_0\)).
Three parallel lines for Metal A, B and C
All three lines have the same gradient, \(h\). The horizontal intercepts are \(f_{0,\text{A}}\), \(f_{0,\text{B}}\) and \(f_{0,\text{C}}\), where \(f_{0,\text{A}} < f_{0,\text{B}} < f_{0,\text{C}}\).
Key features
When \(f < f_0\), no photoelectric effect occurs. When \(f > f_0\), photoelectrons are emitted and acquire kinetic energy.
The higher the threshold frequency of a metal, the higher its work function.
Work function and threshold frequency
\(W = hf_0\)Threshold wavelength: \(W = \frac{hc}{\lambda_0}\)
Short Explanation
Einstein's Photoelectric Theory
Light energy is carried by light particles called photons. Each quantum of light is a discrete packet of energy. The energy of each photon is directly proportional to the frequency of light, \(f\): \(E = hf\), where \(h\) is Planck's constant (\(6.63 \times 10^{-34}\text{ J s}\)).
Photon-Electron Interaction
When a photon hits a metal surface, its energy is fully absorbed by an electron in the metal. This energy is used to release the electron from the metal, and any extra energy becomes the kinetic energy of the emitted photoelectron.
Einstein's Photoelectric Equation
Based on the Principle of Conservation of Energy: \(hf = W + \frac{1}{2}mv_{\text{max}}^2\), or \(hf = W + K_{\text{max}}\).
Work Function and Threshold Frequency
Work function (\(W\)) is the minimum energy required for a photoelectron to be emitted from a metal surface. Threshold frequency (\(f_0\)) is the minimum frequency for a light photon to produce the photoelectric effect. \(W = hf_0\).
Einstein's photoelectric equation
\(hf = W + \frac{1}{2}mv_{\text{max}}^2\)At threshold frequency, \(f_0\), photoelectrons are emitted without kinetic energy: \(0 = hf_0 - W \implies W = hf_0\).
Substituting \(W = hf_0\) gives: \(\frac{1}{2}mv_{\text{max}}^2 = hf - hf_0 = h(f - f_0)\).
| Metal | Threshold frequency, \(f_0\) (Hz) |
|---|---|
| Tantalum (Ta) | \(1.03 \times 10^{14}\) |
| Titanium (Ti) | \(1.05 \times 10^{14}\) |
| Molybdenum (Mo) | \(1.11 \times 10^{14}\) |
| Gold (Au) | \(1.23 \times 10^{14}\) |
| Palladium (Pd) | \(1.24 \times 10^{14}\) |
| Iridium (Ir) | \(1.27 \times 10^{14}\) |
| Platinum (Pt) | \(1.36 \times 10^{14}\) |
Threshold Wavelength
Threshold wavelength (\(\lambda_0\)) is the maximum wavelength of light needed for a metal to emit electrons. \(W = \frac{hc}{\lambda_0}\), or \(\lambda_0 = \frac{hc}{W}\).
Photocell Circuit
A photocell circuit consists of a glass vacuum tube, a semi-cylindrical cathode coated with a light-sensitive metal connected to the negative terminal, and an anode rod fixed along the central axis connected to the positive terminal. When illuminated by light of frequency higher than \(f_0\), photoelectrons are emitted from the cathode and attracted to the positive anode, producing a photoelectric current.
| Photocell metal | Work function, \(W\) (eV) | Threshold frequency, \(f_0\) (Hz) | Maximum wavelength, \(\lambda\) (nm) |
|---|---|---|---|
| Caesium | 2.14 | \(5.16 \times 10^{14}\) | 579 |
| Lithium | 2.50 | \(6.03 \times 10^{14}\) | 496 |
Applications of the Photoelectric Effect
Solar-powered LED road lamps, the Noor Complex Solar Power Plant, automatic door light detectors, image sensors in high-resolution cameras, and International Space Station solar panels all use the photoelectric effect.
Key Relationships
The higher the work function, the shorter the maximum wavelength required to emit electrons. As light intensity increases, the number of emitted photoelectrons per second increases, so the photoelectric current increases.
Try to Answer First
Answer in your mind, then press “Check Answer”.
What is the physical quantity represented by the gradient of a graph of maximum kinetic energy (\(K_{\text{max}}\)) against light frequency (\(f\))?
Check Answer
State the relationship between the work function (\(W\)) and the threshold frequency (\(f_0\)) of a metal.
Check Answer
Violet light of wavelength \(\lambda = 4 \times 10^{-7}\text{ m}\) shines on a metal surface with a work function of \(4.32 \times 10^{-19}\text{ J}\). Calculate the maximum kinetic energy of the emitted photoelectrons. (Given \(h = 6.63 \times 10^{-34}\text{ J s}\), \(c = 3.00 \times 10^8\text{ m s}^{-1}\))
Check Answer
Common Mistakes
- Confusing threshold frequency (\(f_0\)) with threshold wavelength (\(\lambda_0\)): \(f_0\) is the minimum frequency needed for emission, whereas \(\lambda_0\) is the maximum wavelength allowed.
- Believing that increasing the light intensity increases the kinetic energy or speed of the emitted electrons; intensity only increases the number of photoelectrons released per second.
- Forgetting to convert units, such as using electron-volts (eV) directly with SI units (J) in the equation \(hf = W + K_{\text{max}}\) without applying the factor \(1\text{ eV} = 1.60 \times 10^{-19}\text{ J}\).
- Mistakenly assuming different metals have different gradients in the \(K_{\text{max}}\) versus \(f\) graph; all metals yield the exact same gradient, which is Planck's constant (\(h\)).
Concept Misunderstandings
Light energy accumulates gradually over time until an electron gathers enough energy to escape.
Energy transfer occurs via single one-to-one photon-electron interactions instantaneously. If an individual photon has energy \(hf < W\), no electron will be emitted regardless of how long the light shines.
Any light beam will produce a photoelectric current if made bright enough.
If the light's frequency is below the threshold frequency (\(f < f_0\)), photoelectric emission cannot occur, no matter how bright the light is.
All emitted photoelectrons have the exact same kinetic energy.
Photoelectrons emitted from within the metal lose energy colliding with other atoms before escaping, so they have less kinetic energy. Only surface electrons escape with the maximum kinetic energy, \(K_{\text{max}}\).
Summary
- Einstein's Photoelectric Theory establishes that light exists as discrete photons with energy \(E = hf\).
- When absorbed, photon energy releases electrons from the metal surface and supplies them with kinetic energy: \(hf = W + \frac{1}{2}mv_{\text{max}}^2\).
- The threshold frequency \(f_0\) is the minimum frequency for emission, defining the work function \(W = hf_0\), while the threshold wavelength \(\lambda_0 = \frac{hc}{W}\) is the maximum wavelength capable of emission.
- In a graph of \(K_{\text{max}}\) against \(f\), the slope represents Planck's constant \(h\) and the horizontal intercept is \(f_0\).
- Photocells utilize this mechanism to convert incident light into an electric current whose magnitude depends on light intensity, forming the basis for solar panels, automated doors, and digital image sensors.
Short Activity
Objective: Test your understanding of Einstein's photoelectric theory, work function, threshold frequency and applications.
A. Objective Quiz
1 In a graph of maximum kinetic energy of photoelectrons (\(K_{\text{max}}\)) against light frequency (\(f\)) for different metals, which property is identical for all metals?
2 What happens to the emitted photoelectrons when the intensity of light shining on a photocell is increased without changing its frequency?
B. Fill in the Blanks
3 The minimum energy required for a photoelectron to be emitted from a metal surface is known as the .
4 The maximum wavelength of light required to emit photoelectrons from a metal surface is called the wavelength.
C. Matching / Drag and Drop
Drag each term to the correct description. If using a phone, tap the answer first, then tap the matching box.
