FAD1022 L44 — Photons and Photoelectric Effect — Formula Sheet
A complete reference of all formulas, equations, and key relationships from Lecture 44 on Photons and the Photoelectric Effect.
1. Fundamental Constants
| Symbol | Value | Description |
|---|---|---|
| $c$ | $3.00 \times 10^8 \text{ m/s}$ | Speed of light in vacuum |
| $h$ | $6.63 \times 10^{-34} \text{ J}\cdot\text{s}$ | Planck's constant |
| $e$ | $1.60 \times 10^{-19} \text{ C}$ | Elementary charge (magnitude of electron charge) |
| $m_e$ | $9.11 \times 10^{-31} \text{ kg}$ | Mass of electron |
Conversion Factor: $$1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}$$
2. Photon Energy & Properties
Core Photon Energy Equations
The energy of a single photon is determined by its frequency or wavelength:
$$E = hf = \frac{hc}{\lambda}$$
Where:
- $E$ = energy of the photon ($\text{J}$ or $\text{eV}$)
- $h$ = Planck's constant ($6.63 \times 10^{-34} \text{ J}\cdot\text{s}$)
- $f$ = frequency of light ($\text{Hz}$)
- $c$ = speed of light ($3.00 \times 10^8 \text{ m/s}$)
- $\lambda$ = wavelength ($\text{m}$)
Frequency-Wavelength Relationship
For any electromagnetic wave (including photons):
$$c = f\lambda \quad \Longrightarrow \quad f = \frac{c}{\lambda} \quad \text{or} \quad \lambda = \frac{c}{f}$$
Derived Photon Energy Forms
Rearranging the core equation for specific variables:
$$f = \frac{E}{h} \qquad \lambda = \frac{hc}{E} \qquad h = \frac{E}{f}$$
Key Relationships
- Higher frequency $\rightarrow$ Higher photon energy
- Shorter wavelength $\rightarrow$ Higher photon energy
3. Work Function & Threshold Frequency
Work Function ($\phi$)
The minimum energy required to remove an electron from a metal surface.
$$\phi = hf_0$$
Where:
- $\phi$ = work function ($\text{J}$ or $\text{eV}$)
- $f_0$ = threshold (cutoff) frequency ($\text{Hz}$)
Threshold (Cutoff) Frequency
The minimum frequency of incident light required to eject electrons from a specific metal:
$$f_0 = \frac{\phi}{h}$$
Cutoff (Threshold) Wavelength
The maximum wavelength of incident light that can still eject electrons:
$$\lambda_c = \frac{hc}{\phi}$$
Where:
- $\lambda_c$ = cutoff wavelength ($\text{m}$ or $\text{nm}$)
Cutoff Frequency from Cutoff Wavelength
$$f_c = \frac{c}{\lambda_c}$$
4. The Photoelectric Effect — Core Equations
Einstein's Photoelectric Equation
The maximum kinetic energy of an ejected photoelectron equals the photon energy minus the work function:
$$KE_{max} = hf - \phi$$
Alternative forms using wavelength:
$$KE_{max} = \frac{hc}{\lambda} - \phi$$
Where:
- $KE_{max}$ = maximum kinetic energy of ejected electron ($\text{J}$ or $\text{eV}$)
- $hf$ = incident photon energy ($\text{J}$ or $\text{eV}$)
- $\phi$ = work function of the metal ($\text{J}$ or $\text{eV}$)
Maximum Kinetic Energy from Velocity
If the maximum speed of ejected electrons is known:
$$KE_{max} = \frac{1}{2} m_e v_{max}^2$$
Where:
- $m_e$ = mass of electron ($9.11 \times 10^{-31} \text{ kg}$)
- $v_{max}$ = maximum speed of ejected electrons ($\text{m/s}$)
Solving for Work Function
When photon energy and maximum kinetic energy are known:
$$\phi = hf - KE_{max} = \frac{hc}{\lambda} - KE_{max}$$
5. Stopping Potential
Definition
The minimum voltage required to stop the most energetic photoelectrons.
$$KE_{max} = eV_s$$
Where:
- $e$ = elementary charge ($1.60 \times 10^{-19} \text{ C}$)
- $V_s$ = stopping potential ($\text{V}$)
Combined with Photoelectric Equation
$$eV_s = hf - \phi$$
Or solving for stopping potential:
$$V_s = \frac{hf - \phi}{e} = \frac{KE_{max}}{e}$$
6. Emission Conditions (Decision Rules)
The outcome depends on comparing photon energy to the work function:
| Condition | Physical Result |
|---|---|
| $hf < \phi$ | No electrons emitted — photon energy insufficient |
| $hf = \phi$ | Electrons escape with zero kinetic energy ($KE_{max} = 0$) |
| $hf > \phi$ | Electrons emitted with $KE_{max} = hf - \phi$ |
In terms of frequency vs. threshold frequency:
| Condition | Physical Result |
|---|---|
| $f < f_0$ | No photoelectric effect |
| $f = f_0$ | Emission at threshold; $KE_{max} = 0$ |
| $f > f_0$ | Photoelectric effect occurs |
In terms of wavelength vs. cutoff wavelength:
| Condition | Physical Result |
|---|---|
| $\lambda > \lambda_c$ | No photoelectric effect (frequency too low) |
| $\lambda = \lambda_c$ | Emission at threshold; $KE_{max} = 0$ |
| $\lambda < \lambda_c$ | Photoelectric effect occurs |
7. Role of Intensity vs. Frequency
| Property | Controls | Formula Link |
|---|---|---|
| Frequency ($f$) | Whether electrons are emitted | Energy per photon: $E = hf$ |
| Intensity ($I$) | How many electrons are emitted | Number of photons per unit area/time |
Important: Intensity does not affect the kinetic energy of individual electrons — only the number of photons (and thus the number of ejected electrons).
8. Unit Conversion Equations
Joules to Electronvolts
$$E , (\text{eV}) = \frac{E , (\text{J})}{1.60 \times 10^{-19} \text{ J/eV}}$$
Electronvolts to Joules
$$E , (\text{J}) = E , (\text{eV}) \times 1.60 \times 10^{-19} \text{ J/eV}$$
Wavelength Conversions
$$1 \text{ nm} = 10^{-9} \text{ m} \qquad 1 \text{ Å} = 10^{-10} \text{ m}$$
9. Quick Reference — All Equations at a Glance
Photon Energy
$$E = hf = \frac{hc}{\lambda}$$
Wave Relations
$$c = f\lambda$$
Work Function
$$\phi = hf_0 = \frac{hc}{\lambda_c}$$
Threshold / Cutoff Frequency
$$f_0 = \frac{\phi}{h}$$
Cutoff Wavelength
$$\lambda_c = \frac{hc}{\phi}$$
Photoelectric Effect (Einstein)
$$KE_{max} = hf - \phi = \frac{hc}{\lambda} - \phi$$
Kinetic Energy from Speed
$$KE_{max} = \frac{1}{2}m_e v_{max}^2$$
Stopping Potential
$$KE_{max} = eV_s$$
Solving for Work Function
$$\phi = \frac{hc}{\lambda} - KE_{max}$$
Stopping Potential from Photoelectric Equation
$$V_s = \frac{hf - \phi}{e}$$
10. Key Variables Summary
| Variable | Meaning | Common Units |
|---|---|---|
| $E$ | Photon energy | J, eV |
| $h$ | Planck's constant | $6.63 \times 10^{-34} \text{ J}\cdot\text{s}$ |
| $f$ | Frequency | Hz ($\text{s}^{-1}$) |
| $f_0$, $f_c$ | Threshold / cutoff frequency | Hz |
| $\lambda$ | Wavelength | m, nm |
| $\lambda_c$ | Cutoff wavelength | m, nm |
| $c$ | Speed of light | $3.00 \times 10^8 \text{ m/s}$ |
| $\phi$ | Work function | J, eV |
| $KE_{max}$ | Maximum kinetic energy of electron | J, eV |
| $v_{max}$ | Maximum electron speed | m/s |
| $m_e$ | Electron mass | $9.11 \times 10^{-31} \text{ kg}$ |
| $e$ | Elementary charge | $1.60 \times 10^{-19} \text{ C}$ |
| $V_s$ | Stopping potential | V |
| $I$ | Intensity | $\text{W/m}^2$ |
Extracted from: FAD1022 L44 — Photons and Photoelectric Effect Course: FAD1022 - Basic Physics II