FAD1022 L43 — Modern Physics — Formula Sheet
Comprehensive formula sheet extracted from Lecture 43 (Modern Physics: Wave-Particle Duality & Black Body Radiation).
1. Wave-Particle Duality
1.1 Energy Conservation for Incident Radiation
$$\text{Total Incoming Energy} = \text{Absorbed} + \text{Reflected} + \text{Transmitted}$$
For any object receiving electromagnetic radiation, the total incident energy is partitioned into absorbed, reflected, and transmitted components.
1.2 Coefficient Relation
$$\alpha_v + \rho_v + \tau_v = 1$$
| Symbol | Description | Notes |
|---|---|---|
| $\alpha_v$ | Absorptivity | Fraction of incident energy absorbed |
| $\rho_v$ | Reflectivity | Fraction of incident energy reflected |
| $\tau_v$ | Transmissivity | Fraction of incident energy transmitted |
For a perfect black body: $\alpha_v = 1$, which implies $\rho_v = 0$ and $\tau_v = 0$.
1.3 Black Body Condition
$$\alpha_v = 1$$
A black body is an ideal theoretical object that absorbs all incident radiation (reflects nothing, transmits nothing) and emits radiation perfectly.
2. Black Body Radiation
2.1 Planck's Quantum Hypothesis (Energy Quantization)
$$E = hf$$
| Symbol | Description | Value / Units |
|---|---|---|
| $E$ | Energy of a single quantum (photon) | Joules (J) or electron-volts (eV) |
| $h$ | Planck's constant | $6.626 \times 10^{-34} \ \text{J} \cdot \text{s}$ |
| $f$ | Frequency of the radiation | Hertz (Hz) |
Energy is not continuous; it comes in discrete packets called quanta. This resolved the ultraviolet catastrophe.
Alternative form using wavelength:
$$E = \frac{hc}{\lambda}$$
| Symbol | Description | Value / Units |
|---|---|---|
| $c$ | Speed of light in vacuum | $3.00 \times 10^{8} \ \text{m/s}$ |
| $\lambda$ | Wavelength of the radiation | meters (m) |
2.2 Wien's Displacement Law
$$\lambda_{\text{max}} = \frac{b}{T}$$
| Symbol | Description | Value / Units |
|---|---|---|
| $\lambda_{\text{max}}$ | Wavelength at which emission intensity is maximum | meters (m) |
| $b$ | Wien's displacement constant | $2.90 \times 10^{-3} \ \text{m} \cdot \text{K}$ |
| $T$ | Absolute temperature of the black body | Kelvin (K) |
As temperature increases, the peak wavelength shifts toward shorter wavelengths (blue shift). This explains why hot objects first glow red, then white, then blue.
2.3 Stefan-Boltzmann Law
$$\frac{P}{A} = \sigma T^{4}$$
| Symbol | Description | Value / Units |
|---|---|---|
| $P$ | Total power (energy per unit time) radiated | Watts (W) |
| $A$ | Surface area of the black body | square meters ($\text{m}^2$) |
| $\frac{P}{A}$ | Power radiated per unit area (intensity) | $\text{W} \cdot \text{m}^{-2}$ |
| $\sigma$ | Stefan-Boltzmann constant | $5.67 \times 10^{-8} \ \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}$ |
| $T$ | Absolute temperature of the black body | Kelvin (K) |
The total power radiated per unit area by a black body is proportional to the fourth power of its absolute temperature.
Total radiated power (for a black body with surface area $A$):
$$P = \sigma A T^{4}$$
3. Key Physical Constants
| Constant | Symbol | Value | Units |
|---|---|---|---|
| Planck's constant | $h$ | $6.626 \times 10^{-34}$ | $\text{J} \cdot \text{s}$ |
| Wien's displacement constant | $b$ | $2.90 \times 10^{-3}$ | $\text{m} \cdot \text{K}$ |
| Stefan-Boltzmann constant | $\sigma$ | $5.67 \times 10^{-8}$ | $\text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}$ |
| Speed of light | $c$ | $3.00 \times 10^{8}$ | $\text{m/s}$ |
4. Summary Table: Key Laws
| Law | Formula | What It Describes |
|---|---|---|
| Planck's Law | — | Spectral energy distribution of black body radiation across all wavelengths |
| Wien's Law | $\lambda_{\text{max}} = \dfrac{b}{T}$ | Peak emission wavelength shifts inversely with temperature |
| Stefan-Boltzmann Law | $\dfrac{P}{A} = \sigma T^{4}$ | Total radiated power per unit area scales with $T^4$ |
5. Key Relationships & Definitions
Black Body Properties
- Perfect Absorber: $\alpha_v = 1$, $\rho_v = 0$, $\tau_v = 0$
- Perfect Emitter: Emits the maximum possible thermal radiation at any given temperature
- Temperature Dependence: Emission spectrum depends only on temperature, not on material composition
Classical vs. Quantum Physics
| Classical Physics | Quantum Physics |
|---|---|
| Energy is continuous | Energy is quantized: $E = hf$ |
| Rayleigh-Jeans Law predicts ultraviolet catastrophe | Planck's hypothesis resolves ultraviolet catastrophe |
| Deterministic | Probabilistic |
Note: This formula sheet contains all equations, formulas, and key relationships explicitly stated in the source lecture file. Additional quantum mechanical concepts (de Broglie wavelength, photoelectric effect equation, Compton scattering, Heisenberg uncertainty principle) are discussed conceptually in the lecture as part of the wave-particle duality framework, but their governing equations are not provided in the source material for this lecture.