Fundamental Constants in Physics

Why This Matters

Physics isn't just a collection of equations—it's built on a foundation of universal constants that appear again and again across every topic you'll encounter. In AP Physics 2, you're being tested on your ability to recognize why certain constants appear in specific contexts: why Coulomb's constant shows up in electric force calculations, why Planck's constant governs photon energy, and why the elementary charge defines the smallest unit of charge in nature. These constants aren't arbitrary numbers to memorize; they're the bridge between abstract physics principles and real, measurable phenomena.

Understanding fundamental constants means understanding the deep connections between electromagnetism, quantum mechanics, and thermodynamics. When you see $k = 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2$ in a Coulomb's law problem, you should immediately think inverse-square force law and permittivity of free space. When Planck's constant appears, you should recognize energy quantization and wave-particle duality. Don't just memorize values—know what concept each constant illustrates and which equations it belongs to.

---

Electromagnetic Constants

These constants define how electric and magnetic fields behave in vacuum and govern all electrostatic and magnetostatic calculations. They emerge from the fundamental nature of how charges create fields and how fields exert forces.

Elementary Charge (e)

• $e = 1.602 \times 10^{-19}$ C—the smallest indivisible unit of free charge in nature, carried by protons (+e) and electrons (−e)
• Charge quantization means all observable charges are integer multiples of e; you'll never encounter $0.5e$ in any problem
• Directly tested in Coulomb's law problems where you calculate forces between protons, electrons, or ions

Coulomb Constant (k)

• $k = 8.99 \times 10^9$ N·m²/C²—determines the strength of electrostatic force between point charges via $F = k\frac{|q_1 q_2|}{r^2}$
• Related to permittivity by $k = \frac{1}{4\pi\varepsilon_0}$, connecting it to how vacuum "permits" electric fields
• Inverse-square dependence on distance mirrors gravitational force, making comparison questions common on exams

Permittivity of Free Space (ε₀)

• $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m—quantifies how easily vacuum allows electric field lines to pass through
• Appears in capacitance calculations ($C = \varepsilon_0 \frac{A}{d}$ for parallel plates) and electric field equations
• Dielectric constant $\varepsilon_r$ multiplies $\varepsilon_0$ when fields pass through materials other than vacuum

Permeability of Free Space (μ₀)

• $\mu_0 = 4\pi \times 10^{-7}$ N/A²—measures vacuum's ability to support magnetic field lines from moving charges
• Essential for inductance and magnetic field calculations in circuits and electromagnetism
• Connects to speed of light through $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$, unifying electricity, magnetism, and optics

---

Quantum Mechanical Constants

These constants reveal that energy, momentum, and matter behave in discrete, quantized ways at atomic scales. They replace classical continuous values with allowed "steps" that govern photons, electrons, and atomic transitions.

Planck's Constant (h)

• $h = 6.626 \times 10^{-34}$ J·s—the fundamental constant of quantum mechanics, relating photon energy to frequency via $E = hf$
• Quantization signature: whenever you see $h$, think discrete energy levels and wave-particle duality
• Appears in Bohr model (angular momentum $L = n\frac{h}{2\pi}$), Compton scattering ($\Delta\lambda = \frac{h}{m_e c}(1-\cos\theta)$), and de Broglie wavelength ($\lambda = \frac{h}{p}$)

Electron Mass (mₑ)

• $m_e = 9.109 \times 10^{-31}$ kg—determines electron behavior in atoms, fields, and quantum systems
• Compton wavelength $\frac{h}{m_e c}$ sets the scale for photon-electron scattering wavelength shifts
• Bohr model calculations use $m_e$ to determine orbital radii and energy levels in hydrogen

Proton Mass (mₚ)

• $m_p = 1.673 \times 10^{-27}$ kg—roughly 1836 times the electron mass, dominating atomic mass
• Nuclear mass comes almost entirely from protons and neutrons; electrons contribute negligibly
• Force comparisons: the same charge but vastly different mass means protons accelerate much less than electrons in electric fields

---

Speed of Light and Relativity

The speed of light connects space, time, and energy, serving as the ultimate speed limit and appearing in both electromagnetic and quantum contexts.

Speed of Light in Vacuum (c)

• $c = 3.00 \times 10^8$ m/s—the maximum speed for information and energy transfer in the universe
• Electromagnetic connection: $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$ shows light is an electromagnetic wave
• Photon calculations use $c = f\lambda$ to convert between frequency and wavelength; essential for spectral line problems

---

Thermodynamic Constants

These constants bridge the microscopic world of particles with macroscopic measurements like temperature and pressure. They translate between individual molecular behavior and bulk properties you can measure in a lab.

Boltzmann Constant (k_B)

• $k_B = 1.381 \times 10^{-23}$ J/K—converts temperature into average molecular kinetic energy via $\bar{KE} = \frac{3}{2}k_B T$
• Entropy connection: Boltzmann's entropy formula $S = k_B \ln W$ links microscopic states to thermodynamic entropy
• Bridges scales: multiplying by Avogadro's number gives the gas constant ($R = N_A k_B$), connecting particle physics to chemistry

Avogadro's Number (N_A)

• $N_A = 6.022 \times 10^{23}$ mol⁻¹—the number of particles in one mole, enabling atomic-to-macroscopic conversions
• Molar mass connection: allows you to convert between mass in grams and number of atoms or molecules
• Thermodynamics link: appears when relating per-particle constants ($k_B$) to per-mole constants ($R$)

---

Gravitational Constant

While less central to AP Physics 2 than electromagnetism, the gravitational constant provides crucial comparison points for understanding force laws.

Gravitational Constant (G)

• $G = 6.674 \times 10^{-11}$ N·m²/kg²—determines gravitational attraction strength via $F = G\frac{m_1 m_2}{r^2}$
• Inverse-square parallel: same mathematical form as Coulomb's law, making force-law comparisons a common exam topic
• Vastly weaker: electric forces between protons are ~$10^{36}$ times stronger than gravitational forces, explaining why gravity is negligible at atomic scales

---

Quick Reference Table

---

Self-Check Questions

1. Which two constants combine to give the speed of light, and what equation relates them?

2. In the Bohr model, which constants appear in the quantization of angular momentum, and why does this lead to discrete energy levels?

3. Compare the roles of $\varepsilon_0$ and $k$—how are they mathematically related, and when would you use each in a problem?

4. If an FRQ asks you to compare the electric and gravitational forces between a proton and electron, which constants do you need, and why is the ratio so extreme?

5. How does Planck's constant appear differently in photon energy ($E = hf$) versus Compton scattering ($\Delta\lambda = \frac{h}{m_e c}(1-\cos\theta)$)? What physical principle does each equation illustrate?