Key Equations and Constants for Electricity, Magnetism, and Modern Physics

Key Physical Constants and Units

Fundamental Constants

These constants are essential for calculations in electricity, magnetism, and modern physics.

• Planck's constant (h): $h = 6.63 \times 10^{-34} \ \mathrm{J \cdot s}$ or $h = 4.14 \times 10^{-15} \ \mathrm{eV \cdot s}$

• Electron mass (m_e): $m_e = 9.11 \times 10^{-31} \ \mathrm{kg}$

• Atomic mass unit (u): $1 \ \mathrm{u} = 1.66 \times 10^{-27} \ \mathrm{kg} = 931 \ \mathrm{MeV}/c^2$

• Earth's radius (R_E): $R_E = 6.38 \times 10^6 \ \mathrm{m}$

• Earth's mass (M_E): $M_E = 5.98 \times 10^{24} \ \mathrm{kg}$

Electrostatics

Coulomb's Law

Describes the force between two point charges:

• $F = k \frac{Q_1 Q_2}{R^2}$

• Where $k = \frac{1}{4\pi \varepsilon_0}$

Electric Field and Potential

• Electric field: $E = \frac{F}{q}$

• Field due to point charge: $E = k \frac{Q}{R^2}$

• Surface charge: $E = \frac{\sigma}{2\varepsilon_0}$

• Potential energy: $U = qV$

• Work: $W = qEd$

• Potential difference: $V = -\int \vec{E} \cdot d\vec{l}$

• Potential due to point charge: $V = k \frac{Q}{R}$

• Potential energy of two charges: $U = k \frac{Q_1 Q_2}{R}$

Gauss's Law

• $\Phi_E = \int \vec{E} \cdot d\vec{A} = \frac{Q_{encl}}{\varepsilon_0}$

Capacitance and Dielectrics

Capacitance

• $Q = CV$

• $C = \kappa \varepsilon_0 \frac{A}{d}$

• For capacitors in series: $\frac{1}{C_{eq}} = \sum_i \frac{1}{C_i}$

• For capacitors in parallel: $C_{eq} = \sum_i C_i$

• Energy stored: $U = \frac{1}{2} QV$

Electric Current and Circuits

Current and Resistance

• $I = \frac{dQ}{dt}$

• Ohm's Law: $V = IR$

• Resistance: $R = \rho \frac{L}{A}$

• Resistivity temperature dependence: $\rho = \rho_0 [1 + \alpha (T - T_0)]$

• Power: $P = IV$

• Current density: $j = \frac{I}{A} = nev_d$

• Relation to electric field: $E = \rho j$

Circuits

• Series resistors: $R_{eq} = \sum_i R_i$

• Parallel resistors: $\frac{1}{R_{eq}} = \sum_i \frac{1}{R_i}$

• RC time constant: $\tau = RC$

• Charging capacitor: $V = \mathcal{E}(1 - e^{-t/\tau})$

• Discharging capacitor: $V = V_0 e^{-t/\tau}$

Magnetism

Magnetic Force and Fields

• Force on moving charge: $\vec{F} = q\vec{v} \times \vec{B}$

• Force on current: $\vec{F} = I\vec{L} \times \vec{B}$

• Torque on loop: $\tau = NIAB \sin \theta$

• Magnetic dipole: $\vec{\mu} = NIA\hat{n}$

• Potential energy: $U = -\vec{\mu} \cdot \vec{B}$

Biot-Savart and Ampère's Law

• Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$

• Ampère's Law: $\int \vec{B} \cdot d\vec{l} = \mu_0 I_{encl}$

• Field inside solenoid: $B = \mu_0 n I$

Electromagnetic Induction

Faraday's Law and Inductance

• Faraday's Law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$

• Induced emf (moving conductor): $\mathcal{E} = Blv \cos \theta$

• Mutual inductance: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$

• Self-inductance: $\mathcal{E} = -L \frac{dI}{dt}$

• Energy in inductor: $U = \frac{1}{2} LI^2$

AC Circuits

• Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$

• Inductive reactance: $X_L = 2\pi f L$

• Capacitive reactance: $X_C = \frac{1}{2\pi f C}$

• Resonant frequency: $\omega = \frac{1}{\sqrt{LC}}$

Maxwell's Equations and Electromagnetic Waves

• Speed of light: $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

• Wave relation: $c = f \lambda$

• Energy density: $u_E = \frac{1}{2} \varepsilon_0 E^2$, $u_B = \frac{1}{2} \frac{B^2}{\mu_0}$

• Poynting vector: $S = c u$

Geometric Optics

Reflection and Refraction

• Snell's Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$

• Lens equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

• Magnification: $m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}$

Optical Instruments

• Power of lens: $P = \frac{1}{f}$ (in diopters)

• Combination of lenses: $\frac{1}{f_{eq}} = \sum_i \frac{1}{f_i}$

Wave Optics

Interference and Diffraction

• Double-slit: $d \sin \theta = m \lambda$

• Single-slit: $D \sin \theta = m \lambda$

• Diffraction grating: $d \sin \theta = m \lambda$

• Rayleigh criterion: $\theta = 1.22 \frac{\lambda}{D}$

Special Relativity

• Time dilation: $\Delta t = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}}$

• Length contraction: $L = L_0 \sqrt{1 - v^2/c^2}$

• Lorentz factor: $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

• Relativistic momentum: $p = \gamma m_0 v$

• Relativistic energy: $E = \gamma m c^2$

• Energy-momentum relation: $E^2 = (mc^2)^2 + (pc)^2$

Quantum Physics

Photoelectric Effect and Photons

• Photon energy: $E = hf$

• Photoelectric equation: $K_{max} = hf - W_0$

• Photon momentum: $p = \frac{h}{\lambda}$

Atomic Models and Spectra

• Bohr radius: $r_0 = 5.29 \times 10^{-11} \ \mathrm{m}$

• Bohr energy levels: $E_n = -13.6 \frac{Z^2}{n^2} \ \mathrm{eV}$

• Rydberg formula: $\frac{1}{\lambda} = R \left( \frac{1}{n^2} - \frac{1}{m^2} \right)$

Uncertainty Principle

• Position-momentum: $\Delta x \Delta p \geq \frac{h}{4\pi}$

• Energy-time: $\Delta E \Delta t \geq \frac{h}{4\pi}$

• Reduced Planck's constant: $\hbar = \frac{h}{2\pi}$

Table: Selected Physical Constants

Constant	Symbol	Value	Units
Planck's constant	h	6.63 × 10-34	J·s
Electron mass	me	9.11 × 10-31	kg
Atomic mass unit	u	1.66 × 10-27	kg
Earth's radius	RE	6.38 × 106	m
Earth's mass	ME	5.98 × 1024	kg

Additional info:

• Some equations and constants were inferred and expanded for clarity and completeness.

• Where only symbols or fragments were present, standard forms of the equations were provided.

• Table values were reconstructed from context and standard physics references.