Classical and Modern Physics Equations and Concepts

Vocabulary practice flashcards generated from classical mechanics, electrodynamics, optics, thermodynamics, quantum mechanics, and special relativity lecture formulas.

Translational Kinetic Energy

$$\frac{1}{2} m v^2$$

Rotational Kinetic Energy

The kinetic energy associated with the rotation of an object with moment of inertia $I$ and angular velocity $\text{ω}$, expressed as $\frac{1}{2} I \text{ω}^2$.

Parallel Axis Theorem

A theorem stating that the moment of inertia $I$ about any axis parallel to and a distance $r$ from the center-of-mass axis is given by $I = I_{\text{CM}} + M r^2$.

Lagrangian

The function $L(q, \dot{q}, t) = T - U$, defined as the difference between kinetic energy $T$ and potential energy $U$.

Conjugate Momentum

The momentum $p_i$ conjugate to the generalized coordinate $q_i$, defined as $p_i = \frac{\partial L}{\partial \dot{q}_i}$.

Hamiltonian

The function $H(p, q) = \sum_i p_i \dot{q}_i - L$, representing total energy $H = T + U$ when the potential energy $U$ does not depend explicitly on velocities or time.

Reduced Mass

The effective two-body inertial mass used to simplify two-body mechanical problems, defined as $\mu = \frac{m_1 m_2}{m_1 + m_2}$.

Gauss's Law (Electrostatics)

A fundamental field equation of electrostatics stating that the divergence of the electric field is proportional to volumetric charge density, $∇ \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$.

Poisson's Equation

A second-order partial differential equation relating the electric scalar potential $V$ to spatial charge density $\rho$, given by $\nabla^2 V = -\frac{\rho}{\epsilon_0}$.

Laplace's Equation

The partial differential equation satisfied by the electric potential $V$ in charge-free region of space, given by $\nabla^2 V = 0$.

Parallel-Plate Capacitor

A configuration of two parallel conducting plates of area $A$ separated by distance $d$, with capacitance defined as $C = \frac{\epsilon_0 A}{d}$.

Ampere's Law (Magnetostatics)

A fundamental relation in magnetostatics connecting the curl of magnetic field $\mathbf{B}$ to current density $\mathbf{J}$, expressed as $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$.

Cyclotron Radius

The radius $R$ of the circular path executed by a charge $q$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$, given by $R = \frac{m v}{q B}$.

Faraday's Law of Induction

A fundamental Maxwell equation stating that a time-varying magnetic field induces a circulating electric field, expressed as $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$.

Poynting Vector

A vector representing the directional energy flux density of an electromagnetic field, defined as $\mathbf{S} = \frac{1}{\mu_0}(\mathbf{E} \times \mathbf{B})$.

Phase Velocity

The rate at which the phase of a wave propagates in space, defined as $v_{\text{phase}} = \frac{\omega}{k}$.

Group Velocity

The velocity with which the overall shape or envelope of wave amplitudes propagates, defined as $v_{\text{group}} = \frac{d\omega}{dk}$.

Brewster's Angle

The angle of incidence at which light with a particular polarization is transmitted without reflection, given by $\theta_B = \arctan\left(\frac{n_2}{n_1}\right)$.

Snell's Law

The formula describing the relationship between angles of incidence and refraction for waves passing through an interface, $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$.

Partition Function

A key sum over Boltzmann factors of quantum microstates used to derive thermodynamic parameters, given by $Z = \sum_i e^{-\beta E_i}$.

Fundamental Thermodynamic Relation

An expression combining the First and Second Laws of Thermodynamics for a reversible process, written as $dU = T dS - P dV$.

Fermi-Dirac Distribution

The statistical distribution function describing the occupancy probability of quantum states by non-interacting fermions, given by $F_{\text{FD}}(E) = \frac{1}{e^{(E - \mu)/(k_B T)} + 1}$.

Bose-Einstein Distribution

The statistical distribution function describing the occupancy probability of quantum states by non-interacting bosons, given by $F_{\text{BE}}(E) = \frac{1}{e^{(E - \mu)/(k_B T)} - 1}$.

Position and Momentum Operators

In one-dimensional quantum mechanics, position operator $\hat{x} = x$ and linear momentum operator $\hat{p} = -i \hbar \frac{\partial}{\partial x}$.

Heisenberg Uncertainty Principle

The relation expressing the fundamental quantum precision limit for simultaneously measuring position and momentum, $\sigma_x \sigma_p \ge \frac{\hbar}{2}$.

Fine-Structure Constant

A dimensionless fundamental physical constant characterizing the strength of electromagnetic interaction, defined as $\alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c} \approx \frac{1}{137}$.

Lorentz Factor

The scaling factor in special relativity describing time dilation and length contraction, defined as $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$.

Relativistic Energy-Momentum Relation

The relation connecting total relativistic energy $E$, momentum $p$, and rest mass $m$, given by $E^2 = p^2 c^2 + m^2 c^4$.

Compton Shift

The shift in photon wavelength when scattering off a stationary particle, given by $\Delta \lambda = \frac{h}{m c}(1 - \cos(\theta))$.

Fermi Energy (3D Free Electron Gas)

The highest single-particle quantum energy level in a non-interacting Fermi gas at absolute zero temperature, given by $E_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}$.