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Browse All 101 Physics Entries • Version 0.4.6

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Astrophysics

Kepler's Laws of Planetary Motion

Kepler's three laws describe planetary motion around the Sun: orbits are ellipses with the Sun at one focus, planets sweep equal areas in equal times, and the square of orbital period is proportional to the cube of the semi-major axis. These empirical laws provided the foundation for Newton's law of gravitation and remain fundamental to celestial mechanics and orbital dynamics.

Stefan–Boltzmann Law

The Stefan–Boltzmann law gives the radiant energy flux emitted by an ideal black body as the fourth power of its absolute temperature, `j^* = sigma*T^4`. By linking thermodynamic temperature to measurable luminosity, the law is fundamental to stellar astrophysics, climate modelling and furnace engineering.

Atomic Physics

Bohr Model Energy Levels

The Bohr model describes the energy levels of hydrogen atoms through quantized circular orbits, where electrons can only exist at specific energy states. The energy is inversely proportional to the square of the principal quantum number, explaining the discrete line spectra observed in hydrogen.

Bragg's Law

Bragg's law describes the conditions for constructive interference of X-rays diffracted by crystal lattice planes. When X-rays with wavelength λ are incident on parallel crystal planes separated by distance d, constructive interference occurs at specific angles θ where the path difference between rays equals an integer multiple of the wavelength.

Rydberg Formula

The Rydberg formula describes the wavelengths of spectral lines in hydrogen and hydrogen-like atoms. It predicts the wavelength of light emitted when an electron transitions between energy levels, with the wavenumber proportional to the difference in inverse squares of the principal quantum numbers.

Wien's Displacement Law

Wien's displacement law describes the relationship between the temperature of a blackbody and the wavelength at which it emits the maximum intensity of electromagnetic radiation. As temperature increases, the peak wavelength shifts toward shorter wavelengths (higher frequencies), explaining why hot objects glow red, then white, then blue.

Classical Physics

Ampère's Circuital Law

Ampère's law relates the circulation of magnetic field around a closed loop to the electric current passing through the loop. It provides a powerful method for calculating magnetic fields in cases with high symmetry and is one of Maxwell's fundamental equations of electromagnetism.

Angular Momentum

Angular momentum is the rotational analog of linear momentum, representing the quantity of rotation of an object. It depends on the object's moment of inertia, angular velocity, and the distribution of mass relative to the axis of rotation.

Biot-Savart Law

The Biot-Savart law calculates the magnetic field produced by a steady current. It states that the magnetic field contribution from a current element is proportional to the current, the length element, and inversely proportional to the square of the distance.

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. Capacitors store electrical energy in the electric field between conducting plates and are fundamental components in electronic circuits for energy storage, filtering, and timing applications.

Centripetal Force

Centripetal force is the net force directed toward the center of curvature that causes an object to follow a curved path. For uniform circular motion, this force is constant in magnitude but continuously changes direction, providing the centripetal acceleration needed to maintain circular motion.

Conservation of Angular Momentum

Angular momentum is conserved when no external torque acts on a system. This fundamental conservation law explains phenomena from planetary motion to figure skating spins.

Conservation of Energy

Conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In a closed system, the total energy remains constant, though it may change between kinetic, potential, thermal, and other forms.

Conservation of Momentum

Conservation of momentum states that the total momentum of a closed system remains constant over time. This fundamental principle applies to all interactions in the absence of external forces and is a direct consequence of Newton's third law.

Coriolis Force

The Coriolis force is a fictitious force that appears in a rotating reference frame. It causes moving objects to be deflected relative to the Earth's surface or any rotating frame.

Coulomb's Law

Coulomb's law describes the electrostatic force between two point charges, stating that the force is proportional to the product of the charges and inversely proportional to the square of their separation distance. The force acts along the line connecting the charges, being repulsive for like charges and attractive for opposite charges.

Doppler Effect

The Doppler effect describes the change in frequency of a wave when there is relative motion between the source and observer. The observed frequency increases when source and observer approach each other and decreases when they move apart.

Electric Field

Electric field is a vector quantity that describes the electric force per unit charge at any point in space. It provides a way to describe how charges interact with each other through the field they create in the surrounding space.

Electric Potential

Electric potential is the electric potential energy per unit charge at a point in space. It provides a scalar description of the electric field and represents the work required to bring a unit positive charge from infinity to that point.

Euler-Lagrange Equation

The Euler-Lagrange equation provides the equations of motion for a system from its Lagrangian. It states that the physical trajectory makes the action stationary, generalizing Newton's laws to generalized coordinates.

Faraday's Law of Electromagnetic Induction

Faraday's law describes how a changing magnetic flux through a closed loop induces an electromotive force (EMF) in that loop. The induced EMF is proportional to the negative rate of change of magnetic flux, which explains electromagnetic induction phenomena including electric generators, transformers, and inductors.

First Law of Thermodynamics

The first law states that for any closed system the change in internal energy `Delta U` equals heat supplied `Q` minus work done `W`. It enforces macroscopic energy conservation without specifying microscopic details, underpinning calorimetry, engine-cycle analysis and phase-transition studies.

Foucault Pendulum

A Foucault pendulum demonstrates Earth's rotation by the precession of its swing plane. The rate of precession at latitude `phi` is `Omega_E sin phi`, where `Omega_E` is Earth's rotation rate.

Gauss's Law

Gauss's law relates the electric flux through a closed surface to the electric charge enclosed by that surface. It is one of Maxwell's equations and provides a powerful method for calculating electric fields in situations with high symmetry.

Gauss's Law for Gravity

Gauss's law for gravity relates the gravitational field flux through a closed surface to the enclosed mass. Unlike electrostatics, gravitational forces are always attractive, resulting in negative flux and the negative sign in the law.

Gravitational Force (Newton's Law of Universal Gravitation)

Newton's law of universal gravitation states that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. This fundamental force governs planetary motion, tides, and large-scale structure of the universe.

Hamilton's Equations

Hamilton's equations are a set of first-order differential equations governing the evolution of generalized coordinates (`q_i`) and conjugate momenta (`p_i`). They provide a symmetric phase-space formulation equivalent to the Euler-Lagrange equation, often simplifying analysis of conserved quantities and facilitating the transition to quantum mechanics.

Hamilton-Jacobi Equation

The Hamilton-Jacobi equation is a reformulation of classical mechanics as a first-order partial differential equation for the action `S(q_i, t)`. It describes the evolution of Hamilton's principal function `S` such that its spatial gradients equal the momenta.

Hooke's Law

Hooke's law states that the force required to extend or compress a spring is proportional to the displacement from its equilibrium position. This linear relationship forms the foundation for understanding elastic behavior in materials and oscillatory motion.

Kirchhoff's Laws

Kirchhoff's laws are fundamental principles for analyzing electrical circuits. Kirchhoff's Current Law (KCL) states that the sum of currents at any node equals zero, while Kirchhoff's Voltage Law (KVL) states that the sum of voltages around any closed loop equals zero.

Lagrangian Mechanics

Lagrangian mechanics reformulates classical mechanics using the Lagrangian function L = T - V, where T is kinetic energy and V is potential energy. The principle of stationary action states that physical systems evolve along paths that make the action S stationary.

Larmor Formula

The Larmor formula describes the power radiated by a non-relativistic accelerating point charge. It shows that any accelerating charge emits electromagnetic radiation, with power proportional to the square of acceleration.

Lenz's Law

Lenz's law states that the direction of an induced current is such that its magnetic field opposes the change in flux that produced it. This law is a manifestation of energy conservation in electromagnetic systems, ensuring that induced currents always act to resist changes in magnetic flux, requiring work to maintain the changing flux.

Liouville's Theorem

Liouville's theorem relates the time-averaged kinetic energy `T` and potential energy of a bound system. For particles with positions `r_i` and forces `F_i`, it states `<2T> = -`.

Magnetic Dipole Moment

A magnetic dipole moment characterizes the magnetic strength and orientation of a current loop or magnetic object. It arises from circulating currents and determines how the system interacts with external magnetic fields through torques and potential energies.

Magnetic Field

The magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. It is produced by moving charges and changing electric fields, and exerts forces on other moving charges and magnetic dipoles.

Maxwell Equations

Maxwell's equations form the foundation of classical electromagnetism, describing how electric and magnetic fields interact with charges and currents. These four coupled partial differential equations unify electricity and magnetism into a single electromagnetic theory, predicting the existence of electromagnetic waves including light.

Maxwell's Equations (Lorenz Gauge)

In Lorenz gauge, Maxwell's equations for the scalar (phi) and vector (vec A) electromagnetic potentials reduce to decoupled inhomogeneous wave equations driven by charge and current densities. These forms make the causal propagation of electromagnetic influences explicit, simplify relativistic formulations, and are widely used in field quantisation and numerical electrodynamics.

Newton's Laws of Motion

Newton's three laws of motion form the foundation of classical mechanics. The First Law (Law of Inertia) states that objects maintain constant velocity unless acted upon by a net force, defining inertial reference frames.

Ohm's Law

Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance between them. This fundamental relationship forms the basis of electrical circuit analysis.

Principle of Least Action

The principle of least action states that the path taken by a physical system between two configurations is the one that makes the action stationary. The action is defined as the time integral of the Lagrangian function.

Rotational Mechanics

Rotational mechanics describes the motion of objects rotating about an axis, providing the rotational analogs of linear motion concepts. Angular velocity relates rotational speed to period and frequency, while torque is the rotational equivalent of force, causing angular acceleration through the moment of inertia.

Simple Harmonic Motion

Simple harmonic motion describes oscillatory motion where the restoring force is proportional to the displacement from equilibrium. This type of motion is fundamental in physics, appearing in systems from pendulums to quantum oscillators.

Virial Theorem

The virial theorem relates the time-averaged kinetic energy to the potential energy of a system in equilibrium. For systems with power-law potentials V ∝ r^n, it provides a direct relationship between kinetic and potential energies.

Wave Equation

The wave equation is a fundamental partial differential equation that describes the propagation of waves through a medium. It relates the second derivatives of a wave function with respect to time and space coordinates.

Wave Interference

Wave interference occurs when two or more waves overlap in space and time. The superposition principle states that the total wave function is the sum of individual wave functions.

Work-Energy Theorem

The work-energy theorem states that the net work done on an object equals the change in its kinetic energy. This fundamental principle connects the concepts of force, displacement, and energy in classical mechanics.

Condensed Matter Physics

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation describes the relationship between pressure and temperature along phase transition curves. It relates the slope of the coexistence curve to the latent heat and volume change during phase transitions, providing fundamental insight into phase diagrams and enabling prediction of vapor pressures, melting points, and boiling points under different conditions.

Heat Equation

The heat equation describes how temperature changes over time due to thermal diffusion. It is a parabolic partial differential equation that governs heat conduction in materials and is fundamental to understanding thermal transport phenomena.

Ideal Gas Law

The ideal gas law describes the relationship between pressure, volume, temperature, and amount of substance for an ideal gas. It combines Boyle's law, Charles's law, and Gay-Lussac's law into a single equation.

Fluid Dynamics

Archimedes' Principle

Archimedes' principle states that the buoyant force on an object immersed in a fluid equals the weight of the fluid displaced by the object. This fundamental law explains why objects float or sink and forms the foundation of fluid statics, ship design, and density measurements.

Bernoulli's Equation (Inviscid Flow)

Bernoulli's equation asserts that along a streamline in steady, incompressible, inviscid flow the sum of static pressure, kinetic head and gravitational head remains constant. It translates fluid-energy conservation into a simple algebraic relation, connecting pressure differences with velocity changes.

Continuity Equation (Fluid Flow)

The continuity equation expresses conservation of mass in fluid dynamics. It states that the rate of increase of density in a volume plus the net outflow of mass through the volume's boundary is zero.

Navier–Stokes Equation (Incompressible)

The incompressible Navier–Stokes equation balances transient and convective inertia with pressure gradients, viscous diffusion and body forces in a Newtonian fluid. As the fundamental dynamical law for liquids and gases at everyday speeds, it captures phenomena from laminar pipe flow to atmospheric turbulence.

Reynolds Number

The Reynolds number is a dimensionless quantity that characterizes the ratio of inertial forces to viscous forces in fluid flow. It determines whether flow is laminar or turbulent: low Reynolds numbers indicate laminar flow dominated by viscous effects, while high Reynolds numbers indicate turbulent flow dominated by inertial effects.

General Relativity and Quantum Cosmology

Einstein Field Equations

Einstein's field equations describe how matter and energy curve spacetime, forming the foundation of general relativity. They relate the geometry of spacetime (left side) to the matter-energy content (right side), showing that matter tells spacetime how to curve, and curved spacetime tells matter how to move.

Gravitational Time Dilation

Gravitational time dilation describes how time passes more slowly in stronger gravitational fields according to general relativity. Clocks closer to massive objects run slower when observed from weaker gravitational fields, with the effect proportional to the gravitational potential difference.

Relativistic Energy and Momentum

Einstein's special relativity revolutionizes our understanding of energy and momentum at high velocities. By redefining force as the rate of change of relativistic momentum and calculating the work done to accelerate a particle, the theory derives the famous relativistic energy formula E = γmc².

Schwarzschild Radius

The Schwarzschild radius defines the event horizon of a black hole - the critical radius where the escape velocity equals the speed of light. Named after Karl Schwarzschild, who found the first exact solution to Einstein's field equations, it represents the boundary beyond which nothing, not even light, can escape the gravitational pull.

Special Relativity Coordinate Transformations

Einstein's special theory of relativity derives the coordinate transformations between inertial reference frames moving at constant relative velocity. Built on two fundamental postulates - the relativity principle and the constancy of light speed - the theory reveals that space and time are unified into spacetime.

High Energy Physics (Theory)

Dirac Equation

The Dirac equation is a first-order relativistic wave equation that consistently describes spin-1/2 particles such as electrons. By linearly combining space-time derivatives with the Dirac gamma matrices, it reconciles quantum mechanics with special relativity while predicting intrinsic spin and the existence of antiparticles.

Dirac Equation in an Electromagnetic Field

The Dirac equation with minimal coupling to the electromagnetic four-potential extends the free-particle equation to describe spin-1/2 fermions interacting with external fields. By replacing space-time derivatives with gauge-covariant ones, it incorporates electric and magnetic interactions while preserving Lorentz covariance and gauge invariance, forming the quantum-mechanical cornerstone of quantum electrodynamics (QED).

QED Lagrangian Density

The QED Lagrangian density merges the standalone photon field contribution and the free spin-½ fermion action by using the simplest gauge-invariant prescription: replacing ordinary derivatives with a covariant derivative. It therefore fully specifies how light and charged matter propagate and interact under a local U(1) symmetry, and its variation directly yields Maxwell's laws sourced by the fermion current and the Dirac equation in an electromagnetic field.

Materials Science

Crystal Lattice Structure

Crystal lattice structure describes the periodic arrangement of atoms in crystalline solids. The structure is characterized by a unit cell with basis vectors that, when repeated, generates the entire crystal through translational symmetry.

Mathematical Physics

Noether's Theorem

Noether's Theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. This fundamental theorem links continuous symmetries in a system's Lagrangian to conserved quantities.

Mesoscale and Nanoscale Physics

Hall Effect

The Hall effect occurs when a current-carrying conductor is placed in a magnetic field perpendicular to the current direction, resulting in a voltage difference across the conductor. This effect allows determination of charge carrier density, mobility, and type (positive or negative) in materials, making it fundamental for semiconductor characterization and magnetic field sensing.

Nuclear Theory

Alpha Decay

Alpha decay is a type of radioactive decay where an atomic nucleus emits an alpha particle (helium-4 nucleus) consisting of two protons and two neutrons. This process occurs in heavy nuclei to reduce their mass and achieve greater stability.

Nuclear Binding Energy

Nuclear binding energy is the energy required to completely separate a nucleus into its constituent protons and neutrons. It represents the mass defect converted to energy when nucleons bind together.

Nuclear Fusion

Nuclear fusion is the process where light atomic nuclei combine to form heavier nuclei, releasing energy when the products have higher binding energy per nucleon. This process powers stars and is the basis for fusion energy research.

Radioactive Decay Law

The radioactive decay law describes the exponential decrease of unstable nuclei over time. The number of undecayed nuclei decreases exponentially with a characteristic decay constant λ.

Optics

Diffraction Grating

A diffraction grating consists of many parallel slits or lines that diffract light, creating interference patterns with sharp, bright maxima at specific angles. The grating equation determines these angles based on the wavelength and grating spacing, making gratings powerful tools for spectroscopy.

Electromagnetic Waves

Electromagnetic waves are self-propagating oscillations of electric and magnetic fields that travel at the speed of light. They result from Maxwell's equations and include visible light, radio waves, X-rays, and all other forms of electromagnetic radiation.

Poynting Vector

The Poynting vector describes the directional energy flux density of electromagnetic fields, representing the rate of energy flow per unit area. It points in the direction of electromagnetic wave propagation and its magnitude gives the intensity.

Snell's Law of Refraction

Snell's law describes the relationship between the angles of incidence and refraction when light passes from one medium to another with different optical properties. The law states that the product of refractive index and sine of angle remains constant across the interface.

Thin Lens Equation

The thin lens equation relates the focal length of a lens to the object and image distances, forming the foundation of geometric optics. It describes how lenses form images by refraction, enabling the design of optical instruments from eyeglasses to telescopes.

Young's Double Slit Experiment

Young's double slit experiment demonstrates wave interference and the wave nature of light (and matter). When coherent light passes through two parallel slits, it creates an interference pattern with alternating bright and dark fringes.

Quantum Physics

Blackbody Radiation

Planck's law describes the spectral energy density of black-body radiation as a function of frequency and temperature. By quantizing electromagnetic modes with energy quanta `h*nu`, it resolves the ultraviolet catastrophe of classical physics and underpins modern quantum theory and thermal emission models.

Compton Scattering

Compton scattering occurs when a photon collides with a free or loosely bound electron, transferring energy and momentum to the electron. The scattered photon has lower energy (longer wavelength) than the incident photon, providing direct evidence for the particle nature of light.

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle states that the simultaneous precise measurement of position and momentum of a particle is fundamentally impossible. The product of their uncertainties has a quantum mechanical lower bound proportional to Planck's constant.

Hydrogen Atom Energy Levels

The hydrogen atom energy levels describe the quantized energy states of an electron bound to a proton. The energy depends only on the principal quantum number n, with the ground state at -13.

Klein-Gordon Equation

The Klein-Gordon equation is a relativistic wave equation that describes spinless particles. It was the first attempt to merge quantum mechanics with special relativity, predating the Dirac equation.

Pauli Exclusion Principle

The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously. This fundamental quantum mechanical rule explains the structure of atoms, the periodic table, stellar stability, and the behavior of matter at high densities.

Photoelectric Effect

The photoelectric effect demonstrates the particle nature of light, where photons with energy above a threshold frequency can eject electrons from a material. Einstein's explanation of this phenomenon led to the concept of light quanta and earned him the Nobel Prize in Physics.

Planck's Quantum Hypothesis

Planck's quantum hypothesis revolutionized physics by proposing that electromagnetic energy is emitted and absorbed in discrete packets called quanta, with energy proportional to frequency. This fundamental relation ended the ultraviolet catastrophe of classical physics and birthed quantum mechanics.

Quantum Harmonic Oscillator

The quantum harmonic oscillator describes a quantum particle in a parabolic potential. Unlike the classical oscillator, energy levels are discrete and equally spaced, with a zero-point energy of ℏω/2.

Quantum Tunneling

Quantum tunneling is the quantum mechanical phenomenon where particles can pass through potential barriers even when their kinetic energy is less than the barrier height. The transmission probability depends exponentially on the barrier width and height, and is strictly forbidden in classical mechanics.

Schrödinger Equation

The time-dependent Schrödinger equation governs the non-relativistic evolution of a quantum particle's wavefunction ψ(x,t). By promoting energy and momentum to differential operators, it relates the temporal derivative of ψ to the kinetic (−ħ²/2m∇²) and potential (V) terms.

Spin Angular Momentum

Spin angular momentum is an intrinsic quantum mechanical property of particles that has no classical analog. Unlike orbital angular momentum arising from spatial motion, spin is an inherent characteristic like mass or charge.

de Broglie Wavelength

The de Broglie wavelength expresses the wave-particle duality by relating the momentum of any particle to its associated wavelength. This fundamental quantum mechanical relation shows that all matter exhibits wave properties, with the wavelength inversely proportional to momentum.

Statistical Mechanics

Boltzmann Distribution

The Boltzmann distribution describes the probability of finding a system in a particular energy state when in thermal equilibrium at temperature T. This fundamental statistical law governs how energy is distributed among particles in thermal systems and forms the cornerstone of statistical mechanics, explaining phenomena from gas kinetics to chemical equilibrium and phase transitions.

Bose-Einstein Distribution

The Bose-Einstein distribution describes the statistical distribution of bosons (particles with integer spin) in thermal equilibrium. Unlike fermions, bosons can occupy the same quantum state with no restriction, leading to phenomena like Bose-Einstein condensation and stimulated emission.

Carnot Efficiency

Carnot efficiency represents the maximum theoretical efficiency of any heat engine operating between two thermal reservoirs. It depends only on the absolute temperatures of the hot and cold reservoirs, establishing an upper limit that no real heat engine can exceed.

Equipartition Theorem

The equipartition theorem states that each quadratic term in the Hamiltonian contributes (1/2)k_B*T to the average energy in thermal equilibrium. This fundamental principle of classical statistical mechanics explains heat capacities, connects microscopic motion to temperature, and provides the classical limit of quantum systems.

Fermi-Dirac Distribution

The Fermi-Dirac distribution describes the statistical distribution of fermions (particles with half-integer spin) in thermal equilibrium. Unlike classical particles, fermions obey the Pauli exclusion principle, limiting occupancy to one particle per quantum state.

Fourier's Law of Heat Conduction

Fourier's conduction law states that heat-flux density `vec{q}` is proportional to the negative temperature gradient, `vec{q} = -k*nabla T`. It defines thermal conductivity, leads directly to the heat-diffusion equation and quantifies how rapidly solids or stationary fluids conduct heat.

Gibbs Free Energy

Gibbs free energy is a thermodynamic potential that measures the maximum reversible work that can be performed by a system at constant temperature and pressure. It determines the direction of spontaneous processes: reactions proceed when ΔG < 0.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the probability distribution of particle speeds in a classical ideal gas at thermal equilibrium. It shows that most particles have intermediate speeds, with exponentially fewer particles at very high speeds.

Partition Function (Canonical Ensemble)

The partition function is the central quantity in statistical mechanics that connects microscopic energy states to macroscopic thermodynamic properties. It represents the sum over all possible energy states weighted by their Boltzmann factors.

Rayleigh-Jeans Law

The Rayleigh-Jeans law describes the classical limit of blackbody radiation for long wavelengths (low frequencies), where the energy density is proportional to temperature and inversely proportional to the fourth power of wavelength. This classical result agrees with experiments at long wavelengths but fails catastrophically at short wavelengths, leading to the ultraviolet catastrophe that motivated Planck's quantum theory of radiation.

Second Law of Thermodynamics (Clausius Inequality)

The second law of thermodynamics, expressed through the Clausius inequality, states that the integral of heat flow divided by temperature over any closed cycle is non-positive. This leads to the definition of entropy as a state function whose total change in an isolated system is always non-negative.

Third Law of Thermodynamics

The third law of thermodynamics states that the entropy of a perfect crystal approaches zero as temperature approaches absolute zero. This law provides an absolute reference point for entropy calculations and explains why absolute zero is unattainable.