Physic Labs

Laws & Proofs

Physical laws, formulas and axioms — statement, why it holds, and a step-by-step proof where one exists.

All kindsTheoremFormulaAxiomLemmaConjectureDefinition

Motion in a central field, the Kepler problem

  1. Kepler's first law — ellipses
    Each planet traces an ellipse with the Sun at one of its two foci. The Sun is not at the ellipse's center but offset toward a focus.TheoremProvedAnalytical mechanics
  2. Kepler's second law — equal areas
    The radius vector from Sun to planet sweeps equal areas in equal times — so the planet moves faster at perihelion and slower at aphelion.TheoremProvedAnalytical mechanics
  3. Kepler's third law — harmonic
    The square of the orbital period T² is proportional to the cube of the semi-major axis a³ — for all planets orbiting the same star: T² / a³ = 4π² / (G·M) = const with G the gravitational constant and M the star's mass.TheoremProvedAnalytical mechanics

Blackbody radiation and Planck’s law

  1. Stefan–Boltzmann law
    M = σ·T⁴ — the total power radiated per unit area of an ideal black body is proportional to the fourth power of its absolute temperature, with σ = 5.670374×10⁻⁸ W·m⁻²·K⁻⁴.FormulaProvedCondensed matter physics

Electric charge and Coulomb’s law

  1. Coulomb's law
    F=k∣q1q2∣r2F = k\dfrac{|q_1 q_2|}{r^2} — the interaction force between two point charges q1,q2q_1, q_2 separated by rr, with k≈9×109 N⋅m2/C2k \approx 9 \times 10^9\ \text{N·m}^2/\text{C}^2.FormulaProvedElectricity and magnetism

Electromagnetic induction, Faraday–Lenz law

  1. Lenz's law of induction
    The induced current in a closed loop always flows in the direction such that its own magnetic field opposes the flux change that created it — mathematically this is the "−" sign in Faraday's ε = −dΦ/dt.TheoremProvedElectricity and magnetism
  2. The Faraday–Lenz law of electromagnetic induction
    ε=−NdΦdt\varepsilon = -N\dfrac{d\Phi}{dt} — the EMF induced in a closed loop is proportional to the rate of change of magnetic flux Φ\Phi through it (NN is the number of turns); the minus sign (Lenz) shows the induced current always opposes the change that created it.TheoremProvedElectricity and magnetism

Magnetic field and magnetic force

  1. Biot–Savart law
    dB = (μ₀/4π) · I·dℓ × r̂ / r² — a current element I·dℓ produces at distance r a field dB perpendicular to the plane of dℓ and r, falling as 1/r².FormulaProvedElectricity and magnetism

Ohm's law for a circuit segment

  1. Ohm's law
    I = U/R, the current I through a circuit segment is proportional to the voltage U and inversely proportional to its resistance R.FormulaProvedElectricity and magnetism

Ohm’s law for a complete circuit and EMF

  1. Kirchhoff circuit laws
    At every node: total current in equals total current out (Kirchhoff current law — KCL). Around every closed loop: total emf equals total voltage drop (Kirchhoff voltage law — KVL).TheoremProvedElectricity and magnetism

Archimedes’ principle

  1. Archimedes' principle
    FA=ρfluid g VsubmergedF_A = \rho_{\text{fluid}}\, g\, V_{\text{submerged}} — the buoyant force on an object immersed in a fluid equals the weight of the fluid it displaces.TheoremProvedFluid mechanics

Pressure in fluids and atmospheric pressure

  1. Pascal's principle
    Additional pressure applied to a confined fluid is transmitted undiminished to every point of the fluid and the vessel walls — independent of container shape.TheoremProvedFluid mechanics

The Bernoulli equation

  1. The Bernoulli equation
    p+12ρv2+ρgh=constant along a streamlinep + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant along a streamline} — for steady, inviscid, incompressible flow.FormulaProvedFluid mechanics

Gravity and weight

  1. Newton's law of universal gravitation
    F=Gm1m2r2F = G\dfrac{m_1 m_2}{r^2} — the attractive force between two point masses m1,m2m_1, m_2 separated by rr, with gravitational constant G≈6.674×10−11 N⋅m2/kg2G \approx 6.674 \times 10^{-11}\ \text{N·m}^2/\text{kg}^2.FormulaProvedNewtonian mechanics

Newton's second law

  1. Newton's first law (inertia)
    An object remains at rest or in uniform straight-line motion unless acted on by a force, or as long as the net force acting on it is zero.TheoremProvedNewtonian mechanics
  2. Newton's second law
    F⃗=ma⃗\vec{F} = m\vec{a} — F⃗\vec{F} is the net force on the object, mm its inertial mass, and a⃗\vec{a} its acceleration.FormulaProvedNewtonian mechanics
  3. Newton's third law (action-reaction)
    When object A exerts a force on object B, object B simultaneously exerts a force on A that is equal in magnitude, same line of action, but opposite in direction.TheoremProvedNewtonian mechanics

Plane mirrors

  1. Law of reflection
    The angle of reflection equals the angle of incidence (i' = i); incident ray, reflected ray, and normal at the contact point all lie in the same plane.FormulaProvedOptics

Refraction of light, refractive index

  1. Snell–Descartes law of refraction
    n₁·sin i = n₂·sin r — a light ray crossing from a medium of index n₁ to n₂ bends at the interface, where i is the angle of incidence and r the angle of refraction (measured from the normal).FormulaProvedOptics

Standing waves

  1. Condition for a standing wave on a string fixed at both ends
    L=nλ/2L = n\lambda/2, n=1,2,3,…n = 1, 2, 3, \ldots — the string length L must equal an integer number of half-wavelengths for a stable standing wave.FormulaProvedOscillations and waves

The spring–mass oscillator

  1. Hooke's law
    F = −k·x — the elastic force exerted on a body attached to a spring is proportional to the displacement x and opposite to it. The spring constant k (N/m) measures its stiffness.FormulaProvedOscillations and waves

Electrodes and electrode potentials

  1. Faraday's law of electrolysis
    m = (A·I·t) / (n·F) — the mass deposited or dissolved at an electrode is proportional to the charge Q = I·t that passed, where A is the molar mass, n the valence, and F = 96485 C/mol the Faraday constant.FormulaProvedPhysical chemistry

The Heisenberg uncertainty principle

  1. The Heisenberg uncertainty principle
    Δx Δp≥ℏ/2\Delta x \, \Delta p \geq \hbar/2 — the product of the uncertainties in a particle’s position and momentum can never be smaller than the fixed bound ℏ/2\hbar/2.TheoremProvedQuantum mechanics

The Schrödinger equation

  1. The Schrödinger equation
    iℏ ∂tψ=H^ψ=−ℏ22m∇2ψ+Vψi\hbar\,\partial_t\psi = \hat{H}\psi = -\dfrac{\hbar^2}{2m}\nabla^2\psi + V\psi — the fundamental equation describing the time evolution of the quantum wavefunction ψ\psi.FormulaProvedQuantum mechanics

Relativistic momentum and energy, E = mc²

  1. Mass-energy equivalence
    E = mc^2, the rest energy E of an object equals its rest mass m times the speed of light squared.FormulaProvedTheory of relativity

Gas laws

  1. Gay-Lussac law (volume–temperature)
    V/T = const at constant pressure — the volume of a fixed amount of gas is proportional to its absolute temperature T (in kelvins).FormulaProvedThermal physics

Ideal gas equation of state

  1. The ideal gas law
    pV=nRTpV = nRT — with pp pressure, VV volume, nn number of moles, R≈8.314 J/(mol⋅K)R \approx 8.314\ \text{J/(mol·K)}, TT absolute temperature.FormulaProvedThermal physics

The first law of thermodynamics

  1. The first law of thermodynamics
    ΔU=Q−A\Delta U = Q - A — the change in a system’s internal energy equals the heat it absorbs minus the work it performs.TheoremProvedThermal physics

The second law of thermodynamics

  1. The second law of thermodynamics
    The entropy of an isolated system never decreases over time; heat does not spontaneously flow from a colder body to a hotter one without external work.TheoremProvedThermal physics