Laws & Proofs

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

1. Th
   Archimedes' principle

   $F_A = \rho_{\text{fluid}}\, g\, V_{\text{submerged}}$ — the buoyant force on an object immersed in a fluid equals the weight of the fluid it displaces.TheoremProvedFluid mechanics
2. Fo
   Condition for a standing wave on a string fixed at both ends

   $L = n\lambda/2$, $n = 1, 2, 3, \ldots$ — the string length L must equal an integer number of half-wavelengths for a stable standing wave.FormulaProvedOscillations and waves
3. Fo
   Coulomb's law

   $F = k\dfrac{|q_1 q_2|}{r^2}$ — the interaction force between two point charges $q_1, q_2$ separated by $r$, with $k \approx 9 \times 10^9\ \text{N·m}^2/\text{C}^2$.FormulaProvedElectricity and magnetism
4. Fo
   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
5. Th
   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
6. Fo
   Newton's law of universal gravitation

   $F = G\dfrac{m_1 m_2}{r^2}$ — the attractive force between two point masses $m_1, m_2$ separated by $r$, with gravitational constant $G \approx 6.674 \times 10^{-11}\ \text{N·m}^2/\text{kg}^2$.FormulaProvedNewtonian mechanics
7. Fo
   Newton's second law

   $\vec{F} = m\vec{a}$ — $\vec{F}$ is the net force on the object, $m$ its inertial mass, and $\vec{a}$ its acceleration.FormulaProvedNewtonian mechanics
8. Th
   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
9. Fo
   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
10. Fo
    The Bernoulli equation

    $p + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant along a streamline}$ — for steady, inviscid, incompressible flow.FormulaProvedFluid mechanics
11. Th
    The Faraday–Lenz law of electromagnetic induction

    $\varepsilon = -N\dfrac{d\Phi}{dt}$ — the EMF induced in a closed loop is proportional to the rate of change of magnetic flux $\Phi$ through it ($N$ is the number of turns); the minus sign (Lenz) shows the induced current always opposes the change that created it.TheoremProvedElectricity and magnetism
12. Th
    The first law of thermodynamics

    $\Delta U = Q - A$ — the change in a system’s internal energy equals the heat it absorbs minus the work it performs.TheoremProvedThermal physics
13. Th
    The Heisenberg uncertainty principle

    $\Delta x \, \Delta p \geq \hbar/2$ — the product of the uncertainties in a particle’s position and momentum can never be smaller than the fixed bound $\hbar/2$.TheoremProvedQuantum mechanics
14. Fo
    The ideal gas law

    $pV = nRT$ — with $p$ pressure, $V$ volume, $n$ number of moles, $R \approx 8.314\ \text{J/(mol·K)}$, $T$ absolute temperature.FormulaProvedThermal physics
15. Fo
    The Schrödinger equation

    $i\hbar\,\partial_t\psi = \hat{H}\psi = -\dfrac{\hbar^2}{2m}\nabla^2\psi + V\psi$ — the fundamental equation describing the time evolution of the quantum wavefunction $\psi$.FormulaProvedQuantum mechanics
16. Th
    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