Essential Physics Equations

Find physics equations, symbols, and SI units.

Classical mechanics

How forces change motion, how energy is transferred, and why momentum is conserved.

Newton’s second law

∑F=ma\textcolor{#e11d48}{\sum} \textcolor{#e11d48}{\mathbf{F}} = \textcolor{#7c3aed}{m}\textcolor{#2563eb}{\mathbf{a}}

Conservation of linear momentum

∑ipi,initial=∑ipi,final\textcolor{#e11d48}{\sum_{\textcolor{#e11d48}{i}}} \textcolor{#0f766e}{\mathbf{p}_{i,\mathrm{initial}}} = \textcolor{#e11d48}{\sum_{\textcolor{#e11d48}{i}}} \textcolor{#0f766e}{\mathbf{p}_{i,\mathrm{final}}}

Kinetic energy

K=12mv2\textcolor{#be185d}{K} = \frac{1}{2}\textcolor{#7c3aed}{m}\textcolor{#0369a1}{v}^2

Work–energy theorem

Wnet=ΔK\textcolor{#c2410c}{W_{\mathrm{net}}} = \Delta \textcolor{#be185d}{K}

Newton’s law of gravitation

F=Gm1m2r2\textcolor{#e11d48}{F} = \textcolor{#a16207}{G}\frac{\textcolor{#7c3aed}{m_1}\textcolor{#7c3aed}{m_2}}{\textcolor{#4d7c0f}{r}^2}

Oscillations, waves, and optics

Repeating motion, the propagation of disturbances, and the bending of light.

Simple harmonic motion

d2xdt2=−ω2x\frac{d^2\textcolor{#0369a1}{x}}{d\textcolor{#2563eb}{t}^2} = -\textcolor{#c2410c}{\omega}^2 \textcolor{#0369a1}{x}

Wave speed, frequency, and wavelength

vphase=fλ\textcolor{#0369a1}{v_{\mathrm{phase}}} = \textcolor{#7c3aed}{f}\textcolor{#4d7c0f}{\lambda}

Snell’s law of refraction

n1sin⁡θ1=n2sin⁡θ2\textcolor{#0f766e}{n_1}\sin\textcolor{#c2410c}{\theta_1} = \textcolor{#0f766e}{n_2}\sin\textcolor{#c2410c}{\theta_2}

Electromagnetism

Charges, currents, and fields. The four Maxwell equations below connect electricity, magnetism, and light.

Coulomb’s law

F=14πε0∣q1q2∣r2\textcolor{#e11d48}{F} = \frac{1}{4\textcolor{#a16207}{\pi}\textcolor{#a16207}{\varepsilon_0}}\frac{|\textcolor{#c2410c}{q_1}\textcolor{#c2410c}{q_2}|}{\textcolor{#4d7c0f}{r}^2}

Lorentz force

F=q(E+v×B)\textcolor{#e11d48}{\mathbf{F}} = \textcolor{#c2410c}{q}\left(\textcolor{#e11d48}{\mathbf{E}} + \textcolor{#0369a1}{\mathbf{v}}\times\textcolor{#0f766e}{\mathbf{B}}\right)

Gauss’s law for electricity

∯SE⋅dA=Qencε0\textcolor{#e11d48}{\oiint_{\textcolor{#4d7c0f}{S}}} \textcolor{#e11d48}{\mathbf{E}}\cdot \textcolor{#0369a1}{d\mathbf{A}} = \frac{\textcolor{#c2410c}{Q_{\mathrm{enc}}}}{\textcolor{#a16207}{\varepsilon_0}}

Gauss’s law for magnetism

∯SB⋅dA=0\textcolor{#e11d48}{\oiint_{\textcolor{#4d7c0f}{S}}} \textcolor{#0f766e}{\mathbf{B}}\cdot \textcolor{#0369a1}{d\mathbf{A}} = 0

Faraday’s law of induction

∮CE⋅dℓ=−dΦBdt\textcolor{#e11d48}{\oint_{\textcolor{#4d7c0f}{C}}} \textcolor{#e11d48}{\mathbf{E}}\cdot \textcolor{#0369a1}{d\boldsymbol{\ell}} = -\frac{d\textcolor{#be185d}{\Phi_B}}{d\textcolor{#2563eb}{t}}

Ampère–Maxwell law

∮CB⋅dℓ=μ0 Ienc+μ0ε0dΦEdt\textcolor{#e11d48}{\oint_{\textcolor{#4d7c0f}{C}}} \textcolor{#0f766e}{\mathbf{B}}\cdot \textcolor{#0369a1}{d\boldsymbol{\ell}} = \textcolor{#7c3aed}{\mu_0}\,\textcolor{#c2410c}{I_{\mathrm{enc}}} + \textcolor{#7c3aed}{\mu_0}\textcolor{#a16207}{\varepsilon_0}\frac{d\textcolor{#be185d}{\Phi_E}}{d\textcolor{#2563eb}{t}}

Ohm’s law

V=IR\textcolor{#7c3aed}{V} = \textcolor{#c2410c}{I}\textcolor{#0f766e}{R}

Thermodynamics and statistical physics

Heat, work, temperature, and the connection between microscopic states and macroscopic behaviour.

Ideal gas law

PV=nRT\textcolor{#be185d}{P}\textcolor{#7c3aed}{V} = \textcolor{#0f766e}{n}\textcolor{#0f766e}{R}\textcolor{#0369a1}{T}

First law of thermodynamics

ΔU=Q−W\Delta \textcolor{#e11d48}{U} = \textcolor{#c2410c}{Q} - \textcolor{#c2410c}{W}

Second law of thermodynamics

ΔSisolated≥0\Delta \textcolor{#4d7c0f}{S_{\mathrm{isolated}}} \geq 0

Boltzmann’s entropy formula

S=kBln⁡Ω\textcolor{#4d7c0f}{S} = \textcolor{#2563eb}{k_{\mathrm{B}}}\ln\textcolor{#be185d}{\Omega}

Relativity and spacetime

The relationship between mass and energy, the dependence of elapsed time on motion, and gravity as spacetime geometry.

Mass–energy equivalence

E0=mc2\textcolor{#e11d48}{E_0} = \textcolor{#7c3aed}{m}\textcolor{#0369a1}{c}^2

Relativistic energy–momentum relation

E2=p2c2+m2c4\textcolor{#e11d48}{E}^2 = \textcolor{#0f766e}{p}^2\textcolor{#0369a1}{c}^2 + \textcolor{#7c3aed}{m}^2\textcolor{#0369a1}{c}^4

Time dilation

Δt=Δτ1−v2/c2\Delta \textcolor{#2563eb}{t} = \frac{\Delta\textcolor{#2563eb}{\tau}}{\sqrt{1-\textcolor{#0369a1}{v}^2/\textcolor{#0369a1}{c}^2}}

Einstein’s field equations

Gμν+Λgμν=8πGc4Tμν\textcolor{#a16207}{G_{\mu\nu}} + \textcolor{#c2410c}{\Lambda} \textcolor{#2563eb}{g_{\mu\nu}} = \frac{8\textcolor{#a16207}{\pi} \textcolor{#a16207}{G}}{\textcolor{#0369a1}{c}^4}\textcolor{#0369a1}{T_{\mu\nu}}

Quantum physics

Energy quanta, matter waves, probability amplitudes, and the limits on jointly sharp measurements.

Photon energy

E=hf\textcolor{#e11d48}{E} = \textcolor{#a16207}{h}\textcolor{#7c3aed}{f}

De Broglie wavelength

λ=hp\textcolor{#4d7c0f}{\lambda} = \frac{\textcolor{#a16207}{h}}{\textcolor{#0f766e}{p}}

Time-dependent Schrödinger equation

iℏ∂ψ∂t=[−ℏ22m∇2+V]ψ\textcolor{#e11d48}{i}\textcolor{#a16207}{\hbar}\frac{\partial\textcolor{#7c3aed}{\psi}}{\partial \textcolor{#2563eb}{t}} = \left[-\frac{\textcolor{#a16207}{\hbar}^2}{2\textcolor{#7c3aed}{m}}\nabla^2 + \textcolor{#7c3aed}{V}\right]\textcolor{#7c3aed}{\psi}

Born rule for position

P(r∈R)=∫R∣ψ(r,t)∣2 d3r\textcolor{#be185d}{P}(\textcolor{#4d7c0f}{\mathbf{r}}\in\textcolor{#4d7c0f}{\mathcal{R}}) = \textcolor{#e11d48}{\int_{\textcolor{#4d7c0f}{\mathcal{R}}}} |\textcolor{#7c3aed}{\psi}(\textcolor{#4d7c0f}{\mathbf{r}},\textcolor{#2563eb}{t})|^2\,d^3\textcolor{#4d7c0f}{r}

Heisenberg uncertainty principle

Δx Δpx≥ℏ2\Delta \textcolor{#0369a1}{x}\,\Delta \textcolor{#0f766e}{p_x} \geq \frac{\textcolor{#a16207}{\hbar}}{2}