A18
Area A · I
m2 │ L2 │ A = l × b
Acceleration a · I
m s−2 │ L T−2 │ a = dv/dt
Acceleration due to gravity g · I
m s−2 │ L T−2 │ g = GM/R2
Angle θ · I
rad │ — │ θ = arc / radius
Angular velocity ω · I
rad s−1 │ T−1 │ ω = dθ/dt = 2πf
Angular acceleration α · I
rad s−2 │ T−2 │ α = dω/dt
Angular momentum L · I
kg m2 s−1 │ M L2 T−1 │ L = r × p = Iω
Areal velocity dA/dt · I
m2 s−1 │ L2 T−1 │ dA/dt = L/2m
Angular frequency ω · II
rad s−1 │ T−1 │ ω = 2πf
Amplitude A · II
m │ L │ greatest displacement from equilibrium
Acoustic pressure amplitude p₀ · II
Pa │ M L−1 T−2 │ p₀ = Bk s₀ = ρvω s₀
Amount of substance n · III
mol │ mol │ base quantity
Avogadro constant N_A · III
mol−1 │ mol−1 │ N = n N_A
Angle of deviation δ · IV
rad │ — │ δ = i + e − A
Angular magnification M · IV
— │ — │ M = f_o/f_e (telescope)
Aperture D · IV
m │ L │ the clear diameter of the optic
Atomic mass unit u · VI
kg │ M │ 1 u = 1/12 of a 12C atom
Activity A · VI
Bq │ T−1 │ A = λN
B7
Bulk modulus B, K · I
Pa │ M L−1 T−2 │ B = −P / (ΔV/V)
Beat frequency f_b · II
Hz │ T−1 │ f_b = |f₁ − f₂|
Boltzmann constant k_B · III
J K−1 │ M L2 T−2 K−1 │ PV = Nk_BT
Brewster angle θ_B · IV
rad │ — │ tan θ_B = n
Bohr radius a₀ · VI
m │ L │ a₀ = 4πε₀ℏ2/m_e e2
Binding energy E_b · VI
J │ M L2 T−2 │ E_b = Δm c2
Bragg angle θ · VI
rad │ — │ 2d sin θ = nλ
C25
Compressibility k · I
Pa−1 │ M−1 L T2 │ k = 1/B
Coefficient of viscosity η · I
Pa s │ M L−1 T−1 │ F = ηA (dv/dx)
Coefficient of friction μ · I
— │ — │ f = μN
Coefficient of restitution e · I
— │ — │ e = separation speed / approach speed
Centripetal acceleration a_c · I
m s−2 │ L T−2 │ a_c = v2/r = ω2r
Capillary rise h · I
m │ L │ h = 2T cos θ / rρg
Centre of mass r_cm · I
m │ L │ r_cm = Σm r / Σm
Compressibility of a medium 1/B · II
Pa−1 │ M−1 L T2 │ v_sound = √(B/ρ)
Coefficient of linear expansion α · III
K−1 │ K−1 │ Δl = lαΔT
Coefficient of volume expansion γ · III
K−1 │ K−1 │ ΔV = VγΔT
Coefficient of performance COP · III
— │ — │ COP = Q_c/W
Critical angle θ_c · IV
rad │ — │ sin θ_c = 1/n
Coherence length L_c · IV
m │ L │ L_c = λ2/Δλ
Cauchy constant B B · IV
m2 │ L2 │ n = A + B/λ2 (A is a pure number)
Current density J · V
A m−2 │ L−2 A │ J = I/A = nqv_d
Capacitance C · V
F │ M−1 L−2 T4 A2 │ C = Q/V
Conductance G · V
S │ M−1 L−2 T3 A2 │ G = 1/R
Conductivity σ · V
S m−1 │ M−1 L−3 T3 A2 │ σ = 1/ρ = nq2τ/m
Charge-to-mass ratio e/m · V
C kg−1 │ M−1 T A │ e/m from r = mv/qB
Cyclotron frequency f_c · V
Hz │ T−1 │ f_c = qB/2πm
Compton wavelength λ_C · VI
m │ L │ λ_C = h/m_e c
Cross-section σ · VI
m2 │ L2 │ reaction rate = nσv
Carrier concentration n, p · VI
m−3 │ L−3 │ np = n_i2 (mass-action law)
Current gain (common base) α · VI
— │ — │ α = I_C/I_E
Current gain (common emitter) β · VI
— │ — │ β = I_C/I_B = α/(1−α)
D9
Density ρ · I
kg m−3 │ M L−3 │ ρ = m/V
Doppler shift Δf · II
Hz │ T−1 │ f′ = f₀ (v ± v_o)/(v ∓ v_s)
Damping constant b · II
kg s−1 │ M T−1 │ F_damp = −bv
Degrees of freedom f · III
— │ — │ U = (f/2) nRT
Dispersive power ω · IV
— │ — │ ω = (n_v − n_r)/(n_y − 1)
Drift velocity v_d · V
m s−1 │ L T−1 │ I = nAqv_d
Displacement current i_d · V
A │ A │ i_d = ε₀ dΦ_E/dt
de Broglie wavelength λ · VI
m │ L │ λ = h/p = h/mv
Decay constant λ · VI
s−1 │ T−1 │ N = N₀e−λᵗ
E22
Energy E · I
J │ M L2 T−2 │ KE = ½mv2, PE = mgh
Escape velocity v_e · I
m s−1 │ L T−1 │ v_e = √(2GM/R)
Efficiency η · I
— │ — │ useful output / total input
Elastic potential energy U · I
J │ M L2 T−2 │ U = ½kx2
Energy density (elastic) u · I
J m−3 │ M L−1 T−2 │ u = ½ × stress × strain
Excess pressure in a bubble ΔP · I
Pa │ M L−1 T−2 │ ΔP = 4T/R (soap), 2T/R (drop)
End correction e · II
m │ L │ e ≈ 0.6 r
Enthalpy H · III
J │ M L2 T−2 │ H = U + PV
Entropy S · III
J K−1 │ M L2 T−2 K−1 │ ΔS = ∫dQ_rev/T
Emissivity ε · III
— │ — │ ε = 1 for a blackbody
Electric charge q, Q · V
C │ T A │ q = It
Electric current I · V
A │ A │ base quantity; I = dq/dt
Electric field E · V
V m−1 │ M L T−3 A−1 │ E = F/q
Electric potential V · V
V │ M L2 T−3 A−1 │ V = W/q
Electromotive force ε · V
V │ M L2 T−3 A−1 │ ε = W/q at zero current
Electric flux Φ_E · V
V m │ M L3 T−3 A−1 │ Φ_E = ∮E · dA = q/ε₀
Electric dipole moment p · V
C m │ L T A │ p = q d
Energy density (electric) u_E · V
J m−3 │ M L−1 T−2 │ u_E = ½ε₀E2
Energy density (magnetic) u_B · V
J m−3 │ M L−1 T−2 │ u_B = B2/2μ₀
Energy gap E_g · VI
J │ M L2 T−2 │ the forbidden band width
Electron rest mass m_e · VI
kg │ M │ —
Elementary charge e · VI
C │ T A │ the quantum of charge
F6
Force F · I
N │ M L T−2 │ F = ma
Frequency of revolution f, ν · I
Hz │ T−1 │ f = 1/T = ω/2π
Frequency f, ν · II
Hz │ T−1 │ f = 1/T
Focal length f · IV
m │ L │ 1/v − 1/u = 1/f
Fringe width β · IV
m │ L │ β = λD/d
Fine-structure constant α · VI
— │ — │ α = e2/4πε₀ℏc
G5
Gravitational constant G · I
N m2 kg−2 │ M−1 L3 T−2 │ F = Gm₁m₂/r2
Gravitational field strength E_g · I
N kg−1 │ L T−2 │ E_g = F/m
Gravitational potential V_g · I
J kg−1 │ L2 T−2 │ V_g = −GM/r
Gravitational potential energy U · I
J │ M L2 T−2 │ U = −GMm/r
Gas constant R · III
J mol−1 K−1 │ M L2 T−2 K−1 mol−1 │ PV = nRT
H5
Harmonic number n · II
— │ — │ f_n = n f₁
Heat Q · III
J │ M L2 T−2 │ Q = mcΔT
Heat capacity C · III
J K−1 │ M L2 T−2 K−1 │ C = mc
Heat current dQ/dt · III
W │ M L2 T−3 │ dQ/dt = ΔT / R_th
Half-life t₁⁄₂ · VI
s │ T │ t₁⁄₂ = ln2 / λ
I7
Impulse J · I
N s │ M L T−1 │ J = ∫F dt = Δp
Intensity I · II
W m−2 │ M T−3 │ I = P/A ∝ A2
Internal energy U · III
J │ M L2 T−2 │ ΔU = Q − W
Illuminance E_v · IV
lx │ L−2 J │ E_v = Φ_v/A
Inductance L · V
H │ M L2 T−2 A−2 │ NΦ = LI; ε = −L dI/dt
Impedance Z · V
Ω │ M L2 T−3 A−2 │ Z = √(R2 + (X_L − X_C)2)
Ionisation energy E_i · VI
J │ M L2 T−2 │ E_i = 13.6 Z2/n2 eV (hydrogen-like)
K2
Kinetic energy K · I
J │ M L2 T−2 │ K = ½mv2 = p2/2m
Kinematic viscosity ν · I
m2 s−1 │ L2 T−1 │ ν = η/ρ
L7
Length l, x · I
m │ L │ base quantity
Linear mass density μ · I
kg m−1 │ M L−1 │ μ = m/l
Latent heat L · III
J kg−1 │ L2 T−2 │ Q = mL
Luminous intensity I_v · IV
cd │ J │ base quantity
Luminous flux Φ_v · IV
lm │ J │ Φ_v = I_v Ω
Least distance of distinct vision D · IV
m │ L │ the near point of a normal eye
Lorentz factor γ · VI
— │ — │ γ = 1/√(1 − v2/c2)
M19
Mass m · I
kg │ M │ base quantity
Momentum p · I
kg m s−1 │ M L T−1 │ p = mv
Moment of inertia I · I
kg m2 │ M L2 │ I = Σ m r2
Mechanical advantage MA · I
— │ — │ load / effort
Mach number M · II
— │ — │ M = v_object / v_sound
Molar specific heat C · III
J mol−1 K−1 │ M L2 T−2 K−1 mol−1 │ Q = nCΔT
Mean free path λ · III
m │ L │ λ = 1/(√2 nπd2)
Molar mass M · III
kg mol−1 │ M mol−1 │ M = m/n
Magnification m · IV
— │ — │ m = v/u = h′/h
Mobility μ · V
m2 V−1 s−1 │ M−1 T2 A │ μ = v_d/E
Magnetic field B · V
T │ M T−2 A−1 │ F = qv × B
Magnetic flux Φ_B · V
Wb │ M L2 T−2 A−1 │ Φ_B = ∫B · dA
Magnetic dipole moment m · V
A m2 │ L2 A │ m = NIA
Magnetising field H · V
A m−1 │ L−1 A │ B = μ₀(H + M)
Magnetisation M · V
A m−1 │ L−1 A │ M = m/V
Magnetic susceptibility χ · V
— │ — │ M = χH; μ_r = 1 + χ
Mutual inductance M · V
H │ M L2 T−2 A−2 │ ε₂ = −M dI₁/dt
Mass defect Δm · VI
kg │ M │ Δm = Zm_p + Nm_n − M
Mean life τ · VI
s │ T │ τ = 1/λ = t₁⁄₂/ln2
N2
Numerical aperture NA · IV
— │ — │ NA = n sin θ
Nuclear radius R · VI
m │ L │ R = R₀A^(1/3), R₀ = 1.2 fm
O1
Optical path length Δ · IV
m │ L │ Δ = n × geometric path
P17
Potential energy U · I
J │ M L2 T−2 │ U = mgh (near Earth)
Power P · I
W │ M L2 T−3 │ P = dW/dt = F · v
Pressure P, p · I
Pa │ M L−1 T−2 │ P = F/A
Poisson's ratio σ · I
— │ — │ lateral strain / longitudinal strain
Phase φ · II
rad │ — │ y = A sin(ωt + φ)
Phase difference Δφ · II
rad │ — │ Δφ = (2π/λ)·Δx
Particle velocity (wave) u · II
m s−1 │ L T−1 │ u = ∂y/∂t = −v (∂y/∂x)
Power of a lens P · IV
D │ L−1 │ P = 1/f (f in metres)
Path difference Δx · IV
m │ L │ Δx = d sin θ
Potential difference ΔV · V
V │ M L2 T−3 A−1 │ ΔV = W/q = IR
Permittivity of free space ε₀ · V
F m−1 │ M−1 L−3 T4 A2 │ F = q₁q₂/4πε₀r2
Permeability of free space μ₀ · V
H m−1 │ M L T−2 A−2 │ B = μ₀I/2πr
Power factor cos φ · V
— │ — │ P = V_rms I_rms cos φ
Poynting vector S · V
W m−2 │ M T−3 │ S = (E × B)/μ₀
Planck constant h · VI
J s │ M L2 T−1 │ E = hν
Photon energy E · VI
J │ M L2 T−2 │ E = hν = hc/λ
Photon momentum p · VI
kg m s−1 │ M L T−1 │ p = h/λ = E/c
Q1
Quality factor Q · II
— │ — │ Q = ω₀ / Δω = 2π × (energy stored / energy lost per cycle)
R19
Relative density — · I
— │ — │ ρ / ρ_water
Reynolds number Re · I
— │ — │ Re = ρvD/η
Radius of gyration K · I
m │ L │ I = MK2
Ratio of specific heats γ · III
— │ — │ γ = C_P/C_V = 1 + 2/f
RMS speed v_rms · III
m s−1 │ L T−1 │ v_rms = √(3RT/M)
Refractive index n, μ · IV
— │ — │ n = c/v = sin i / sin r
Radius of curvature R · IV
m │ L │ R = 2f (mirror)
Resolving power (telescope) 1/dθ · IV
rad−1 │ — │ dθ = 1.22 λ/D
Relative permittivity ε_r, K · V
— │ — │ C = K C₀
Resistance R · V
Ω │ M L2 T−3 A−2 │ R = V/I
Resistivity ρ · V
Ω m │ M L3 T−3 A−2 │ R = ρl/A
Relative permeability μ_r · V
— │ — │ μ = μ_r μ₀
Reactance X · V
Ω │ M L2 T−3 A−2 │ X_L = ωL, X_C = 1/ωC
Radiation pressure p_rad · V
Pa │ M L−1 T−2 │ p = I/c (absorbed), 2I/c (reflected)
Reduced Planck constant ℏ · VI
J s │ M L2 T−1 │ ℏ = h/2π; mvr = nℏ
Rydberg constant R · VI
m−1 │ L−1 │ 1/λ = RZ2(1/n_f2 − 1/n_i2)
Rest energy E₀ · VI
J │ M L2 T−2 │ E₀ = m₀c2
Rectifier ripple frequency f_r · VI
Hz │ T−1 │ f_r = f (half-wave), 2f (full-wave)
Radiation dose D · VI
Gy │ L2 T−2 │ D = energy absorbed / mass
S13
Speed v · I
m s−1 │ L T−1 │ v = distance / time
Stress σ · I
Pa │ M L−1 T−2 │ σ = F/A
Strain ε · I
— │ — │ ε = Δl/l
Shear modulus η, G · I
Pa │ M L−1 T−2 │ G = shear stress / shear strain
Surface tension T, S · I
N m−1 │ M T−2 │ T = F/l = energy per area
Solid angle Ω · I
sr │ — │ Ω = area / r2
Spring constant k · I
N m−1 │ M T−2 │ F = −kx
Speed on a string v · II
m s−1 │ L T−1 │ v = √(T/μ)
Sound level β · II
dB │ — │ β = 10 log₁₀(I/I₀)
Specific heat capacity c · III
J kg−1 K−1 │ L2 T−2 K−1 │ Q = mcΔT
Stefan–Boltzmann constant σ · III
W m−2 K−4 │ M T−3 K−4 │ P = σεAT4
Stopping potential V₀ · VI
V │ M L2 T−3 A−1 │ eV₀ = K_max
Speed of light c · VI
m s−1 │ L T−1 │ c = 1/√(ε₀μ₀)
T11
Time t · I
s │ T │ base quantity
Torque τ · I
N m │ M L2 T−2 │ τ = r × F = Iα
Terminal velocity v_t · I
m s−1 │ L T−1 │ v_t = 2r2(ρ−σ)g / 9η
Time period T · II
s │ T │ T = 1/f = 2π/ω
Temperature T · III
K │ K │ base quantity
Thermal conductivity K, k · III
W m−1 K−1 │ M L T−3 K−1 │ dQ/dt = KA (dT/dx)
Thermal resistance R_th · III
K W−1 │ M−1 L−2 T3 K │ R = x/KA
Thermal efficiency η · III
— │ — │ η = W/Q_h = 1 − T_c/T_h (Carnot)
Temperature coefficient of resistance α · V
K−1 │ K−1 │ R = R₀(1 + αΔT)
Turns ratio N_s/N_p · V
— │ — │ V_s/V_p = N_s/N_p
Threshold frequency ν₀ · VI
Hz │ T−1 │ ν₀ = φ/h
V3
Volume V · I
m3 │ L3 │ V = l × b × h
Velocity v · I
m s−1 │ L T−1 │ v = dr/dt
Volume flow rate Q · I
m3 s−1 │ L3 T−1 │ Q = Av
W8
Weight W · I
N │ M L T−2 │ W = mg
Work W · I
J │ M L2 T−2 │ W = F · d
Wavelength λ · II
m │ L │ λ = v/f
Wave number k · II
rad m−1 │ L−1 │ k = 2π/λ
Wave speed v · II
m s−1 │ L T−1 │ v = fλ = ω/k
Wien's constant b · III
m K │ L K │ λ_max T = b
Water equivalent w · III
kg │ M │ w = mc / c_water
Work function φ · VI
J │ M L2 T−2 │ K_max = hν − φ
Y1
Young's modulus Y, E · I
Pa │ M L−1 T−2 │ Y = stress / longitudinal strain
SI base units
Length — metre m [L]
the path light travels in vacuum in 1/299 792 458 of a second — so c is exact and the metre is derived from it
Mass — kilogram kg [M]
fixed by giving the Planck constant the exact value 6.626 070 15×10−34 J s; the Paris cylinder was retired in 2019
Time — second s [T]
9 192 631 770 periods of the caesium-133 hyperfine transition
Electric current — ampere A [A]
fixed by giving the elementary charge the exact value 1.602 176 634×10−19 C
Thermodynamic temperature — kelvin K [K]
fixed by giving the Boltzmann constant the exact value 1.380 649×10−23 J K−1
Amount of substance — mole mol [mol]
exactly 6.022 140 76×1023 elementary entities
Luminous intensity — candela cd [J]
a source emitting 540×1012 Hz radiation at 1/683 watt per steradian
SI prefixes
T tera 1012 · G giga 109 · M mega 106 · k kilo 103 · h hecto 102 · da deca 101 · d deci 10−1 · c centi 10−2 · m milli 10−3 · µ micro 10−6 · n nano 10−9 · p pico 10−12 · f femto 10−15 · a atto 10−18
Physical constants32
c 2.997 924 58×108 m s−1
Speed of light in vacuum
h 6.626 070 15×10−34 J s
Planck constant
ℏ 1.054 571 82×10−34 J s
Reduced Planck constant
e 1.602 176 634×10−19 C
Elementary charge
k_B 1.380 649×10−23 J K−1
Boltzmann constant
N_A 6.022 140 76×1023 mol−1
Avogadro constant
R 8.314 462 618 J mol−1 K−1
Molar gas constant
G 6.674 30×10−11 N m2 kg−2
Gravitational constant
g₀ 9.806 65 m s−2
Standard gravity
ε₀ 8.854 187 813×10−12 F m−1
Permittivity of free space
μ₀ 1.256 637 062×10−6 H m−1
Permeability of free space
m_e 9.109 383 70×10−31 kg
Electron rest mass
m_p 1.672 621 924×10−27 kg
Proton rest mass
m_n 1.674 927 498×10−27 kg
Neutron rest mass
u 1.660 539 067×10−27 kg
Atomic mass unit
a₀ 5.291 772 109×10−11 m
Bohr radius
R_∞ 1.097 373 157×107 m−1
Rydberg constant
α 7.297 352 569×10−3 —
Fine-structure constant
λ_C 2.426 310 24×10−12 m
Compton wavelength (electron)
μ_B 9.274 010 08×10−24 J T−1
Bohr magneton
σ 5.670 374 419×10−8 W m−2 K−4
Stefan–Boltzmann constant
b 2.897 771 955×10−3 m K
Wien displacement constant
F 9.648 533 21×104 C mol−1
Faraday constant
V_m 2.241 396 954×10−2 m3 mol−1
Molar volume at STP
e/m_e 1.758 820 01×1011 C kg−1
Electron charge-to-mass ratio
hc 1239.84 eV nm
Photon energy conversion
M_E 5.972×1024 kg
Earth's mass
R_E 6.371×106 m
Earth's mean radius
S 1361 W m−2
Solar constant
v 343 m s−1
Speed of sound in air (20 °C)
I₀ 1×10−12 W m−2
Threshold of hearing intensity
atm 1.013 25×105 Pa
Standard atmosphere
Same dimensions, different physics
M L2 T−2 — Work · Energy · Heat · Torque · Moment of a couple
M L T−1 — Momentum · Impulse
M L−1 T−2 — Pressure · Stress · Young's modulus · Bulk modulus · Energy density
T−1 — Frequency · Angular velocity · Angular frequency · Decay constant · Activity · Velocity gradient
M L2 T−1 — Angular momentum · Planck constant · Action
L T−2 — Acceleration · Gravitational field strength
L2 T−2 — Latent heat · Specific heat × temperature · Gravitational potential · Absorbed dose · (speed)2
— — Strain · Refractive index · Relative density · Coefficient of friction · Angle · Poisson's ratio · Reynolds number · Efficiency · Magnification
M L2 T−3 — Power · Heat current · Rate of doing work
M T−3 — Intensity · Poynting vector · Irradiance · Radiant emittance
M T−2 — Surface tension · Spring constant · Force per length
M L2 T−3 A−1 — Potential difference · EMF · Stopping potential
How it is measured
Distance by parallax — D = b/θ
Sight the same object from two places a known basis b apart. Its direction shifts by the parallax angle θ, in radians, and the distance follows at once. Every stellar distance out to a few hundred light years is measured this way, with Earth's orbit as the basis.
Size from angular width — d = Dθ
The same triangle read the other way: once the distance is known, an object's angular width gives its true size. θ belongs in radians — feeding degrees straight in is the commonest slip in the topic.
Vernier callipers — LC = 1 MSD − 1 VSD = MSD/N
N vernier divisions are ruled to span N−1 main-scale divisions, so the two scales differ by one Nth of a division — and that difference is the least count. Reading = main-scale reading + (coinciding vernier division × LC), then the zero error is removed.
Screw gauge — LC = pitch / circular divisions
One full turn advances the spindle by the pitch, so one division of the circular scale is that much smaller a displacement. Read the zero error with the faces closed and subtract it with its sign — a negative zero error is added back.
The second, defined — 9 192 631 770 periods (exact)
The second is that many periods of the caesium-133 hyperfine transition. Nothing about the Earth's rotation enters the definition any more, which is why leap seconds have to be inserted by hand.
The metre, defined — 1 m = light in 1/299 792 458 s
Since 1983 c has been exact by definition, so every length measurement is really a time measurement. That is also why c carries no uncertainty while G, which must still be weighed, is the worst-known constant in physics.
Dimensional analysis
Only like may be added. Every term in an equation must carry the same dimensions. x = x₀ + v₀t + ½at2 passes; x = v₀ + at2 does not. This is the first check to run and it costs five seconds.
Arguments of functions are pure numbers. sin θ, ln x, e^{−t/τ} — whatever sits inside must be dimensionless. If you find a sin(vt) you have already lost a length somewhere.
Dimensionless does not mean unitless. An angle in radians and a strain are both dimensionless, but radians still have to be radians when a formula assumes them. Degrees in ω = 2πf is the classic loss.
A constant may carry dimensions. G, k_B, h and ε₀ all do. Checking an equation means checking the constants too, not assuming they are pure numbers.
Dimensional analysis cannot find a pure number. It gives T ∝ √(l/g) for a pendulum; it can never give you the 2π. The method fixes the powers, never the coefficient.
It cannot handle a sum of unknown terms. If the true answer is a + b with a and b of the same dimensions, no dimensional argument separates them. It works when the answer is a single product of powers.
Deriving a formula from powers. Assume the quantity is a product of powers of the relevant variables, equate the exponents of M, L, T on both sides, and solve. Three base dimensions means three equations — so at most three unknown exponents.
Checking a converted answer. Convert every input to SI first, work the algebra, then convert once at the end. Mixing centimetres into an SI formula is not a dimensional error, so no check will catch it.