Practice · quick reference

Units, dimensions and constants.

208 quantities, each with its SI unit, its dimensional formula, the relation it comes from, and a number to hold it against. Plus 32 constants, the prefixes, and three calculators so you never have to leave the page to convert something.

Free, all of it. Jump to — the calculators · the table · SI base units · how it is measured · prefixes · constants · the rules · same dimensions, different physics.

Work it out here

three calculators · 184 units · free

Everything below runs in the page, offline, on the same table the rest of this document is built from — so a converter answer and the table always agree.

Tool 01

Convert a unit

Pick a quantity, type a value, and every other unit in that family updates at once. Click any row to measure from it instead.

▸ The theory in three lines

Every conversion is a multiplication by 1. Because 1 inch is 2.54 cm, the ratio (2.54 cm / 1 inch) equals one — so you may multiply by it freely, and the unwanted unit cancels. Chain as many of those as you need.

A scale with an offset is not a ratio. Celsius and Fahrenheit have different zeros, so they convert by v × f + o, never by a factor alone — which is why 0 °C is 32 °F and not 0 °F. Kelvin, Celsius, Fahrenheit and Rankine are the only four here that behave that way.

Convert everything to SI first, do the algebra, and convert once at the end. A centimetre left inside an SI formula is not a dimensional error, so no dimensional check will ever catch it.

Equivalents {{ famDim }} · SI: {{ famSI }}
{{ convRows }}
Tool 02

Check the dimensions

Write an expression in base dimensions — M L T A K mol J, with *, /, ^ or superscripts, brackets and +. Put an = in it and both sides are compared.

▸ The theory in three lines

Every quantity is a product of powers of seven base dimensions — mass M, length L, time T, current A, temperature K, amount mol, luminous intensity J. Velocity is L T−1; force is M L T−2.

The rule is that only like may be added: every term in an equation carries the same dimensions, and whatever sits inside sin, ln or ex must be a pure number. An equation that fails this is wrong, with no exceptions.

An equation that passes may still be wrong. The method fixes the powers and never the coefficient — it gives T ∝ √(l/g) for a pendulum and can never give you the 2π. To derive a relation, assume the answer is a product of powers of the variables that matter, equate the exponents on both sides, and solve; three base dimensions means at most three unknowns.

Insert a quantity’s dimensions
{{ dimPalette }}
{{ dimOut }}
Tool 03

Propagate an error

Give each measurement its value and uncertainty. Products carry fractional errors weighted by their powers; sums and differences carry absolute ones.

▸ The theory in three lines

Adding or subtracting: the ABSOLUTE errors add. Δ(a ± b) = Δa + Δb. They add even when the values subtract — taking the difference of two close numbers is how a good measurement becomes a bad one.

Multiplying, dividing or raising to a power: the FRACTIONAL errors add, each weighted by its power. ΔZ/Z = Σ |n| · Δx/x. A quantity that appears squared contributes twice its own error, which is why the radius almost always dominates a volume.

Then quote the answer to its error: round the uncertainty up to one significant figure, and give the value to that same decimal place. 4.8261 ± 0.0374 is reported as 4.83 ± 0.04 — the digits after it are noise.

{{ errRows }}
{{ errOut }}

The table

208 quantities · alphabetical

Alphabetical, because that is how you look something up. Filter by Part when you want one syllabus at a time. A dash in the dimensions column means the quantity is a pure number — and that a dimensional check can never validate it.

Quantity
SI unit
Dimensions
Defining relation
A sense of scale

A

18
Area
A · I
m2square metre
L2
A = l × b
a desk top is 1 m21 hectare = 104 m2 · 1 barn = 10−28 m2
Acceleration
a · I
m s−2metre per second squared
L T−2
a = dv/dt
a car pulls 3; a fighter pilot blacks out near 90 m s−21 g = 9.81 m s−2
Acceleration due to gravity
g · I
m s−2metre per second squared
L T−2
g = GM/R2
9.81 on Earth, 1.62 on the Moon, 274 on the Sun1 gal = 10−2 m s−2
Angle
θ · I
radradian
—
θ = arc / radius
a full turn is 2π rad1 rad = 57.30° · 1° = 0.01745 rad
Angular velocity
ω · I
rad s−1radian per second
T−1
ω = dθ/dt = 2πf
a CD spins at 50 rad s−11 rpm = 0.1047 rad s−1
Angular acceleration
α · I
rad s−2radian per second squared
T−2
α = dω/dt
a spinning-up drill reaches 500 rad s−2
Angular momentum
L · I
kg m2 s−1kilogram square metre per second
M L2 T−1
L = r × p = Iω
the electron's is ℏ = 1.05×10−34 J s= J s — the dimensions of Planck's constant
Areal velocity
dA/dt · I
m2 s−1square metre per second
L2 T−1
dA/dt = L/2m
Earth sweeps 2.2×1015 m2 s−1Kepler's second law says it is constant
Angular frequency
ω · II
rad s−1radian per second
T−1
ω = 2πf
mains AC is 314 rad s−1same dimensions as frequency — the 2π is the whole difference
Amplitude
A · II
mmetre
L
greatest displacement from equilibrium
an eardrum moves 10−11 m at the hearing thresholdin sound often quoted as a pressure amplitude in Pa
Acoustic pressure amplitude
p₀ · II
Papascal
M L−1 T−2
p₀ = Bk s₀ = ρvω s₀
2×10−5 Pa at the hearing threshold
Amount of substance
n · III
molmole
mol
base quantity
18 g of water is 1 mol1 mol = 6.022×1023 entities
Avogadro constant
N_A · III
mol−1per mole
mol−1
N = n N_A
6.022×1023
Angle of deviation
δ · IV
radradian
—
δ = i + e − A
a 60° crown prism deviates about 38°
Angular magnification
M · IV
—dimensionless
—
M = f_o/f_e (telescope)
binoculars are 8 or 10
Aperture
D · IV
mmetre
L
the clear diameter of the optic
the human pupil opens to 7 mmthe f-number is f/D
Atomic mass unit
u · VI
kgkilogram
M
1 u = 1/12 of a 12C atom
a proton is 1.007 u1 u = 931.5 MeV/c2 = 1.661×10−27 kg
Activity
A · VI
Bqbecquerel
T−1
A = λN
a human body is about 5000 Bq1 Ci = 3.7×1010 Bq

B

7
Bulk modulus
B, K · I
Papascal
M L−1 T−2
B = −P / (ΔV/V)
water 2.2×109 Paits reciprocal is compressibility
Beat frequency
f_b · II
Hzhertz
T−1
f_b = |f₁ − f₂|
two strings 3 Hz apart throb three times a second
Boltzmann constant
k_B · III
J K−1joule per kelvin
M L2 T−2 K−1
PV = Nk_BT
1.381×10−23 — k_BT at room temperature is 1/40 eV= 8.617×10−5 eV K−1
Brewster angle
θ_B · IV
radradian
—
tan θ_B = n
56.3° for glass in air
Bohr radius
a₀ · VI
mmetre
L
a₀ = 4πε₀ℏ2/m_e e2
5.29×10−11 m = 0.529 Å1 Å = 10−10 m
Binding energy
E_b · VI
Jjoule
M L2 T−2
E_b = Δm c2
the B/A peak is 8.79 MeV at 56Fequoted in MeV
Bragg angle
θ · VI
radradian
—
2d sin θ = nλ
copper Kα off NaCl reflects near 15°

C

25
Compressibility
k · I
Pa−1reciprocal pascal
M−1 L T2
k = 1/B
water 4.6×10−10 Pa−1
Coefficient of viscosity
η · I
Pa spascal second
M L−1 T−1
F = ηA (dv/dx)
water 10−3; honey 10 Pa s1 poise = 0.1 Pa s
Coefficient of friction
μ · I
—dimensionless
—
f = μN
rubber on dry road 0.8; ice on ice 0.03
Coefficient of restitution
e · I
—dimensionless
—
e = separation speed / approach speed
a superball 0.9; putty 0
Centripetal acceleration
a_c · I
m s−2metre per second squared
L T−2
a_c = v2/r = ω2r
a fairground ride pulls 30 m s−2
Capillary rise
h · I
mmetre
L
h = 2T cos θ / rρg
water rises 15 mm in a 1 mm tube
Centre of mass
r_cm · I
mmetre
L
r_cm = Σm r / Σm
a high jumper's can pass under the bar
Compressibility of a medium
1/B · II
Pa−1reciprocal pascal
M−1 L T2
v_sound = √(B/ρ)
air 7×10−6 Pa−1
Coefficient of linear expansion
α · III
K−1per kelvin
K−1
Δl = lαΔT
steel 1.2×10−5; invar 1.2×10−6 K−1β = 2α, γ = 3α
Coefficient of volume expansion
γ · III
K−1per kelvin
K−1
ΔV = VγΔT
mercury 1.8×10−4; an ideal gas 1/273 K−1
Coefficient of performance
COP · III
—dimensionless
—
COP = Q_c/W
a domestic fridge gives 3–5
Critical angle
θ_c · IV
radradian
—
sin θ_c = 1/n
water 48.8°; glass 41.8°; diamond 24.4°usually quoted in degrees
Coherence length
L_c · IV
mmetre
L
L_c = λ2/Δλ
a filtered lamp 30 μm; a laser kilometres
Cauchy constant B
B · IV
m2square metre
L2
n = A + B/λ2 (A is a pure number)
crown glass: A ≈ 1.51, B ≈ 4×10−15 m2
Current density
J · V
A m−2ampere per square metre
L−2 A
J = I/A = nqv_d
household wiring runs at 5×106 A m−2
Capacitance
C · V
Ffarad
M−1 L−2 T4 A2
C = Q/V
a circuit capacitor 100 nF; a supercapacitor 10 F1 μF = 10−6 F
Conductance
G · V
Ssiemens
M−1 L−2 T3 A2
G = 1/R
a 100 Ω resistor is 10 mS1 S = 1 Ω−1 (the old mho)
Conductivity
σ · V
S m−1siemens per metre
M−1 L−3 T3 A2
σ = 1/ρ = nq2τ/m
copper 5.9×107 S m−1
Charge-to-mass ratio
e/m · V
C kg−1coulomb per kilogram
M−1 T A
e/m from r = mv/qB
the electron's is 1.759×1011 C kg−1
Cyclotron frequency
f_c · V
Hzhertz
T−1
f_c = qB/2πm
an electron in 1 T circles at 28 GHzindependent of speed — that is why cyclotrons work
Compton wavelength
λ_C · VI
mmetre
L
λ_C = h/m_e c
2.426×10−12 m — why Compton needed X-rays
Cross-section
σ · VI
m2square metre
L2
reaction rate = nσv
thermal neutron capture on 235U: 585 barn1 barn = 10−28 m2
Carrier concentration
n, p · VI
m−3per cubic metre
L−3
np = n_i2 (mass-action law)
intrinsic silicon 1.5×1016 m−3
Current gain (common base)
α · VI
—dimensionless
—
α = I_C/I_E
typically 0.98
Current gain (common emitter)
β · VI
—dimensionless
—
β = I_C/I_B = α/(1−α)
typically 50–200

D

9
Density
ρ · I
kg m−3kilogram per cubic metre
M L−3
ρ = m/V
water 1000; air 1.2; gold 19300 kg m−31 g cm−3 = 103 kg m−3
Doppler shift
Δf · II
Hzhertz
T−1
f′ = f₀ (v ± v_o)/(v ∓ v_s)
an ambulance passing at 20 m s−1 drops the pitch 12 %
Damping constant
b · II
kg s−1kilogram per second
M T−1
F_damp = −bv
a shock absorber gives 103 kg s−1
Degrees of freedom
f · III
—dimensionless
—
U = (f/2) nRT
monatomic 3; diatomic 5 (7 with vibration)
Dispersive power
ω · IV
—dimensionless
—
ω = (n_v − n_r)/(n_y − 1)
crown 0.017; flint 0.033
Drift velocity
v_d · V
m s−1metre per second
L T−1
I = nAqv_d
0.1 mm s−1 in household wire — slower than a snail
Displacement current
i_d · V
Aampere
A
i_d = ε₀ dΦ_E/dt
equals the conduction current in a charging capacitor
de Broglie wavelength
λ · VI
mmetre
L
λ = h/p = h/mv
a 100 eV electron: 0.12 nm; a cricket ball: 10−34 m
Decay constant
λ · VI
s−1per second
T−1
N = N₀e−λᵗ
carbon-14: 3.8×10−12 s−1

E

22
Energy
E · I
Jjoule
M L2 T−2
KE = ½mv2, PE = mgh
a chocolate bar holds 106 J1 kWh = 3.6×106 J · 1 cal = 4.186 J
Escape velocity
v_e · I
m s−1metre per second
L T−1
v_e = √(2GM/R)
11.2 km s−1 from Earth; 2.4 from the Moon
Efficiency
η · I
—dimensionless
—
useful output / total input
a car engine 25 %; a motor 90 %usually quoted as a percentage
Elastic potential energy
U · I
Jjoule
M L2 T−2
U = ½kx2
a drawn bow stores 100 J
Energy density (elastic)
u · I
J m−3joule per cubic metre
M L−1 T−2
u = ½ × stress × strain
a stretched steel wire holds 105 J m−3same dimensions as pressure
Excess pressure in a bubble
ΔP · I
Papascal
M L−1 T−2
ΔP = 4T/R (soap), 2T/R (drop)
a 1 cm soap bubble holds 29 Pa extra
End correction
e · II
mmetre
L
e ≈ 0.6 r
a 2 cm pipe adds 6 mm of effective length
Enthalpy
H · III
Jjoule
M L2 T−2
H = U + PV
vaporising a mole of water costs 4.07×104 J
Entropy
S · III
J K−1joule per kelvin
M L2 T−2 K−1
ΔS = ∫dQ_rev/T
melting 1 kg of ice raises S by 1220 J K−1
Emissivity
ε · III
—dimensionless
—
ε = 1 for a blackbody
polished silver 0.02; soot 0.95
Electric charge
q, Q · V
Ccoulomb
T A
q = It
a lightning bolt carries 15 C1 e = 1.602×10−19 C
Electric current
I · V
Aampere
A
base quantity; I = dq/dt
an LED 20 mA; a kettle 10 A; a starter motor 200 A
Electric field
E · V
V m−1volt per metre
M L T−3 A−1
E = F/q
air breaks down at 3×106 V m−1= N C−1
Electric potential
V · V
Vvolt
M L2 T−3 A−1
V = W/q
a AA cell 1.5 V; mains 230 V; a power line 400 kV1 V = 1 J C−1
Electromotive force
ε · V
Vvolt
M L2 T−3 A−1
ε = W/q at zero current
a lead-acid cell 2.1 Va voltage, despite the name — never a force
Electric flux
Φ_E · V
V mvolt metre
M L3 T−3 A−1
Φ_E = ∮E · dA = q/ε₀
1 C enclosed gives 1.13×1011 V m= N m2 C−1
Electric dipole moment
p · V
C mcoulomb metre
L T A
p = q d
a water molecule is 6.2×10−30 C m1 debye = 3.34×10−30 C m
Energy density (electric)
u_E · V
J m−3joule per cubic metre
M L−1 T−2
u_E = ½ε₀E2
at air's breakdown field, 40 J m−3
Energy density (magnetic)
u_B · V
J m−3joule per cubic metre
M L−1 T−2
u_B = B2/2μ₀
a 1 T field holds 4×105 J m−3
Energy gap
E_g · VI
Jjoule
M L2 T−2
the forbidden band width
silicon 1.1 eV; germanium 0.7; diamond 5.5quoted in eV
Electron rest mass
m_e · VI
kgkilogram
M
—
9.109×10−31 kg= 0.511 MeV/c2 = 5.486×10−4 u
Elementary charge
e · VI
Ccoulomb
T A
the quantum of charge
1.602×10−19 C

F

6
Force
F · I
Nnewton
M L T−2
F = ma
a 100 g apple weighs about 1 N1 dyne = 10−5 N · 1 kgf = 9.81 N
Frequency of revolution
f, ν · I
Hzhertz
T−1
f = 1/T = ω/2π
a car wheel at 100 km h−1 turns 15 times a second1 rpm = 1/60 Hz
Frequency
f, ν · II
Hzhertz
T−1
f = 1/T
hearing spans 20 Hz–20 kHz1 kHz = 103 Hz
Focal length
f · IV
mmetre
L
1/v − 1/u = 1/f
a reading lens 250 mm; a phone camera 4 mm
Fringe width
β · IV
mmetre
L
β = λD/d
a classroom Young's slit gives 1 mm fringes
Fine-structure constant
α · VI
—dimensionless
—
α = e2/4πε₀ℏc
1/137.036 — a pure number nobody can derive

G

5
Gravitational constant
G · I
N m2 kg−2newton square metre per square kilogram
M−1 L3 T−2
F = Gm₁m₂/r2
6.674×10−11 — the worst-known constant in physics
Gravitational field strength
E_g · I
N kg−1newton per kilogram
L T−2
E_g = F/m
9.81 N kg−1 at the surfacenumerically the same as g
Gravitational potential
V_g · I
J kg−1joule per kilogram
L2 T−2
V_g = −GM/r
−6.3×107 J kg−1 at Earth's surface
Gravitational potential energy
U · I
Jjoule
M L2 T−2
U = −GMm/r
−6.3×107 J for 1 kg at Earth's surface
Gas constant
R · III
J mol−1 K−1joule per mole kelvin
M L2 T−2 K−1 mol−1
PV = nRT
8.314 — and R = N_A k_B= 0.0821 L atm mol−1 K−1

H

5
Harmonic number
n · II
—dimensionless
—
f_n = n f₁
a closed pipe only allows odd n
Heat
Q · III
Jjoule
M L2 T−2
Q = mcΔT
boiling a kettle takes 5×105 J1 cal = 4.186 J · 1 BTU = 1055 J
Heat capacity
C · III
J K−1joule per kelvin
M L2 T−2 K−1
C = mc
a 1 kg pan of water is 4186 J K−1
Heat current
dQ/dt · III
Wwatt
M L2 T−3
dQ/dt = ΔT / R_th
a person radiates 100 W
Half-life
t₁⁄₂ · VI
ssecond
T
t₁⁄₂ = ln2 / λ
carbon-14 5730 yr; a free neutron 611 s

I

7
Impulse
J · I
N snewton second
M L T−1
J = ∫F dt = Δp
a tennis serve ≈ 4 N s in 5 mssame dimensions as momentum — deliberately
Intensity
I · II
W m−2watt per square metre
M T−3
I = P/A ∝ A2
the hearing threshold is 10−12 W m−2; sunlight 1361
Internal energy
U · III
Jjoule
M L2 T−2
ΔU = Q − W
a mole of monatomic gas at 300 K holds 3740 J
Illuminance
E_v · IV
lxlux
L−2 J
E_v = Φ_v/A
an office 500 lx; full sun 105 lx1 lx = 1 lm m−2
Inductance
L · V
Hhenry
M L2 T−2 A−2
NΦ = LI; ε = −L dI/dt
a small choke 1 mH; a large coil 10 H1 H = 1 Wb A−1 = 1 V s A−1
Impedance
Z · V
Ωohm
M L2 T−3 A−2
Z = √(R2 + (X_L − X_C)2)
a speaker is 8 Ω
Ionisation energy
E_i · VI
Jjoule
M L2 T−2
E_i = 13.6 Z2/n2 eV (hydrogen-like)
hydrogen 13.6 eV; He+ 54.4 eVquoted in eV

K

2
Kinetic energy
K · I
Jjoule
M L2 T−2
K = ½mv2 = p2/2m
a car at 25 m s−1 carries 4×105 J
Kinematic viscosity
ν · I
m2 s−1square metre per second
L2 T−1
ν = η/ρ
water 10−6 m2 s−11 stokes = 10−4 m2 s−1

L

7
Length
l, x · I
mmetre
L
base quantity
a doorway is 2 m; a hydrogen atom 10−10 m1 Å = 10−10 m · 1 ly = 9.46×1015 m
Linear mass density
μ · I
kg m−1kilogram per metre
M L−1
μ = m/l
a guitar string ≈ 5×10−4 kg m−1
Latent heat
L · III
J kg−1joule per kilogram
L2 T−2
Q = mL
ice 3.34×105; water vapour 2.26×106 J kg−11 cal g−1 = 4186 J kg−1
Luminous intensity
I_v · IV
cdcandela
J
base quantity
a candle is about 1 cd
Luminous flux
Φ_v · IV
lmlumen
J
Φ_v = I_v Ω
a 10 W LED bulb gives 800 lm
Least distance of distinct vision
D · IV
mmetre
L
the near point of a normal eye
0.25 m by convention
Lorentz factor
γ · VI
—dimensionless
—
γ = 1/√(1 − v2/c2)
γ = 2 at 0.866 c; γ = 7 at 0.99 c

M

19
Mass
m · I
kgkilogram
M
base quantity
a litre of water is 1 kg; an electron 9.1×10−31 kg1 u = 1.66×10−27 kg · 1 t = 103 kg
Momentum
p · I
kg m s−1kilogram metre per second
M L T−1
p = mv
a thrown cricket ball ≈ 5 kg m s−1= N s
Moment of inertia
I · I
kg m2kilogram square metre
M L2
I = Σ m r2
a bicycle wheel ≈ 0.15 kg m2
Mechanical advantage
MA · I
—dimensionless
—
load / effort
a car jack gives 50
Mach number
M · II
—dimensionless
—
M = v_object / v_sound
Concorde flew at M = 2
Molar specific heat
C · III
J mol−1 K−1joule per mole kelvin
M L2 T−2 K−1 mol−1
Q = nCΔT
a monatomic gas: C_V = 12.5, C_P = 20.8
Mean free path
λ · III
mmetre
L
λ = 1/(√2 nπd2)
air at STP: 68 nm; a good vacuum: metres
Molar mass
M · III
kg mol−1kilogram per mole
M mol−1
M = m/n
water 0.018 kg mol−1usually quoted in g mol−1
Magnification
m · IV
—dimensionless
—
m = v/u = h′/h
a microscope reaches 1000negative means inverted
Mobility
μ · V
m2 V−1 s−1square metre per volt second
M−1 T2 A
μ = v_d/E
electrons in silicon 0.135 m2 V−1 s−1
Magnetic field
B · V
Ttesla
M T−2 A−1
F = qv × B
Earth 5×10−5 T; an MRI 3 T; a neutron star 108 T1 gauss = 10−4 T
Magnetic flux
Φ_B · V
Wbweber
M L2 T−2 A−1
Φ_B = ∫B · dA
a transformer core carries 10−3 Wb1 Wb = 1 T m2 = 1 V s
Magnetic dipole moment
m · V
A m2ampere square metre
L2 A
m = NIA
an electron's is 9.27×10−24 A m2= J T−1
Magnetising field
H · V
A m−1ampere per metre
L−1 A
B = μ₀(H + M)
a solenoid at 1000 turns m−1 and 1 A gives 1000 A m−11 oersted = 79.6 A m−1
Magnetisation
M · V
A m−1ampere per metre
L−1 A
M = m/V
saturated iron reaches 1.7×106 A m−1same units as H
Magnetic susceptibility
χ · V
—dimensionless
—
M = χH; μ_r = 1 + χ
water −9×10−6; aluminium +2.2×10−5
Mutual inductance
M · V
Hhenry
M L2 T−2 A−2
ε₂ = −M dI₁/dt
a transformer's is millihenries
Mass defect
Δm · VI
kgkilogram
M
Δm = Zm_p + Nm_n − M
helium-4 loses 0.0304 uusually in u
Mean life
τ · VI
ssecond
T
τ = 1/λ = t₁⁄₂/ln2
a muon lives 2.2 μsalways longer than the half-life

N

2
Numerical aperture
NA · IV
—dimensionless
—
NA = n sin θ
an oil-immersion objective reaches 1.4
Nuclear radius
R · VI
mmetre
L
R = R₀A^(1/3), R₀ = 1.2 fm
uranium is 7.4 fm across the radius1 fm = 10−15 m

O

1
Optical path length
Δ · IV
mmetre
L
Δ = n × geometric path
5 mm of glass is 7.5 mm of optical path

P

17
Potential energy
U · I
Jjoule
M L2 T−2
U = mgh (near Earth)
a 1 kg book on a 1 m shelf holds 9.8 J
Power
P · I
Wwatt
M L2 T−3
P = dW/dt = F · v
a person sustains 100 W; a car engine 105 W1 hp = 746 W
Pressure
P, p · I
Papascal
M L−1 T−2
P = F/A
atmosphere 105 Pa; a car tyre 2×105 Pa gauge1 atm = 1.013×105 Pa · 1 bar = 105 Pa · 1 torr = 133 Pa
Poisson's ratio
σ · I
—dimensionless
—
lateral strain / longitudinal strain
most metals 0.25–0.35; rubber near 0.5
Phase
φ · II
radradian
—
y = A sin(ωt + φ)
quadrature is π/21 cycle = 2π rad = 360°
Phase difference
Δφ · II
radradian
—
Δφ = (2π/λ)·Δx
a half-wavelength gap gives π
Particle velocity (wave)
u · II
m s−1metre per second
L T−1
u = ∂y/∂t = −v (∂y/∂x)
an air molecule moves μm s−1 while sound runs at 343 m s−1never the same as the wave speed v
Power of a lens
P · IV
Ddioptre
L−1
P = 1/f (f in metres)
the relaxed eye is 60 D; reading glasses +2 D1 D = 1 m−1
Path difference
Δx · IV
mmetre
L
Δx = d sin θ
one wavelength of green light: 550 nma phase difference of 2π per λ
Potential difference
ΔV · V
Vvolt
M L2 T−3 A−1
ΔV = W/q = IR
a nerve membrane holds 70 mV
Permittivity of free space
ε₀ · V
F m−1farad per metre
M−1 L−3 T4 A2
F = q₁q₂/4πε₀r2
8.854×10−121/4πε₀ = 8.99×109 N m2 C−2
Permeability of free space
μ₀ · V
H m−1henry per metre
M L T−2 A−2
B = μ₀I/2πr
1.257×10−6= 4π×10−7 T m A−1
Power factor
cos φ · V
—dimensionless
—
P = V_rms I_rms cos φ
an uncorrected motor runs at 0.7
Poynting vector
S · V
W m−2watt per square metre
M T−3
S = (E × B)/μ₀
sunlight at Earth is 1361 W m−2
Planck constant
h · VI
J sjoule second
M L2 T−1
E = hν
6.626×10−34 — the dimensions of angular momentum= 4.136×10−15 eV s · ℏ = h/2π
Photon energy
E · VI
Jjoule
M L2 T−2
E = hν = hc/λ
a green photon is 2.25 eVλ(nm) = 1240 / E(eV)
Photon momentum
p · VI
kg m s−1kilogram metre per second
M L T−1
p = h/λ = E/c
a green photon carries 1.2×10−27 kg m s−1

Q

1
Quality factor
Q · II
—dimensionless
—
Q = ω₀ / Δω = 2π × (energy stored / energy lost per cycle)
a tuning fork 103; a quartz crystal 106

R

19
Relative density
— · I
—dimensionless
—
ρ / ρ_water
gold 19.3; ice 0.92also called specific gravity
Reynolds number
Re · I
—dimensionless
—
Re = ρvD/η
below ~2000 laminar, above ~3000 turbulent
Radius of gyration
K · I
mmetre
L
I = MK2
for a disc it is R/√2
Ratio of specific heats
γ · III
—dimensionless
—
γ = C_P/C_V = 1 + 2/f
monatomic 1.67; air 1.40
RMS speed
v_rms · III
m s−1metre per second
L T−1
v_rms = √(3RT/M)
nitrogen at 300 K moves at 517 m s−1
Refractive index
n, μ · IV
—dimensionless
—
n = c/v = sin i / sin r
water 1.33; glass 1.5; diamond 2.42
Radius of curvature
R · IV
mmetre
L
R = 2f (mirror)
the cornea's is 7.8 mm
Resolving power (telescope)
1/dθ · IV
rad−1per radian
—
dθ = 1.22 λ/D
a 60 mm scope resolves 2.3 arcsec
Relative permittivity
ε_r, K · V
—dimensionless
—
C = K C₀
air 1.0006; water 80; a conductor ∞also called the dielectric constant
Resistance
R · V
Ωohm
M L2 T−3 A−2
R = V/I
a filament lamp 400 Ω; a human body 105 Ω dry
Resistivity
ρ · V
Ω mohm metre
M L3 T−3 A−2
R = ρl/A
copper 1.7×10−8; glass 1012 Ω m
Relative permeability
μ_r · V
—dimensionless
—
μ = μ_r μ₀
aluminium 1.000022; soft iron 5000
Reactance
X · V
Ωohm
M L2 T−3 A−2
X_L = ωL, X_C = 1/ωC
a 1 μF cap at 50 Hz is 3183 Ω
Radiation pressure
p_rad · V
Papascal
M L−1 T−2
p = I/c (absorbed), 2I/c (reflected)
sunlight presses 4.5 μPa
Reduced Planck constant
ℏ · VI
J sjoule second
M L2 T−1
ℏ = h/2π; mvr = nℏ
1.055×10−34= 6.582×10−16 eV s
Rydberg constant
R · VI
m−1per metre
L−1
1/λ = RZ2(1/n_f2 − 1/n_i2)
1.097×107 m−1R_H = 13.6 eV in energy units
Rest energy
E₀ · VI
Jjoule
M L2 T−2
E₀ = m₀c2
an electron 0.511 MeV; a proton 938 MeVquoted in MeV
Rectifier ripple frequency
f_r · VI
Hzhertz
T−1
f_r = f (half-wave), 2f (full-wave)
100 Hz on a 50 Hz full-wave supply
Radiation dose
D · VI
Gygray
L2 T−2
D = energy absorbed / mass
a chest X-ray gives 0.1 mGy1 rad = 0.01 Gy · dose equivalent in sievert

S

13
Speed
v · I
m s−1metre per second
L T−1
v = distance / time
walking 1.4; highway 25; sound 343 m s−11 km h−1 = 5/18 m s−1
Stress
σ · I
Papascal
M L−1 T−2
σ = F/A
steel yields near 2.5×108 Pasame dimensions as pressure
Strain
ε · I
—dimensionless
—
ε = Δl/l
steel breaks by 0.2 % strainoften quoted as a percentage
Shear modulus
η, G · I
Papascal
M L−1 T−2
G = shear stress / shear strain
steel 8×1010 Paalso called modulus of rigidity
Surface tension
T, S · I
N m−1newton per metre
M T−2
T = F/l = energy per area
water 0.072 N m−1; mercury 0.47= J m−2 · 1 dyne cm−1 = 10−3 N m−1
Solid angle
Ω · I
srsteradian
—
Ω = area / r2
a whole sphere is 4π sr
Spring constant
k · I
N m−1newton per metre
M T−2
F = −kx
a pen spring ≈ 100; a car suspension 3×104 N m−1
Speed on a string
v · II
m s−1metre per second
L T−1
v = √(T/μ)
a guitar's top string carries 400 m s−1
Sound level
β · II
dBdecibel
—
β = 10 log₁₀(I/I₀)
a whisper 30 dB; a jet 140 dBI₀ = 10−12 W m−2
Specific heat capacity
c · III
J kg−1 K−1joule per kilogram kelvin
L2 T−2 K−1
Q = mcΔT
water 4186; iron 450 J kg−1 K−11 cal g−1 °C−1 = 4186 J kg−1 K−1
Stefan–Boltzmann constant
σ · III
W m−2 K−4watt per square metre kelvin4
M T−3 K−4
P = σεAT4
5.67×10−8
Stopping potential
V₀ · VI
Vvolt
M L2 T−3 A−1
eV₀ = K_max
UV on zinc gives about 1 V
Speed of light
c · VI
m s−1metre per second
L T−1
c = 1/√(ε₀μ₀)
a foot per nanosecond, near enoughexactly 299 792 458 m s−1 by definition

T

11
Time
t · I
ssecond
T
base quantity
a heartbeat is 1 s; the universe 4×1017 s1 yr = 3.16×107 s
Torque
τ · I
N mnewton metre
M L2 T−2
τ = r × F = Iα
a car engine gives 200 N msame dimensions as energy — but never call it a joule
Terminal velocity
v_t · I
m s−1metre per second
L T−1
v_t = 2r2(ρ−σ)g / 9η
a raindrop falls at 9 m s−1; a skydiver 55
Time period
T · II
ssecond
T
T = 1/f = 2π/ω
a 1 m pendulum swings in 2.0 s
Temperature
T · III
Kkelvin
K
base quantity
room 293 K; the Sun's core 1.5×107 KT(K) = θ(°C) + 273.15 · °F = 9/5 °C + 32
Thermal conductivity
K, k · III
W m−1 K−1watt per metre kelvin
M L T−3 K−1
dQ/dt = KA (dT/dx)
copper 400; glass 1; air 0.024 W m−1 K−1
Thermal resistance
R_th · III
K W−1kelvin per watt
M−1 L−2 T3 K
R = x/KA
a double-glazed pane ≈ 0.3 K W−1 per m2adds in series like electrical resistance
Thermal efficiency
η · III
—dimensionless
—
η = W/Q_h = 1 − T_c/T_h (Carnot)
a power station reaches 40 %
Temperature coefficient of resistance
α · V
K−1per kelvin
K−1
R = R₀(1 + αΔT)
copper 3.9×10−3; a thermistor is negative
Turns ratio
N_s/N_p · V
—dimensionless
—
V_s/V_p = N_s/N_p
a phone charger is about 1:20
Threshold frequency
ν₀ · VI
Hzhertz
T−1
ν₀ = φ/h
caesium 5.1×1014 Hz — green light

V

3
Volume
V · I
m3cubic metre
L3
V = l × b × h
a fridge is 0.3 m31 litre = 10−3 m3
Velocity
v · I
m s−1metre per second
L T−1
v = dr/dt
same magnitudes as speed, with a direction1 knot = 0.514 m s−1
Volume flow rate
Q · I
m3 s−1cubic metre per second
L3 T−1
Q = Av
a garden tap gives 2×10−4 m3 s−11 L s−1 = 10−3 m3 s−1

W

8
Weight
W · I
Nnewton
M L T−2
W = mg
a 70 kg person weighs 687 Na force, not a mass — never kg
Work
W · I
Jjoule
M L2 T−2
W = F · d
lifting 1 kg by 1 m costs 9.8 J1 erg = 10−7 J · 1 eV = 1.602×10−19 J
Wavelength
λ · II
mmetre
L
λ = v/f
green light 550 nm; middle-C sound 1.3 m1 Å = 10−10 m · 1 nm = 10−9 m
Wave number
k · II
rad m−1radian per metre
L−1
k = 2π/λ
green light 1.1×107 rad m−1spectroscopists use ν̄ = 1/λ in cm−1
Wave speed
v · II
m s−1metre per second
L T−1
v = fλ = ω/k
sound in air 343; in steel 5000 m s−1
Wien's constant
b · III
m Kmetre kelvin
L K
λ_max T = b
2.898×10−3 m K — the Sun peaks at 500 nm
Water equivalent
w · III
kgkilogram
M
w = mc / c_water
a copper calorimeter of 100 g is 9 g of water
Work function
φ · VI
Jjoule
M L2 T−2
K_max = hν − φ
caesium 2.1 eV; platinum 5.6 eValmost always quoted in eV

Y

1
Young's modulus
Y, E · I
Papascal
M L−1 T−2
Y = stress / longitudinal strain
steel 2×1011; rubber 107 Paoften quoted in GPa

The SI base units

seven of them · everything else is derived

Since 2019 every one is fixed by giving a constant of nature an exact value, rather than by an artefact in a vault.

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

How a length or a time is actually measured

6 methods · callipers to caesium

A unit is a definition; a reading is a measurement. These are the six a paper assumes you can do — two triangles, two instruments, and the two definitions everything else is calibrated against.

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.
b = 3×1011 m and θ = 1 arcsec give 1 parsec = 3.08×1016 m
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.
the Moon subtends 0.0093 rad at 3.84×108 m, so it is 3.5×106 m across
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.
a millimetre main scale with 10 vernier divisions reads to 0.1 mm
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.
0.5 mm pitch over 50 divisions reads to 0.01 mm
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.
an atomic clock holding this drifts under 1 s in 100 million years
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.
laser ranging fixes the Moon's distance to a few centimetres

SI prefixes

tera to atto
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 constants

32 values

CODATA values, to the precision a paper ever needs. Several are now exact by definition — the note says which.

Speed of light in vacuum
c
2.997 924 58×108 m s−1
exact by definition
Planck constant
h
6.626 070 15×10−34 J s
exact; = 4.136×10−15 eV s
Reduced Planck constant
ℏ
1.054 571 82×10−34 J s
h/2π
Elementary charge
e
1.602 176 634×10−19 C
exact by definition
Boltzmann constant
k_B
1.380 649×10−23 J K−1
exact; = 8.617×10−5 eV K−1
Avogadro constant
N_A
6.022 140 76×1023 mol−1
exact by definition
Molar gas constant
R
8.314 462 618 J mol−1 K−1
= N_A k_B, so also exact
Gravitational constant
G
6.674 30×10−11 N m2 kg−2
the least precisely known of them all
Standard gravity
g₀
9.806 65 m s−2
a defined conventional value
Permittivity of free space
ε₀
8.854 187 813×10−12 F m−1
1/4πε₀ = 8.987 551 79×109 N m2 C−2
Permeability of free space
μ₀
1.256 637 062×10−6 H m−1
≈ 4π×10−7; c = 1/√(ε₀μ₀)
Electron rest mass
m_e
9.109 383 70×10−31 kg
= 0.510 999 MeV/c2
Proton rest mass
m_p
1.672 621 924×10−27 kg
= 938.272 MeV/c2 = 1836 m_e
Neutron rest mass
m_n
1.674 927 498×10−27 kg
= 939.565 MeV/c2
Atomic mass unit
u
1.660 539 067×10−27 kg
= 931.494 MeV/c2
Bohr radius
a₀
5.291 772 109×10−11 m
= 0.529 Å
Rydberg constant
R_∞
1.097 373 157×107 m−1
hcR_∞ = 13.6057 eV
Fine-structure constant
α
7.297 352 569×10−3 —
= 1/137.035 999
Compton wavelength (electron)
λ_C
2.426 310 24×10−12 m
h/m_e c
Bohr magneton
μ_B
9.274 010 08×10−24 J T−1
eℏ/2m_e
Stefan–Boltzmann constant
σ
5.670 374 419×10−8 W m−2 K−4
P = σεAT4
Wien displacement constant
b
2.897 771 955×10−3 m K
λ_max T = b
Faraday constant
F
9.648 533 21×104 C mol−1
= N_A e
Molar volume at STP
V_m
2.241 396 954×10−2 m3 mol−1
22.414 L at 273.15 K, 101.325 kPa
Electron charge-to-mass ratio
e/m_e
1.758 820 01×1011 C kg−1
—
Photon energy conversion
hc
1239.84 eV nm
λ(nm) = 1240/E(eV) — worth memorising
Earth's mass
M_E
5.972×1024 kg
—
Earth's mean radius
R_E
6.371×106 m
—
Solar constant
S
1361 W m−2
irradiance at the top of the atmosphere
Speed of sound in air (20 °C)
v
343 m s−1
∝ √T
Threshold of hearing intensity
I₀
1×10−12 W m−2
the 0 dB reference
Standard atmosphere
atm
1.013 25×105 Pa
exact by definition

Dimensional analysis, honestly

what the method can and cannot do

The check is free and catches a whole class of error in five seconds. It also has hard limits, and knowing them is the difference between using it and trusting it.

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.

Same dimensions, different physics

12 families

Matching dimensions never prove two quantities are the same thing. These are the pairs that cost marks — the units side of the concept-trap index.

M L2 T−2
Work · Energy · Heat · Torque · Moment of a couple
Torque is r×F and work is F·d — same dimensions, different physics. Torque is never quoted in joules; it is newton metres, and the distinction is the vector product.
M L T−1
Momentum · Impulse
Deliberate: impulse IS the change in momentum, so the equality of dimensions is the content of the theorem.
M L−1 T−2
Pressure · Stress · Young's modulus · Bulk modulus · Energy density
Energy per volume and force per area are the same thing written two ways — J m−3 = Pa exactly.
T−1
Frequency · Angular velocity · Angular frequency · Decay constant · Activity · Velocity gradient
All are per-second. Hz, rad s−1 and Bq are the same dimension with different names kept apart on purpose, so a reader knows which is meant.
M L2 T−1
Angular momentum · Planck constant · Action
Why ℏ can be an angular momentum in mvr = nℏ without any fudge — it already has those dimensions.
L T−2
Acceleration · Gravitational field strength
g is both, which is why N kg−1 and m s−2 are numerically identical.
L2 T−2
Latent heat · Specific heat × temperature · Gravitational potential · Absorbed dose · (speed)2
Energy per unit mass, in every case.
—
Strain · Refractive index · Relative density · Coefficient of friction · Angle · Poisson's ratio · Reynolds number · Efficiency · Magnification
Dimensionless is not a small class. A dimensional check can never validate a formula whose answer is a pure number.
M L2 T−3
Power · Heat current · Rate of doing work
One quantity under three names, depending on what is flowing.
M T−3
Intensity · Poynting vector · Irradiance · Radiant emittance
Power per area, however the energy arrives.
M T−2
Surface tension · Spring constant · Force per length
N m−1 = J m−2 — surface tension read as energy per area is often the faster route.
M L2 T−3 A−1
Potential difference · EMF · Stopping potential
An EMF is a voltage. The name is a nineteenth-century mistake that stuck.

A term you want defined rather than measured is in the A–Z glossary; the mistakes are in concept traps; the formulas are on the thirty chapter cards. Five more calculators — kinematics and projectiles, lenses and mirrors, circuit networks, hydrogen levels and significant figures — are on the calculators page. This page also prints — the appendix.