From: http://physics.nist.gov/constants Fundamental Physical Constants — Frequently used constants Relative std. Quantity Symbol Value Unit uncert. ur −1 speed of light in vacuum c, c0 299 792 458 m s (exact) −7 −2 magnetic constant µ0 4π × 10 NA =12.566 370 614... × 10−7 NA−2 (exact) 2 −12 −1 electric constant 1/µ0c 0 8.854 187 817... × 10 Fm (exact) Newtonian constant of gravitation G 6.6742(10) × 10−11 m3 kg−1 s−2 1.5 × 10−4 Planck constant h 6.626 0693(11) × 10−34 Js 1.7 × 10−7 h/2π ¯h 1.054 571 68(18) × 10−34 Js 1.7 × 10−7 elementary charge e 1.602 176 53(14) × 10−19 C 8.5 × 10−8 −15 −8 magnetic flux quantum h/2e Φ0 2.067 833 72(18) × 10 Wb 8.5 × 10 2 −5 −9 conductance quantum 2e /h G0 7.748 091 733(26) × 10 S 3.3 × 10 −31 −7 electron mass me 9.109 3826(16) × 10 kg 1.7 × 10 −27 −7 proton mass mp 1.672 621 71(29) × 10 kg 1.7 × 10 −10 proton-electron mass ratio mp/me 1836.152 672 61(85) 4.6 × 10 2 −3 −9 fine-structure constant e /4π0¯hc α 7.297 352 568(24) × 10 3.3 × 10 inverse fine-structure constant α−1 137.035 999 11(46) 3.3 × 10−9 2 −1 −12 Rydberg constant α mec/2hR∞ 10 973 731.568 525(73) m 6.6 × 10 23 −1 −7 Avogadro constant NA,L 6.022 1415(10) × 10 mol 1.7 × 10 −1 −8 Faraday constant NAeF96 485.3383(83) C mol 8.6 × 10 molar gas constant R 8.314 472(15) J mol−1 K−1 1.7 × 10−6 −23 −1 −6 Boltzmann constant R/NA k 1.380 6505(24) × 10 JK 1.8 × 10 Stefan-Boltzmann constant (π2/60)k4/¯h3c2 σ 5.670 400(40) × 10−8 Wm−2 K−4 7.0 × 10−6 Non-SI units accepted for use with the SI electron volt: (e/C) J eV 1.602 176 53(14) × 10−19 J 8.5 × 10−8 (unified) atomic mass unit = m = 1 m(12 1.660 538 86(28) × 10−27 1.7 × 10−7 1u u 12 C) u kg −3 −1 =10 kg mol /NA Page 1 Source: Peter J. Mohr and Barry N. Taylor, CODATA Recommended Values of the Fundamental Physical Constants: 2002, to be published in an archival journal in 2004. From: http://physics.nist.gov/constants Fundamental Physical Constants — Non-SI units Relative std. Quantity Symbol Value Unit uncert. ur electron volt: (e/C)J eV 1.602 176 53(14) × 10−19 J 8.5 × 10−8 (unified) atomic mass unit: = m = 1 m(12 1.660 538 86(28) × 10−27 1.7 × 10−7 1u u 12 C) u kg −3 −1 =10 kg mol /NA Natural units (n.u.) n.u. of velocity: −1 speed of light in vacuum c, c0 299 792 458 m s (exact) n.u. of action: reduced Planck constant (h/2π)¯h 1.054 571 68(18) × 10−34 Js 1.7 × 10−7 in eV s 6.582 119 15(56) × 10−16 eV s 8.5 × 10−8 in MeV fm ¯hc 197.326 968(17) MeV fm 8.5 × 10−8 n.u. of mass: −31 −7 electron mass me 9.109 3826(16) × 10 kg 1.7 × 10 2 −14 −7 n.u. of energy mec 8.187 1047(14) × 10 J 1.7 × 10 in MeV 0.510 998 918(44) MeV 8.6 × 10−8 −22 −1 −7 n.u. of momentum mec 2.730 924 19(47) × 10 kgms 1.7 × 10 in MeV/c 0.510 998 918(44) MeV/c 8.6 × 10−8 −15 −9 n.u. of length (¯h/mec) λC 386.159 2678(26) × 10 m 6.7 × 10 2 −21 −9 n.u. of time ¯h/mec 1.288 088 6677(86) × 10 s 6.7 × 10 Atomic units (a.u.) a.u. of charge: elementary charge e 1.602 176 53(14) × 10−19 C 8.5 × 10−8 a.u. of mass: −31 −7 electron mass me 9.109 3826(16) × 10 kg 1.7 × 10 a.u. of action: reduced Planck constant (h/2π)¯h 1.054 571 68(18) × 10−34 Js 1.7 × 10−7 a.u. of length: −10 −9 Bohr radius (bohr) (α/4πR∞) a0 0.529 177 2108(18) × 10 m 3.3 × 10 a.u. of energy: −18 −7 Hartree energy (hartree) Eh 4.359 744 17(75) × 10 J 1.7 × 10 2 2 2 (e /4π0a0 =2R∞hc = α mec ) VECTOR IDENTITIES4 Notation: f, g, are scalars; A, B, etc., are vectors; T is a tensor; I is the unit dyad. (1) A · B × C = A × B · C = B · C × A = B × C · A = C · A × B = C × A · B (2) A × (B × C)=(C × B) × A =(A · C)B − (A · B)C (3) A × (B × C)+B × (C × A)+C × (A × B)=0 (4) (A × B) · (C × D)=(A · C)(B · D) − (A · D)(B · C) (5) (A × B) × (C × D)=(A × B · D)C − (A × B · C)D (6) ∇(fg)=∇(gf)=f∇g + g∇f (7) ∇·(fA)=f∇·A + A ·∇f (8) ∇×(fA)=f∇×A + ∇f × A (9) ∇·(A × B)=B ·∇×A − A ·∇×B (10) ∇×(A × B)=A(∇·B) − B(∇·A)+(B ·∇)A − (A ·∇)B (11) A × (∇×B)=(∇B) · A − (A ·∇)B (12) ∇(A · B)=A × (∇×B)+B × (∇×A)+(A ·∇)B +(B ·∇)A (13) ∇2f = ∇·∇f (14) ∇2A = ∇(∇·A) −∇×∇×A (15) ∇×∇f =0 (16) ∇·∇×A =0 If e1,e2,e3 are orthonormal unit vectors, a second-order tensor T can be written in the dyadic form T ij eiej (17) = i,j T In cartesian coordinates the divergence of a tensor is a vector with components ∇·T i ji j (18) ( ) = j (∂T /∂x ) [This definition is required for consistency with Eq. (29)]. In general (19) ∇·(AB)=(∇·A)B +(A ·∇)B (20) ∇·(fT )=∇f·T +f∇·T 4 The r operator The rop erator In cartesian co ordinates x y z holds f f f r e e e gradf rf e e e x y z x y z x y z x y z a f f a a f x y z div a r a r f x y z x y z a a a a a a y z x x y z e e e rot a ra x y z y z z x x y In cylinder co ordinates rz holds f f f r e e e gradf e e e r z r z r r z r r z a f f f a a f a r z r r f div a r r r z r r r r z a a a a a a a r z z r rot a e e e r z r z z r r r r In spherical co ordinates rholds r e e e r r r r sin f f f e e e gradf r r r r sin a a a a a r r div a r r r r tan r sin a a a a a a r rot a e e r r r tan r sin r sin r r a a a r e r r r f f f f f r f r r r r r tan r sin General orthonormal curvelinear co ordinates u v w can b e obtained from cartesian co ordinates by the transformation x xu v w The unit vectors are then given by x x x e e e u v w h u h v h w where the factors h set the norm to Then holds i f f f e e e gradf u v w h u h v h w h h a h h a h h a div a u v w h h h u v w h a h a h a h a w u v w rot a e e u v h h v w h h w u h a h a v u e w h h u v h h h h h h f f f r f h h h u h u v h v w h w Relationships of the SI derived units with special names and symbols and the SI base units Derived units SI BASE UNITS without special SI DERIVED UNITS WITH SPECIAL NAMES AND SYMBOLS names Solid lines indicate multiplication, broken lines indicate division kilogram kg newton (kg·m/s2) pascal (N/m2)gray (J/kg) sievert (J/kg) 3 N Pa Gy Sv MASS m FORCE PRESSURE, ABSORBED DOSE VOLUME STRESS DOSE EQUIVALENT meter m 2 m joule(N·m) watt (J/s) becquerel (1/s)hertz (1/s) LENGTH J W Bq Hz AREA ENERGY,WORK, POWER, ACTIVITY FREQUENCY QUANTITY OF HEAT HEAT FLOW RATE second s m/s (OF A RADIONUCLIDE) TIME VELOCITY katal (mol/s) weber (V·s) henry(Wb/A) tesla (Wb/m2) kat Wb H T 2 mole m/s CATALYTIC MAGNETIC INDUCTANCE MAGNETIC mol ACTIVITY FLUX FLUX DENSITY ACCELERATION AMOUNT OF SUBSTANCE coulomb (A·s) volt (W/A) C V ampere A ELECTRIC POTENTIAL, CHARGE ELECTROMOTIVE ELECTRIC CURRENT FORCE degree (K) farad(C/V) ohm (V/A) siemens (1/W) kelvin K Celsius °C F W S CELSIUS CAPACITANCE RESISTANCE CONDUCTANCE THERMODYNAMIC TEMPERATURE TEMPERATURE t/°C=T/K – 273.15 candela 2 steradian radian cd lux (lm/m ) lumen (cd·sr) 2 2 (m/m = 1) lx lm sr (m /m =1) rad LUMINOUS INTENSITY ILLUMINANCE LUMINOUS SOLID ANGLE PLANE ANGLE FLUX The diagram above shows graphically how the 22 SI derived units with special names and symbols are related to the seven SI base units.