Relations between SI and other units 1.1.3



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1.1.3 Relations between SI and other units

CGS units

Many books and papers on electromagnetism were written before the general adoption of SI. The following table gives the relations between the units, and the conversion factor by which the number expressing the value of the quantity in CGS units must be multiplied to express its value in SI units.

Electric and magnetic units

Quantity and symbol

SI unit

Conversion factors

Name and symbol

Defining equation

CGSm

CGSe

Electric current, I . .

ampere

F_z = 10⁻⁷I²(dN/dz)^‡

10/c^†

Current density, J . .

A m⁻²

10⁵

10⁵/c

Electromotive force, potential

difference, V . .

volt

P^§ = IV

10⁻⁸

10⁻⁸c

Electric field strength, E

V m⁻¹

E = V/l

10⁻⁶

10⁻⁶c

Resistance, R . .

ohm

R = V/I

10⁻⁹

10⁻⁹c²

Conductance, G . .

siemens

G = 1/R

10⁹

10⁹/c²

Volume resistivity, ρ .

Ω m

10⁻²

10⁻¹¹c²

Conductivity, γ .

S m⁻¹

γ = 1/ρ = J/E

10²

10¹¹/c²

Electric charge, Q .

coulomb

Q = It

10/c

10.

Capacitance, C .

farad

C = Q/V

10⁹

10⁹/c²

11.

Electric flux density, D

C m⁻²

D = Q/l^2||

10⁵

10⁵/c

12.

Permittivity, ε .

F m⁻¹

ε = D/E

10¹¹/4πc²

13.

Magnetic field strength, H

A m⁻¹

H/dl = nI

10³/4π

14.

Magnetic flux, .

weber

E = d/dt

10⁻⁸

15.

Magnetic flux density, B

tesla

B =

/l^{2 ||}

10⁻⁴

16.

Inductance L, M .

henry

M =

10⁻⁹

17.

Permeability, μ .

H m⁻¹

μ = B/H

4π × 10⁻⁷

^†c = velocity of light in free space in cm s⁻¹ = 2.997 924 58 × 10¹⁰
^‡N denotes Neumann’s integral for two linear circuits each carrying the current I; F_z is the force between the two circuits in any
direction z, the circuits being in a vacuum; r is the distance between the vector elements ds₁, ds₂.

	N =			ds₁.ds₂
				r

^§ P denotes power.
^|| l² denotes area.

The proportionality constants in the following equations are determined by use of the defining equations (1) to (17) in the table above.

For the mutual inductance M of two linear circuits, for which Neumann’s integral is N, in a medium of permeability μ:

M =	μ	N ...(18)
	4π

For the force F_Z due to currents I₁ and I₂ in the above two circuits:

F_Z = I₁I₂	dM	=	μ	I₁I₂	dN	...(19)
	dz		4π		dz

For the capacitance δC of an element of volume of length δl and cross-sectional area δA in a uniform dielectric medium of permittivity ε, the electric field having the direction of l:

δC = ε	δA	...(20)
	δl

In the SI, Maxwell’s equations for propagation in a medium are

× H = + J, − × E = …(21)

where the dot means d/dt, J is the current density, and the other quantities are as in the table. The velocity of propagation of an electromagnetic wave in the medium is given by

v² = 1/με …(22)

In free space, where J is zero, and where D and B are linked with E and H by the electric and magnetic constants ε₀ and μ₀, the equations may be written

c² × B = , − × E = …(23)

The velocity of propagation c is given by

c² = 1/μ₀ε₀…(24)

The values of μ₀, ε₀ and of the characteristic impedance Z₀ of free space are

μ₀ = 4π × 10⁻⁷ H m⁻¹ = 1.256 637 μH m⁻¹

ε₀ = 8.854 19 pF m⁻¹

Z₀ = √(μ₀/ε₀) = 376.730 Ω

Coulomb's law of force F between electric charges Q, Q' a distance r apart, and its analogue for magnetic poles , ', can be deduced from Maxwell’s equations. The expressions are

F =	QQ'	...(25)
	4 π ε r²

F =	'	...(26)
	4 π μ r²

Conversion factors

The following table gives conversion factors for British and other units which are still used in some fields, but which, for the most part, are likely to disappear in course of time. Exact conversion factors are in bold type.

Unit name and symbol

SI equivalent

Reciprocal

Length

ångström . . . . . . . . . . . . .

0.1

fermi . . . . . . . . . . . . . . .

micron . . . . . . . . . . . . . .

μm

yard . . . . . . . . . . . . . . .

0.914 4

1.093 61

foot . . . . . . . . . . . . . . .

30.48

0.032 808 4

inch . . . . . . . . . . . . . . .

2.54

0.393 701

statute mile . . . . . . . . . . .

1.609 344

0.621 371

nautical mile (international) . . . .

1.852

0.539 957

fathom (6 ft) . . . . . . . . . .

1.828 8

0.546 807

X unit . . . . . . . . . . . . .

0.100 2

astronomical unit (AU) . . . . .

0.149 6

parsec . . . . . . . . . . . .

30.857

Area

hectare . . . . . . . . . . . . .

hm²

barn . . . . . . . . . . . .

100

fm²

square yard . . . . . . . . . .

yd²

0.836 127

m²

1.195 99

square foot . . . . . . . . . .

ft²

9.290 30

dm²

0.107 639

square inch . . . . . . . . . .

in²

6.451 6

cm²

0.155 000

square statute mile . . . . . . .

2.589 99

km²

0.386 102

acre . . . . . . . . . . . . . .

0.404 686

hm²

2.471 05

Volume

stere . . . . . . . . . . . . . .

m³

litre . . . . . . . . . . . . . .

dm³

cubic yard . . . . . . . . . . .

yd³

0.764 555

m³

1.307 95

cubic foot . . . . . . . . . . .

ft³

0.028 3168

m³

35.314 7

cubic inch . . . . . . . . . . . .

in³

16.387 1

cm³

0.061 023 7

gallon (UK) . . . . . . . . . . .

gal

4.546 09

dm³

0.219 969

gallon (US) = 231 in³. . . . . .

3.785 41

dm³

0.264 172

pint (UK) . . . . . . . . . . .

0.568 261

dm³

1.759 75

barrel (US) = 42 gal (US)

bbl

0.159 0

m³

6.290

Plane angle

right angle = π/2 rad . . . . . . .

1.570 796

rad

0.636 620

degree = 1/90 right angle . . . .

π/180

rad

57.295 8

minute = (1/60)° . . . . . . .

π/10 800

rad

3437.75

second = (1/60)´ . . . . . . .

π/648

mrad

206.265

grade = 0.01 right angle . . . . .

π/200

rad

63.662 0

Mass

tonne . . . . . . . . . . . . .

metric carat . . . . . . . . .

0.2

pound (avdp) . . . . . . . . .

0.453 592 37

2.204 62

grain . . . . . . . . . . . . .

0.064 798 91

15.432 4

ounce (avdp) . . . . . . . . .

28.349 5

0.035 274 0

ounce (troy) = 480 gr . . . . .

31.103 5

0.032 150 7

ton = 2240 lb . . . . . . . .

1.016 05

0.984 207

slug . . . . . . . . . . . . . .

14.593 9

0.068 521 8

Specific volume

cubic foot per pound . . . . . .

ft³ lb⁻¹

0.062 428

m³ kg⁻¹

16.018 5

Density

pound per cubic foot . . . . . .

lb ft⁻³

16.018 5

kg m⁻³

0.062 428

pound per cubic inch . . . . . .

lb in⁻³

27.679 9

Mg m⁻³

0.036 127 3

pound per gallon (UK) . . . . .

lb gal⁻¹

99.776 3

kg m⁻³

0.010 022 4

slug per cubic foot . . . . . . .

slug ft⁻³

0.515 379

Mg m⁻³

1.940 32

Consumption, yield

gallon (UK) per mile . . . . . .

2.824 8

dm³ km⁻¹

0.354 01

bushel per acre . . . . . . . . .

0.089 87

m³ ha⁻¹

11.127

Moment of inertia

lb in² . . . . . . . . . . . . .

2.926 40

kg cm²

0.341 717

slug ft² . . . . . . . . . . . . .

1.355 82

kg m²

0.737 562

Moment of section

in⁴ . . . . . . . . . . . . .

41.623 1

cm⁴

0.024 025 1

Time

minute . . . . . . . . . . . . .

min

hour = 60 min . . . . . . . . .

3600

day = 24 h . . . . . . . . . . .

86 400

Velocity

foot per second . . . . . . . .

ft s⁻¹

0.304 8

m s⁻¹

3.280 84

1.097 28

km h⁻¹

0.911 344

mile per hour . . . . . . . .

mile h⁻¹

1.609 344

km h⁻¹

0.621 371

knot (international) . . . . . .

1.852

km h⁻¹

0.539 957

Acceleration

gal . . . . . . . . . . .. . . . . .

Gal

cm s⁻²

Force

dyne . . . . . . . . . . . . . . . .

dyn

μN

0.1

poundal . . . . . . . . . . . . . .

pdl

0.138 255

7.233 01

pound-force . . . . . . . . . . . .

lbf

4.448 22

0.224 809

ton-force . . . . . . . . . . . .

tonf

9.964 02

0.100 361

kilogram-force . . . . . . . . . . .

kgf

9.806 65

0.101 972

Force per unit length

lbf in⁻¹ . . . . . . . . . . . . .

0.175 127

kN m⁻¹

5.710 15

tonf ft⁻¹ . . . . . . . . . . . . .

32.690 3

kN m⁻¹

0.030 590 1

Torque

pound-force foot . . . . . . . . .

lbf ft

1.355 82

N m

0.737 562

Pressure, stress

bar . . . . . . . . . . . . . . . .

bar

0.1

MPa

lbf in⁻² . . . . . . . . . . . . . .

6.894 76

kPa

0.145 038

lbf ft⁻² . . . . . . . . . . . . . .

47.880 3

0.020 885 4

tonf in⁻². . . . . . . . . . . . . .

15.444 3

MPa

0.064 749 0

standard atmosphere . . . . . . .

atm

0.101 325

MPa

9.869 23

torr	. . . . . . . . . . .
mmHg

torr

0.133 322

kPa

7.500 62

inHg

3.386 39

kPa

0.295 300

Work, energy, heat

erg . . . . . . . . . . . . .

erg

0.1

μJ

electronvolt . . . . . . . . .

0.160 219

6.241 46

calorie (IT) . . . . . . . . .

cal

4.186 8

0.238 846

British thermal unit . . . . .

Btu

1.055 06

0.947 817

therm = 10⁵ Btu . . . . .

0.105 506

9.478 17

foot poundal . . . . . . . . .

ft pdl

0.042 140 1

23.730 4

foot pound-force . . . . . .

ft lbf

1.355 82

0.737 562

Power

horsepower = 550 ft lbf s⁻¹ . . .

0.745 700

1.341 02

cheval vapeur = 75 m kgf s⁻¹ . .

ch, CV

0.735 499

1.359 62

Temperature

degree Celsius . . . . . . . . .

°C

t °C . . . . . . . . . . . . . .

273.15 + t

degree Fahrenheit . . . . . . .

°F

5/9

1.8

t °F . . . . . . . . . . . . . .

5(t − 32)/9

°C

degree Rankine . . . . . . . . .

5/9

1.8

Heat transmission

Btu h⁻¹ . . . . . . . . . . . . .

0.293 071

3.412 14

Viscosity (dynamic)

poise . . . . . . . . . . . . .

0.1

Pa s

lb ft⁻¹ s⁻¹ = pdl s ft⁻² . . . .

1.488 16

Pa s

0.671 969

slug ft⁻¹ s⁻¹ = lbf s ft⁻² . . .

47.880 3

Pa s

0.020 885 4

Viscosity (kinematic)

stokes . . . . . . . . . . . . .

cm² s⁻¹

ft² s⁻¹ . . . . . . . . . . . . .

9.290 30

dm² s⁻¹

0.107 639

Electric stress

kilovolt per inch	. . . . . . .
volt per mil

kV in⁻¹

0.393 701

kV cm⁻¹

2.54

Magnetism

gauss . . . . . . . . . . . . . .

Gs, G

100

μT

0.01

oersted . . . . . . . . . . . . .

(1/4π)

kA m⁻¹

4π

maxwell . . . . . . . . . . . . .

nWb

0.1

gamma . . . . . . . . . . . . .

Light

stilb . . . . . . . . . . . . . . .

kcd m⁻²

0.1

phot . . . . . . . . . . . . . . .

klx

0.1

Radioactivity

curie . . . . . . . . . . . . . .

GBq

0.027 027

röntgen . . . . . . . . . . . . .

0.258

mC kg⁻¹

3.875 97

rad . . . . . . . . . . . . . .

rad

mGy

0.1

rem . . . . . . . . . . . . . .

rem

mSv

0.1

References

(1) BS 350:1983, Conversion Factors.
(2) BS 5555:1993 (ISO 1000–1992) SI units.
(3) Quantities, Units and Symbols (1975, 1981) The Royal Society.
(4) SI—The International System of Units (1993), 6th edn (translation of a BIPM publication), HMSO.

P. Vigoureux

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