A guide to the different approaches used to define measurement units in the physics of magnetism and electrostatics.

- ↓ Multiple systems
- ↓ The laws of Ampère and Coulomb
- ↓ The approach taken by the CGS system
- ↓ The approach taken by the SI
- ↓ The evolution of unit systems
- ↓ The ratio of the e.s.u. to the e.m.u.
- ↓ Dimensional calculations
- ↓ Turns of wire
- ↓ Some other conversions
- ↓ Using units and symbols
- ↓ A note on spelling

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See also ...

[↑ Producing wound components]
[↑ The terminology of electromagnetism]
[→ Air coils]
[↓ Ampère's Law of Force]
[W
A Dictionary of Units of Measurement]
[W
Unit Converter]

Our methods of measurement define who we are and what we value.

Ken Alder,* The Measure of All Things*

Units are no longer taught extensively. Their blandness excites less than drying paint. Only the heart of a true pedant beats faster at the prospect of a good, long argument over the merits of his preferred unit system. They are also irrelevant in the sense that the universe operates in exactly the same way regardless of humanity's attempts to document it. Following Thomson, however, there is something reassuring about numbers. Drop a mass, m, on your foot and you worry. Drop nine grams and you don't.

In magnetism, unfortunately, you quickly encounter the problem of its
unit systems. That's right: system**s**, plural. You can understand
that the world has different unit systems for length or mass. One half
talks of pounds, feet and inches while the other says grams, millimetres
and metres*. People by
their millions find that the older 'imperial' or British system works
for them (despite the occasional mishap). They conclude it not to be
worth effort and expense changing.

Magnetism, you might think, would be different. It's only scientists
and engineers who calculate it, and that it would be easy for them to
concur about units. Having more than one set of units makes life hard
for beginners in a subject which is quite difficult enough already,
thank you very much. Sadly, scientists have spent at least seventy
years** disagreeing** about the units for electromagnetism. Even
were everyone to adopt the 'new' SI units the need for familiarity
with the older CGS systems would remain in order to understand the large
number of original works published in the latter. We'll spend a couple
of paragraphs on the historical reasons for this schism because it also
sheds light on the science. To this end, we also ignore all post-2019
developments since they do not explain how diversity arose.

[↑ Top of page]

The trouble started with the
definitions for the units of electric current and electric charge. Three
simple, but fundamental, equations are crucial here. The first is *Ampère's Force Law* from
which may be derived the electromechanical force, F_{m}, between
two thin, parallel wires (see fig USA). The length of the wires, s, has
to be much greater than their separation, a, and the wires must be thin
in comparison with a. These conditions ensure that -

- The field produced by each wire decays at a rate proportional
to 1 / a,

and

- Field lines produced by wire 1 which intersect wire 2 do so at right angles both to wire 2 and a line joining 1 to 2 normally, and vice versa.

If s > 15a then the error will be below 1%. The Force Law
predicts the force on these wires to be -

where the* magnetic force constant*, k_{m} or
κ_{m}, is a constant whose value depends upon the unit
system used to measure the currents, forces and distances. In other
words, having decided how to define distance and force, you can then
choose a k_{m} that defines how large your unit of current will
be. The factor of 2 here is not intrinsic to Ampère's Force Law
but arises through its application to the particular case of parallel
wires.
Some writers incorporate it into k_{m}, but most do not.

The second equation is * Coulomb's Law* which
gives the electrostatic force, F_{e}, between two isolated
particles, separated by a distance, r, and carrying electric charges
Q_{1} and Q_{2} -

F_{e} = k_{e} Q_{1} Q_{2} /
r^{2}

Coulomb's Law

Where k_{e} is a constant whose value depends upon the unit
system used to measure the charges, forces and distances. In other
words, having decided how to define distance and force, you can then
choose a k_{e} that defines how large your unit of charge will
be. Or, at least you could do if you had not already decided upon
k_{m}. Why? To understand this we need to introduce our third
equation which expresses the fact that electric charge, Q, and current,
I, are dependent quantities -

Q = I × T

Equation USQ

where T is the time, in seconds, for which a given current, I, flows.
Equation USQ must hold regardless of the unit system chosen; one unit
of charge being represented by the passage of one unit of current for
one second. You should now be able to see that it is possible either to
invent some convenient value for k_{m} and then calculate
k_{e} or else invent some convenient value for k_{e} and
then calculate k_{m}. You cannot choose both k_{e} and
k_{m} independently because that would violate Equation USQ.

Performing both Ampere's and Coulomb's experiments (Weber and Kohlrausch, 1856) gives you a
value for the relationship between k_{e} and k_{m} -

where c is the speed of light in a vacuum. This equation is considered further below.

That's the basic physics out of the way. However, the freedom to select
an arbitrary value for either k_{e} or k_{m} leads to
the profusion of unit systems from which we suffer today. Consider next
how the choices just described are made in the two major unit systems:
the CGS and the SI.

[↑ Top of page]

Every electromagnetic quantity may be defined with reference to the
fundamental units of Length, Mass, and Time.

James Clerk Maxwell,* A Treatise on
Electricity and Magnetism*

In the unit system called *CGS* (for **C**entimetres
**G**rams **S**econds) force is measured in *dynes*: the
force required to accelerate a mass of one gram at one centimetre per
second squared.

System | Quantity name | Quantity symbol | Unit name | Unit symbol | Equivalent |
---|---|---|---|---|---|

SI | force (mechanical) | F | newton | N | 1.0×10^{5} dyn |

CGS | force (mechanical) | F | dyne | dyn | 1.0×10^{-5} N |

However, the CGS system doesn't like to pick winners and losers in the
k_{m} & k_{e} game.
Instead it defines two subsystems known as *electromagnetic units*
(e.m.u. or CGSm) and *electrostatic units* (e.s.u. or CGSe).

[↑ Top of page]

In the e.m.u. subsystem it is Monsieur Ampère
who is the winner, but k_{mCGSm}
takes the value of 1.0.
That is, two wires one centimetre apart carrying 1 e.m.u. of current
experience a force of two dynes for each centimetre of their length.
What is the value of this current in SI units?
Rewriting Ampère's
Force Law shows -

I = √(F_{SI} a_{SI} /
(2 k_{mSI} s_{mSI}))

Equation USL

This is consistently in SI units so we must translate the CGS force and
length before substituting them -

I = √(2×10^{-5} ×
0.01 / (2×10^{-7} × 0.01))
= 10 amps

Equation USM

In CGS electromagnetic units this is sometimes called the *abampere*,
or the *biot* or sometimes just 'one e.m.u. of current'.

System | Quantity name | Quantity symbol | Unit name | Unit symbol | Equivalent |
---|---|---|---|---|---|

SI | current | I | ampere | A | 0.1 abampere |

e.m.u. | current | I | abampere | - | 10 amps |

Also important here is the speed of light in a vacuum, c_{CGS}
≅ 2.998×10^{10} centimetres per second.
Using equation USJ tells us that
k_{eCGSm} is equal to c_{CGS}^{2} ≅
8.988×10^{20}. Also, via equation USQ, -

System | Quantity name | Quantity symbol |
Unit name | Unit symbol | Equivalent |
---|---|---|---|---|---|

SI | charge | Q | coulomb | C | 0.1 |

e.m.u. | charge | Q | abcoulomb | - | 10 coulombs |

Sometimes no name is given for this unit other than an 'e.m.u. unit of
charge'. You can, with the assistance of figure USU, continue
from here to derive values in the rest of the e.m.u. such as potential
(1 *abvolt* = 1.0×10^{-8} volts), capacitance
resistivity and so on. The abvolt is so small in comparison with
customary potentials because the dyne is feeble. Trouble brews faster
yet.

[↑ Top of page]

In the CGS e.s.u. it is Monsieur Coulomb who gets first prize and a
value of
k_{eCGSe} equal to 1.0.
Coulomb's Law now looks like -

Coulomb's Law (in CGSe)

That means two charges of one
electrostatic unit each, separated by one centimetre, experience a force
of 1 dyne. How much is this
quantity of charge in SI units: coulombs? First we re-arrange Coulomb's Law -

Equation USB

Substituting equation USJ -

This is consistently in SI so we must translate the 1 dyne and one centimetre
before substituting -

Q_{SI} =
√(10^{-5}(0.01)^{2} /
c_{SI}^{2} 2×10^{7} / 2)

Equation USD

Which simplifies to -

Equation USE

In CGS electrostatic units this is sometimes called the *statcoulomb*
and sometimes just 'one e.s.u. of charge'.

System | Quantity name | Quantity symbol |
Unit name | Unit symbol | Equivalent |
---|---|---|---|---|---|

SI | electrostatic charge | Q | coulombs | Q | ≅ 3×10^{9} statcoulombs |

e.s.u. | electrostatic charge | Q | statcoulombs | - | ≅ 3.336×10^{-10} coulombs |

Using equation USJ in reverse tells us that
k_{mCGSe} is equal to 1/c_{CGS}^{2} ≅
1.113×10^{-21}. Also, via equation USQ, -

System | Quantity name | Quantity symbol |
Unit name | Unit symbol | Equivalent |
---|---|---|---|---|---|

SI | current | I | ampere | A | ≅ 3×10^{9} statamperes |

e.s.u. | current | I | statampere | - | ≅ 3.336×10^{-10} amps |

Sometimes this quantity is called 'an esu per second of current'. Other
e.s.u. units follow on. For example the *statvolt* is equal to
about 299.8 volts.

Combining Equation USE with the result that 1 abamp = 10 amperes gives -

System | Quantity name | Quantity symbol |
Unit name | Unit symbol | Equivalent |
---|---|---|---|---|---|

e.m.u. | current | I | abamperes | - |
c_{CGS} statamperes |

e.s.u. | current | I | statampere | - | 1 /
c_{CGS} abampere |

This curious factor of c is considered further below. It's worth pointing out here that the 'equivalence' shown between e.m.u. current and e.s.u. current does not extend to the units, which differ by that velocity: c. Ditto, of course, for the other quantities.

The e.m.u. and e.s.u. systems were principally the work of Gauss and Weber.

[↑ Top of page]

The e.m.u. and e.s.u. subsystems described
above are not adopted in their pure forms. Instead a composite system
is devised called *Gaussian Units*. This takes a 'pick and mix'
selection from the e.m.u. and the e.s.u. . From the former it takes the
'magnetic units' of field strength, flux, flux density, magnetization
etc.. From the latter it takes the 'electric units' charge, current,
permittivity etc.. See table USS below.

The Gaussian unit of magnetic flux density, the *gauss*, is defined
using the Force Law -

F_{CGS} = I_{CGSm} s_{CGS}
B_{GSN} dynes

Equation USF

Equation USF says that a wire 1 centimetre long, sitting in a flux
density equal to one gauss and carrying a current of 1 abampere
experiences a force equal to 1 dyne. What is this flux density in the SI?
The SI defining relationship is -

B_{SI} = F_{SI} / (I_{SI} s_{SI})

The Motor Equation

Substituting the current, force and length values
converted from their CGS definitions -

B_{SI} = 10^{-5} / (10 × 0.01) =
10^{-4} tesla

Equation USI

The Gaussian unit of magnetic field strength, the* oersted*, was defined as equal to
unity (1.0) at the centre of a circular loop of thin wire, radius one
centimetre, carrying 1/(2π) abampere of current. Modern definitions rest on Maxwell's
equation for H, but, under steady current conditions, it all boils down
to Ampère's circuital law (and it's more refined sibling the
Biot-Savart equation). Let's convert the oersted into SI using the
result for the field at the centre of a current loop:

H_{SI} = I_{SI} / (2 a) = (10 / (2π))
/ (2 × 0.01) = 250 / π amps per metre

Equation USO

where a is the radius of the loop.

Quantity Name | ESU dimensions | EMU dimensions | ||||
---|---|---|---|---|---|---|

cm | g | s | cm | g | s | |

Electric charge | +3/2 | +1/2 | -1 | +1/2 | +1/2 | +0 |

Electric potential (voltage) | +1/2 | +1/2 | -1 | +3/2 | +1/2 | -2 |

Magnetic flux | +1/2 | +1/2 | +0 | +3/2 | +1/2 | -1 |

Current | +3/2 | +1/2 | -2 | +1/2 | +1/2 | -1 |

Electric flux density | -1/2 | +1/2 | -1 | -3/2 | +1/2 | +0 |

Electric field strength | -1/2 | +1/2 | -1 | +1/2 | +1/2 | -2 |

Magnetic flux density | -3/2 | +1/2 | +0 | -1/2 | +1/2 | -1 |

Magnetic field strength | +1/2 | +1/2 | -2 | -1/2 | +1/2 | -1 |

Capacitance | +1 | +0 | +0 | -1 | +0 | +2 |

Inductance | -1 | +0 | +2 | +1 | +0 | +0 |

Permittivity | +0 | +0 | +0 | -2 | +0 | +2 |

Permeability | -2 | +0 | +2 | +0 | +0 | +0 |

Resistance | -1 | +0 | +1 | +1 | +0 | -1 |

A most important fact to grasp is that the Gaussian has no separate
concepts of absolute and
relative permeability. If you take the Gaussian flux density (in
gausses) and divide by the field strength (in oersteds) then you still
get a result which is called permeability. That result, however, is
numerically equal to what in the SI is the ** relative**
permeability. So, for a vacuum, oersteds/gausses = 1.0 and for iron it
might be 8000. The same calculation in the SI, using
teslas/(amps per metre), would yield values lower by a factor
of 4π×10^{-7} and are designated ** absolute**. It
follows that, Gaussian permeability being a dimensionless ratio (just
like 'relative permeability'), the fundamental units for flux density
and field strength are the same:
cm^{-½} g^{½} s^{-1}
.

Another trap for the unwary is the use of the symbol μ_{0} by
some authors (eg
Bozorth) to represent initial permeability. This is not, repeat
not, the permeability of a vacuum as used by authors working in the SI.
Oh, me 'ead :-(

The Gaussian system, adopted in 1881, was (despite its name) principally the work of Heinrich Hertz and Maxwell. Its success meant that the term 'CGS units' become a synonym for it.

[↑ Top of page]

I often say that when you can measure what you are speaking about, and
express it in numbers, you know something about it ...

William Thomson,* Popular
Lectures*

In the SI, distance is measured in metres and force is measured in
newtons (the force which when applied to a mass of one kilogram accelerates it at one
metre per second squared). Now the important
bit: Monsieur Ampère is again the winner and k_{mSI} is
given the value of 10^{-7}. In the SI current is measured in
amperes -

One ampere is that constant
electric current which, if maintained in two straight parallel
conductors of infinite length, of negligible cross-section, and placed
one metre apart in vacuum, would produce between these conductors
a force equal to 2×10^{-7} newton per metre of length.

If you were wondering* why* the constant is 2×10^{-7}
then the short answer is that this definition of the amp is compatible
with earlier definitions of it. The ampere was
in use long before the SI. However, you also have a field strength (H) of one ampere per metre at the
centre of a circular loop of wire of diameter one metre when it carries
one ampere of current. See the Biot-Savart page for more details.
Anyway, the runner up, Monsieur Coulomb, has to define his unit of
electric charge via Equation USQ and then equation USJ gives the
electrostatic force constant -

In passing you might note that this means that two charges, of one
coulomb each, placed one metre apart exert a force upon each other that
exceeds the weight of two fully laden oil super-tankers. Do not try
this at home. One ampere is a significant amount of current, yet it
still produces a feeble magnetostatic force. One ampere second, i.e.
one coulomb, produces an enormous electrostatic force. That is why
Monsieur Ampère and Monsieur Coulomb do not see eye to eye and
why science has had such a hard time reconciling all this Gallic
discord.

It's easy to be confused by this stark definition of the ampere, in
terms of the mechanical forces produced by it, into thinking that it is
a* derived* unit from the* base* units of mass, length and
time in the same way that joules, watts or pascals are. This indeed is
how earlier systems treated current. The SI produces instead, depending
on your point of view, either a stroke of genius or a fudge. it says
"OK, we have said that you can tell how large an amp is in terms of
force and length; but forget that we told you that. As far as the rest
of the SI goes we treat the ampere as a** fundamental** unit which
cannot be decomposed into more basic ones - in the same way that the
second or the kilogram cannot".

Immediately, however, we hit a problem. Re-arranging ampere's force law slightly -

I^{2} = a F_{m} / (2 k_{m} s)

Equation USR

On the left we have amperes^{2} (A^{2}). On the right
we just have the old mechanical units. The equation is not dimensionally
balanced. The workaround is to say that k_{mSI} is not just a constant
but a quantity with units of its own. Remembering that the length (s)
and the separation (a) both have dimensions of metres, inspection of
Equation USR tells us that the units of k_{mSI} must be newtons
per ampere squared (N A^{-2}).

Another problem with the SI is that, after its introduction in 1960, a number of 'loose ends' were left concerning its application to magnetics. This led to the Kennelly and Sommerfeld subsystems of units. Each of these is based on a different model of the way magnetism works. The Kennelly subsystem makes use of the concept of 'magnetic poles' to model the interaction of materials with a field at an atomic level. The Sommerfeld system ascribes 'dipole moments' instead to circulating electric currents. The end result is that different definitions (and units) are used for magnetization.

On a brighter note, modern authors appear to favour the Sommerfeld system - as will these web pages.

[↑ Top of page]

The amp, the volt and ohm agree

to live together peacefully ...

(As not drawn by) Joseph Barbera,* The Truce
Hurts*

Devising a unit system is difficult because of the conflicting requirements which a good one should meet -

- The units should be of a size that is appropriate for use in
science and engineering and in everyday ('customary') use by the wider
commercial and public community.

- The units should be defined in a manner that is both precise
and capable of being reproduced anywhere in the world.

- The units should be coherent. In a coherent system all the
subsidiary or derived units can be expressed as combinations of the
basic or fundamental units without the introduction of any
multiplier.

Compromise is unavoidable. First to adopt a customary metric system was
France in 1799. Progress towards its adoption elsewhere has been
distinctly less rapid now that those opposed to it are no longer
decapitated. Anyway, the CGS system enjoyed wide acceptance until the
electricity industry decided that the e.m.u. abvolt was too small for practical
use and succeeded in introducing a value 10^{8} times larger:
our present value for the volt. The ohm and the amp were also
re-scaled. Electricians were happy, but coherence with mechanical units
was lost. That is the fundamental reason why k_{mSI} takes the value it does:
it's what's needed to make electrical units a 'sensible' size for the
installers of electric heat and light.

In case anybody was still able to understand the state of the unit system it was decided in 1932 that what had been called the gauss would thereafter be known as the oersted. Only since then has the gauss been a unit of flux density.

In 1948 it was decided that the centimetre, the gram and the erg (the CGS unit of energy) should be replaced by the larger metre, kilogram and joule. This was called the MKSA system and it restored coherence (as foreseen shortly after 1901 by Giovanni Giorgi).

In 1960 the radian, steradian, kelvin and the candela were added to the MKSA system to form the International System of Units, abbreviated to SI (Système International d'Unités). Only minor tweaks to the SI followed, apart from the introduction in 1971 of the mole.

Why then, a hundred years after Giorgi's proposal, does everyone not use a coherent system? The reasons vary. They include, for example, a reluctance to specify flux density in teslas because Carl Friedrich Gauss was a greater intellect than Nikola Tesla. More importantly the SI permeability and permittivity are altogether nastier affairs compared to the CGS equivalents.

That, though, is a fundamental difference - because the SI includes the
ampere as a base unit, which CGS does not. It also explains why the
dimensions of magnetic induction in the SI are
different from those of magnetizing force. People with
knowledge of the B-field and H-field only in the SI tend to see them as
physically different. That the two exist for the sake of computational
convenience alone is more obvious in Gaussian Units. Some have argued
(Larson) that the unit system is
in a mess because physics is in a mess. They say that ascribing units
to k_{mSI} is
not science but computational legerdemain. Perish the thought.

So, the answer to our original question "Why use different unit
systems?" is not simple reluctance to change. It is, as Ken Alder says,
about what you hold to be important. Pragmatists favour the SI's
utilitarian approach to calculation, want all the factors of c kept out
of sight, accept the embarrassing units for k_{m} and will never
replace a 3 abampere fuse. OTOH, philosophically minded physicists want
only the three base quantities of the Gaussian system to better reflect
the underlying science. They dream of a visit to Woolworths where
they enquire of the sales assistant

"Excuse me, have you any 600,000,000 ergs per second light bulbs?"

"Why, certainly, sir. We sell all our electrical goods by Gaussian units because in advanced texts on electrodynamics we find that the tensor calculus equations which unify the electric and magnetic fields are just so much clearer!"

"Great, I'll take 59,958,491,600 / c of them, please."

"While you're here, may I interest you in our 0.005 statvolt AA cells?"

Stuck in between these groups are the engineers; we must cope both with text books written by the latter and problems set by the former. What system should we adopt? It's anybody's gauss. Whatever your own view may be, don't expect a truce soon. However, fear not. 1799 has gone and the Bureau no longer knocks on your door should you make inadvertent mention of the pre-revolutionary. In fact, it explicitly recognises that, for good or ill, non-SI units will endure.

[↑ Top of page]

This velocity, therefore, which indicates the relation between
electrostatic and electromagnetic phenomena, is a natural quantity of
definite magnitude, and the measurement of this quantity is one of the
most important researches in electricity.

James Clerk Maxwell,* A Treatise on
Electricity and Magnetism*

Although not essential to understanding unit systems, this topic does
relate to equation
USJ above. Examine the first entry in table USS above: that
for electric charge, and divide the electrostatic units by the
electromagnetic units. You obtain a velocity: cm s^{-1}. You
obtain the same for current, electric flux density and
magnetic field strength. For magnetic flux, magnetic flux density,
voltage, and electric field strength you obtain the reciprocal of a
velocity: cm^{-1} s. For capacitance and permittivity you obtain
the square of a velocity: cm^{2} s^{-2}. Finally, for
permeability, inductance and resistance you obtain the reciprocal of the
square of a velocity: cm^{-2} s^{2}.

Fundamentally, what this boils down to is that a dimensional analysis of
the ratio k_{e} / k_{m},
via a rearrangement of Ampère's Force Law and Coulomb's Law, shows that it has units of
velocity squared. This seems a bit curious since nothing in either
Ampère's or Coulomb's experiments needs to move (apart from the
electrons themselves). Even more curious is the
experimental value of this ratio -

Equation USW

where c is the speed of light in a
vacuum. This fact lent real proof to the proposal made many years
previously by Leonard Euler that light had
something to do with electromagnetism. It can be shown that the
magnetic force is just the electrostatic force after adjustment by the
relativistic Lorentz transform. You may
already know that the speed of light features heavily in the theory of
relativity too.

The constants k_{eSI} and k_{mSI} are not used directly in
the rest of SI electromagnetism. Instead new constants are defined as -

Equation USX

Equation USY

Introducing the factor of π at this early stage (called
*rationalisation*) prevents it from infesting later formulæ
as in the Gaussian
system. Note that because k_{mSI} is defined as exactly
10^{-7} that μ_{0} is also exact.
Since 1983 the speed of light has been defined to be **exactly**
2.997 924 58×10^{8} m s^{-1
}. Therefore k_{eSI} and ε_{0} also acquire
exact values. Although μ_{0} has retained the units of
k_{mSI} these are translated into the equivalent H
m^{-1}.

Trivia point: designation of the letter c is from the word* celerity*: quickness, or
rapidity of motion. Its Latin root also gives* accelerate*.

[↑ Top of page]

An advantage of the SI is the ease with which *dimensional calculations*
can be performed. To assist these table USF below lists the base units
for a few quantities which are not listed elsewhere in these web pages.

Quantity name | Quantity symbol |
Unit name | Unit symbol | Base Units | |||
---|---|---|---|---|---|---|---|

kg | m | s | A | ||||

Angle (planar) | θ | radian | rad | 0 | 0 | 0 | 0 |

Angular momentum | L | joule seconds | J s | 1 | 2 | -1 | 0 |

Capacitance | C | farad | F | -1 | -2 | 4 | 2 |

Electro motive force | E | volt | V | 1 | 2 | -3 | -1 |

Energy | W | joule | J | 1 | 2 | -2 | 0 |

Force (mechanical) | F | newton | N | 1 | 1 | -2 | 0 |

Frequency | f | hertz | Hz | 0 | 0 | -1 | 0 |

Moment of inertia | I | kilogram metres^{2} | kg m^{2} | 1 | 2 | 0 | 0 |

Permittivity | ε | farads per metre | F m^{-1} | -1 | -3 | 4 | 2 |

Power | P | watt | W | 1 | 2 | -3 | 0 |

Resistance | R | ohm | Ω | 1 | 2 | -3 | -2 |

Torque | T | newton metres | N m | 1 | 2 | -2 | 0 |

Example: Flux, mmf and reluctance are related via the equation

is the equation dimensionally correct?

On the right hand side we have F_{m} in amps,
divided by R_{m} in per henry,

A / (A^{2} s^{2} kg^{-1}
m^{-2}) = A (A^{-2} s^{-2} kg m^{2})
= kg m^{2} s^{-2} A^{-1}

and these are the base units for the left hand side of the equation: flux in webers.

Dimensional calculations are not merely party tricks which enable you to check your working. When you can follow the process by which the higher level (derived) units are composed from the simpler, more intuitive base units then you obtain a better understanding of quantities like resistance or inductance. Even the strangest and most abstract quantities eventually boil down to metres, kilograms, seconds and amps. Is there anything abstract or hard to grasp about them? Drop a kilogram on your foot. These pages are particular about units because of this.

You could also impress your friends by mentioning your 10 million kilogram metre squared per second cubed per ampere squared resistor. Uhh ... well, maybe not.

Dimensional analysis, in the form we use today, was first proposed by
Maxwell.

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As mentioned in the section on current, many engineers prefer that the number of turns is included explicitly in the units of magneto-motive force. They say, for example, that the coil in figure TMX carries '12 ampere turns' of MMF, or '12 A t'. Similarly, inductance factor has units of henries per turn squared.

Now, units come in one of two flavours: basic (like kilograms) or derived (like ohms). The former group, of which the SI has seven, are starting points for the units system, and cannot be decomposed into more fundamental ones. The latter group, on the other hand, is each made up from a combination of two or more of the basic units.

The question arises, then, into which category you fit the unit for turns. The established view is that turns aren't really part of either. Instead, they are regarded as having 'unit dimensions'. That is, they have the same units as the digit '1'. Informally, you say they are 'dimensionless'.

So, as far as dimensional calculations are concerned -

1^{3} = 1^{2} = 1^{1} = 1^{0}
= 1 = t^{0} = t^{1} = t^{2}

Equation UNA

etc. And -

nH t^{-2} = nH 1 = nH

Equation UNB

From which you see that, in a technical sense, turns are redundant.
Their inclusion, or omission, whenever or wherever you like, makes no
difference to the dimensional validity of an equation. You could, for
example, specify MMF as 'amperes per turn squared' and be no more and no
less correct than 'ampere turns'.

Quantity name | Base Units | ||||
---|---|---|---|---|---|

kg | m | s | A | t | |

Capacitance | -1 | -2 | 4 | 2 | 2 |

Inductance | 1 | 2 | -2 | -2 | -2 |

Magneto motive force | 0 | 0 | 0 | 1 | 1 |

Electro motive force | 1 | 2 | -3 | -1 | -1 |

Permittivity | -1 | -3 | 4 | 2 | 2 |

Resistance | 1 | 2 | -3 | -2 | -2 |

"That's not very helpful.", you complain, "Yah, boo, sucks to the SI ! Let's just treat turns as an eighth basic unit alongside amps, seconds and the rest." Well, as figure USU shows, all the electric units are related in a kind of 'family tree'; with amperes as the common ancestor. A change to one quantity has a 'knock-on' effect with the units from which and to which it relates. If you go tinkering with MMF then you face a dilemma. Either you have to accept that, in at least one equation, you will have dimensional inconsistency, or you must make the same alteration to all the other electric quantities, substituting 'A t' wherever 'A' appears originally -

If you do this then the physicists will ask if you have gone mad. They
don't care about turns. Whether a coil has four turns carrying three
amps or three turns carrying four amps then it's all just '12 amps' to
them. The magnetic field will be identical - so quit fussing. Consider
the 'dual' case of a battery. No one is confused if it's described as
having '9 volts'. No one imagines that each** cell** within it has
an EMF of 9 volts. We accept that it's the overall potential that's
being specified, without a special unit called the 'volt cell'. If
engineers were consistent then they wouldn't need 'ampere turns' to talk
about MMF.

We can only stutter "Yes, but ...", or else eschew the ampere, use Gaussian Units and the 'gilbert' for MMF instead. Dohhh!

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If you have a quantity expressed in CGS units then multiply by k to find the equivalent in SI units.

Quantity name | CGS Unit | k | SI Unit |
---|---|---|---|

magnetomotive force | gilbert | 2.5/π | ampere |

magnetic field strength | oersted | 250/π | amp per metre |

magnetic flux density | gauss | 10^{-4} | tesla |

magnetic flux | maxwell | 10^{-8} | weber |

magnetization | abamps per cm |
10^{3} | amps per metre |

Energy product | gauss-oersted | 7.962×10^{-3} | J m^{-3} |

Converting formulae between the two systems requires care. You may also see flux being specified in 'lines' - this is synonymous with maxwells.

Much data on susceptibility is still only available in CGS units. In CGS the susceptibility is defined as

χ = (μ - 1) / (4 π)

Equation USY

Where μ is the permeability. If this surprises you then recall
that CGS permeability also differs from SI permeability (by a factor of
4 π×10^{-7}).

Where you have a susceptibility expressed in CGS units then multiply by the factor tabulated below to obtain the susceptibility in the corresponding SI units.

CGS type | bulk | mass | molar | |||
---|---|---|---|---|---|---|

CGS symbol | κ, χ | χ_{ρ} |
χ_{M} | |||

CGS units | 1 | cm^{3} g^{-1} |
cm^{3} mol^{-1} | |||

SI type | SI symbol | SI units | ||||

bulk | χ | 1 | 4 π | 4 π×10^{-3} ρ |
4 π×10^{-6} ρ / M | |

mass | χ_{ρ} | m^{3} kg^{-1} | 4 π / ρ | 4 π×10^{-3} |
4 π×10^{-6} / M | |

molar | χ_{M} | m^{3} mol^{-1} | 4 π M / ρ | 4 π×10^{-3} M |
4 π×10^{-6} |

where ρ is the density of the substance in kg m^{-3} and
M is the molar mass in kg mol^{-1}.
For example, the element copper, Cu, has a molar mass of 0.06355 kg
mol^{-1} and a density of 8940 kg m^{-3}, so

bulk | mass | molar | |
---|---|---|---|

CGS | -7.68E-07 | -8.59E-08 | -5.46E-06 |

SI | -9.65E-06 | -1.08E-09 | -6.86E-11 |

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Although the following tips on the usage of units and symbols are not specific to electromagnetism they gain importance where the range of exponents is large or the reader may be unfamiliar with the quantity.

Refer to the four parts comprising a measurement as follows:

Quantity name | Quantity symbol |
Unit name | Unit symbol |
---|---|---|---|

Length | l | metre | m |

Resistance | R | ohm | Ω |

Magnetic flux | Φ | weber | Wb |

An interval of time might be 20 seconds, a mass may be forty kilograms, a resistance ten ohms, a temperature 600 kelvins and an energy ninety joules. The point is that we write all these unit names without a capital letter - even where the unit is named after a scientist.

Strictly speaking, the m, Ω and Wb above are symbols and not
abbreviations. Thus you don't use a full stop after one (as you would
an abbreviation) nor need you follow them by an s to denote a plural.
Like other symbols you can put them in formulas. Where the unit name
is that of a scientist you then capitalize the unit symbol, e.g.
C = 1.8×10^{-6} F.

You need to include a space between the numerical value and the unit symbol. For example, L = 2.753 mH rather than L = 2.753mH. Also, L = 2.753 millihenries is better than L = 2.753 milli henries or L = 2.753 milli-henries.

You sometimes see tables headed like that on the left

River name | length, d x 10 ^{6} m |
---|---|

Nile | 6.7 |

Amazon | 6.3 |

Mississippi-Missouri | 6.2 |

Congo | 4.3 |

Niger | 4.2 |

Paraná | 4.0 |

Murray-Darling | 3.7 |

Volga | 3.7 |

Danube | 3.0 |

River name | length, d / (10 ^{6} m) |
---|---|

Nile | 6.7 |

Amazon | 6.3 |

Mississippi-Missouri | 6.2 |

Congo | 4.3 |

Niger | 4.2 |

Paraná | 4.0 |

Murray-Darling | 3.7 |

Volga | 3.7 |

Danube | 3.0 |

where the writer has tried to indicate that you need to multiply
the values in the length column by one million before using them.
Other writers use the same scheme to indicate that the data values
have already been multiplied by the given factor. For river lengths
it's perfectly obvious what the correct interpretation is but for
many quantities in science and engineering the ambiguity may be fatal.
Avoid it with the type of heading in USTb:
This you read like a dimensionally consistent equation
. So, for the Nile:

d / (10^{6} m) = 6.7

Equation USV

which we can re-arrange

d = 6.7×10^{6} m

Equation USP

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It's illiterate, Lewis.

Colin Dexter,* "Ghost in the Machine"*

adopted spelling | non-adopted spelling | complain to |
---|---|---|

metre | meter | William the Conqueror |

magnetized | magnetised | Endeavour Morse |

In the age of the search engine your spelling acquires an importance beyond the purely academic. Here's how to irritate anyone from the USA or who despises pedantry -

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E-mail:
R.Clarke@surrey.ac.uk

Last modified: 2019 October 7^{th}.