# CHAPTER X.

## DIMENSIONS OF ELECTRIC UNITS.

620.] Every electromagnetic quantity may be defined with reference to the fundamental units of Length, Mass, and Time. If we begin with the definition of the unit of electricity, as given in Art. 65, we may obtain definitions of the units of every other electromagnetic quantity, in virtue of the equations into which they enter along with quantities of electricity. The system of units thus obtained is called the Electrostatic System.

If, on the other hand, we begin with the definition of the unit magnetic pole, as given in Art. 374, we obtain a different system of units of the same set of quantities. This system of units is not consistent with the former system, and is called the Electromagnetic System.

We shall begin by stating those relations between the different units which are common to both systems, and we shall then form a table of the dimensions of the units according to each system.

621.] We shall arrange the primary quantities which we have to consider in pairs. In the first three pairs, the product of the two quantities in each pair is a quantity of energy or work. In the second three pairs, the product of each pair is a quantity of energy referred to unit of volume.

## First Three Pairs.

### Electrostatic Pair.

 Symbol. (1) Quantity of electricity e (2) Line-integral of electromotive force, or electric potential E

### Magnetic Pair.

(3) Quantity of free magnetism, or strength of a pole . ${\displaystyle m}$

(4) Magnetic potential . . . . . . . ${\displaystyle \Omega }$

### Electrokinetic Pair.

(5) Electrokinetic momentum of a circuit . . . ${\displaystyle p}$

(6) Electric current . . . . . . . ${\displaystyle C}$

## Second Three Pairs.

### Electrostatic Pair.

(7) Electric displacement (measured by surface-density) . ${\displaystyle {\mathfrak {D}}}$

(8) Electromotive force at a point . . . . ${\displaystyle {\mathfrak {E}}}$

### Magnetic Pair.

(9) Magnetic induction . . . . . . ${\displaystyle {\mathfrak {B}}}$

(10) Magnetic force . . . . . . . ${\displaystyle {\mathfrak {H}}}$

### Electrokinetic Pair.

(11) Intensity of electric current at a point . . . ${\displaystyle {\mathfrak {C}}}$

(12) Vector potential of electric currents . . . ${\displaystyle {\mathfrak {A}}}$

622.] The following relations exist between these quantities. In the first place, since the dimensions of energy are ${\displaystyle \left[{\frac {L^{2}M}{T^{2}}}\right]}$, and those of energy referred to unit of volume ${\displaystyle \left[{\frac {M}{LT^{2}}}\right]}$, we have the following equations of dimensions:

 ${\displaystyle [e\,E]=[mz,\Omega ]=[p\,C]=\left[{\frac {L^{2}M}{T^{2}}}\right],}$ (1)
 ${\displaystyle [{\mathfrak {D}}\,{\mathfrak {E}}]=[{\mathfrak {B}}\,{\mathfrak {H}}]=[{\mathfrak {C}}\,{\mathfrak {A}}]=\left[{\frac {M}{LT^{2}}}\right].}$ (2)

Secondly, since e, p and ${\displaystyle {\mathfrak {A}}}$ are the time-integrals of C, E, and ${\displaystyle {\mathfrak {E}}}$ respectively

 ${\displaystyle \left[{\frac {e}{C}}\right]=\left[{\frac {p}{E}}\right]=\left[{\frac {\mathfrak {A}}{\mathfrak {E}}}\right]=[T].}$ (3)

Thirdly, since E, Ω, and p are the line-integrals of ${\displaystyle {\mathfrak {E}}}$, ${\displaystyle {\mathfrak {H}}}$, and ${\displaystyle {\mathfrak {A}}}$ respectively,

 ${\displaystyle \left[{\frac {e}{\mathfrak {E}}}\right]=\left[{\frac {\Omega }{\mathfrak {H}}}\right]=\left[{\frac {p}{\mathfrak {A}}}\right]=[L].}$ (4)

Finally, since e, C, and m are the surface-integrals of ${\displaystyle {\mathfrak {D}}}$, ${\displaystyle {\mathfrak {G}}}$, and ${\displaystyle {\mathfrak {B}}}$ respectively,

 ${\displaystyle \left[{\frac {e}{\mathfrak {D}}}\right]=\left[{\frac {C}{\mathfrak {E}}}\right]=\left[{\frac {m}{\mathfrak {B}}}\right]=\left[L^{2}\right].}$ (5)
623.] These fifteen equations are not independent, and in order to deduce the dimensions of the twelve units involved, we require one additional equation. If, however, we take either ${\displaystyle e}$ or ${\displaystyle m}$ as an independent unit, we can deduce the dimensions of the rest in terms of either of these.
 (1) ${\displaystyle [e]}$ ${\displaystyle {}=[e]}$ ${\displaystyle {}=\left[{\frac {L^{2}M}{mT}}\right]}$. (2) ${\displaystyle [E]}$ ${\displaystyle {}=\left[{\frac {L^{2}M}{eT^{2}}}\right]}$ ${\displaystyle {}=\left[{\frac {m}{T}}\right]}$. (3) and (5) ${\displaystyle [p]=[m]}$ ${\displaystyle {}=\left[{\frac {L^{2}M}{eT}}\right]}$ ${\displaystyle {}=[m]}$. (4) and (6) ${\displaystyle [C]=[\Omega ]}$ ${\displaystyle {}=\left[{\frac {e}{T}}\right]}$ ${\displaystyle {}=\left[{\frac {L^{2}M}{mT^{2}}}\right]}$. (7) ${\displaystyle [{\mathfrak {D}}]}$ ${\displaystyle {}=\left[{\frac {e}{L^{2}}}\right]}$ ${\displaystyle {}=\left[{\frac {M}{mT}}\right]}$. (8) ${\displaystyle [{\mathfrak {E}}]}$ ${\displaystyle {}=\left[{\frac {LM}{eT^{2}}}\right]}$ ${\displaystyle {}=\left[{\frac {m}{LT}}\right]}$. (9) ${\displaystyle [{\mathfrak {B}}]}$ ${\displaystyle {}=\left[{\frac {M}{eT}}\right]}$ ${\displaystyle {}=\left[{\frac {m}{L^{2}}}\right]}$. (10) ${\displaystyle [{\mathfrak {H}}]}$ ${\displaystyle {}=\left[{\frac {e}{LT}}\right]}$ ${\displaystyle {}=\left[{\frac {LM}{mT^{2}}}\right]}$. (11) ${\displaystyle [{\mathfrak {C}}]}$ ${\displaystyle {}=\left[{\frac {e}{L^{2}T}}\right]}$ ${\displaystyle {}=\left[{\frac {M}{mT^{2}}}\right]}$. (12) ${\displaystyle [{\mathfrak {A}}]}$ ${\displaystyle {}=\left[{\frac {LM}{eT}}\right]}$ ${\displaystyle {}=\left[{\frac {m}{L}}\right]}$.

624.] The relations of the first ten of these quantities may be exhibited by means of the following arrangement:

 ${\displaystyle e}$, ${\displaystyle {\mathfrak {D}}}$, ${\displaystyle {\mathfrak {H}}}$, ${\displaystyle C}$ and ${\displaystyle \Omega }$. ${\displaystyle E}$ ${\displaystyle {\mathfrak {E}}}$, ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle m}$ and ${\displaystyle p}$. ${\displaystyle m}$ and ${\displaystyle p}$, ${\displaystyle {\mathfrak {B}}}$, ${\displaystyle {\mathfrak {E}}}$, ${\displaystyle E}$. ${\displaystyle C}$ and ${\displaystyle \Omega }$ ${\displaystyle {\mathfrak {H}}}$, ${\displaystyle {\mathfrak {H}}}$, ${\displaystyle e}$.

The quantities in the first line are derived from ${\displaystyle e}$ by the same operations as the corresponding quantities in the second line are derived from ${\displaystyle m}$. It will be seen that the order of the quantities in the first line is exactly the reverse of the order in the second line. The first four of each line have the first symbol in the numerator. The second four in each line have it in the denominator.

All the relations given above are true whatever system of units we adopt.

625.] The only systems of any scientific value are the electrostatic and the electromagnetic system. The electrostatic system is founded on the definition of the unit of electricity, Arts. 41, 42, and may be deduced from the equation,

${\displaystyle {\mathfrak {E}}={\frac {e}{L^{2}}}}$,

which expresses that the resultant force ${\displaystyle {\mathfrak {E}}}$ at any point, due to the action of a quantity of electricity ${\displaystyle e}$ at a distance ${\displaystyle L}$, is found by dividing ${\displaystyle e}$ by ${\displaystyle L^{2}}$. Substituting the equations of dimension (1) and (8), we find

${\displaystyle \left[{\frac {LM}{eT^{2}}}\right]=\left[{\frac {e}{L^{2}}}\right]}$, ${\displaystyle \left[{\frac {m}{LT}}\right]=\left[{\frac {M}{mT}}\right]}$,

whence

${\displaystyle [e]=[L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}$, ${\displaystyle m=[L^{\frac {1}{2}}M^{\frac {1}{2}}]}$,

in the electrostatic system.

The electromagnetic system is founded on a precisely similar definition of the unit of strength of a magnetic pole, Art. 374, leading to the equation

${\displaystyle {\mathfrak {H}}={\frac {m}{L^{2}}}}$.

whence

${\displaystyle \left[{\frac {e}{LT}}\right]=\left[{\frac {M}{eT}}\right]}$. ${\displaystyle \left[{\frac {LM}{mT^{2}}}\right]=\left[{\frac {m}{L^{2}}}\right]}$,

and

${\displaystyle [e]=[L^{\frac {1}{2}}M^{\frac {1}{2}}]}$, ${\displaystyle [m]=[L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}$,

in the electromagnetic system. From these results we find the dimensions of the other quantities.

626.]

 Table of Dimensions. Dimensions in Symbol Electrostatic System Electromagnetic System Quantity of electricity ${\displaystyle e}$ ${\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}$ ${\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}]}$. Line-integral of electromotive force ${\displaystyle E}$ ${\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-1}]}$ ${\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-2}]}$. Quantity of magnetism ${\displaystyle m}$ ${\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}]}$ ${\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}$. Electrokinetic momentum of a circuit ${\displaystyle p}$ Electric current ${\displaystyle C}$ ${\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-2}]}$ ${\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-1}]}$. Magnetic potential ${\displaystyle \Omega }$ .mw-parser-output .wst-rule{background-color:black;color:black;width:auto;margin:2px auto 2px auto;height:1px} Electric displacement ${\displaystyle {\mathfrak {D}}}$ ${\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-1}]}$ ${\displaystyle [L^{-{\frac {3}{2}}}M^{\frac {1}{2}}]}$. Surface-density Electromotive force at a point ${\displaystyle {\mathfrak {E}}}$ ${\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-1}]}$ ${\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-2}]}$. Magnetic induction ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle [L^{-{\frac {3}{2}}}M^{\frac {1}{2}}]}$ ${\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-1}]}$. Magnetic force ${\displaystyle {\mathfrak {H}}}$ ${\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-2}]}$ ${\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-2}]}$. Strength of current at a point ${\displaystyle {\mathfrak {C}}}$ ${\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-2}]}$ ${\displaystyle [L^{-{\frac {3}{2}}}M^{\frac {1}{2}}T^{-1}]}$. Vector potential ${\displaystyle {\mathfrak {V}}}$ ${\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}]}$ ${\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-1}]}$.
627.] We have already considered the products of the pairs of these quantities in the order in which they stand. Their ratios are in certain cases of scientific importance. Thus
 Symbol. Electrostatic System. Electromagnetic System. ${\displaystyle {\frac {e}{E}}={}}$ capacity of an accumulator ${\displaystyle q}$ ${\displaystyle [L]}$ ${\displaystyle \left[{\frac {T^{2}}{L}}\right]}$. ${\displaystyle {\frac {P}{C}}={}}$ coefficient of self-induction of a circuit, or electromagnetic capacity ${\displaystyle L}$ ${\displaystyle \left[{\frac {T^{2}}{L}}\right]}$ ${\displaystyle [L]}$. ${\displaystyle {\frac {\mathfrak {D}}{\mathfrak {E}}}={}}$ specific inductive capacity of dielectric ${\displaystyle K}$ ${\displaystyle [0]}$ ${\displaystyle \left[{\frac {T^{2}}{L^{2}}}\right]}$. ${\displaystyle {\frac {\mathfrak {B}}{\mathfrak {H}}}={}}$ magnetic inductive capacity ${\displaystyle \mu }$ ${\displaystyle \left[{\frac {T^{2}}{L^{2}}}\right]}$ ${\displaystyle [0]}$. ${\displaystyle {\frac {E}{C}}={}}$ resistance of a conductor ${\displaystyle R}$ ${\displaystyle \left[{\frac {T}{L}}\right]}$ ${\displaystyle {\frac {L}{T}}}$. ${\displaystyle {\frac {\mathfrak {E}}{\mathfrak {C}}}={}}$ specific resistance of a substance ${\displaystyle r}$ ${\displaystyle [T]}$ ${\displaystyle \left[{\frac {L^{2}}{T}}\right]}$.

628.] If the units of length, mass, and time are the same in the two systems, the number of electrostatic units of electricity contained in one electromagnetic unit is numerically equal to a certain velocity, the absolute value of which does not depend on the magnitude of the fundamental units employed. This velocity is an important physical quantity, which we shall denote by the symbol ${\displaystyle v}$.

Number of Electrostatic Units in one Electromagnetic Unit.

For ${\displaystyle e}$, ${\displaystyle C}$, ${\displaystyle \Omega }$, ${\displaystyle {\mathfrak {D}}}$, ${\displaystyle {\mathfrak {H}}}$, ${\displaystyle C}$, …… ${\displaystyle v}$.

For ${\displaystyle m}$, ${\displaystyle p}$, ${\displaystyle E}$, ${\displaystyle {\mathfrak {B}}}$, ${\displaystyle {\mathfrak {E}}}$, ${\displaystyle {\mathfrak {A}}}$, …… ${\displaystyle {\frac {1}{v}}}$.

For electrostatic capacity, dielectric inductive capacity, and conductivity, ${\displaystyle v^{2}}$.

For electromagnetic capacity, magnetic inductive capacity, and resistance, ${\displaystyle {\frac {1}{v^{2}}}}$.

Several methods of determining the velocity ${\displaystyle v}$ will be given in Arts. 768–780.

In the electrostatic system the specific dielectric inductive capacity of air is assumed equal to unity. This quantity is therefore represented by ${\displaystyle {\frac {1}{v^{2}}}}$ in the electromagnetic system. In the electromagnetic system the specific magnetic inductive capacity of air is assumed equal to unity. This quantity is therefore represented by ${\displaystyle {\frac {1}{v^{2}}}}$ in the electrostatic system.

Practical System of Electric Units.

629.] Of the two systems of units, the electromagnetic is of the greater use to those practical electricians who are occupied with electromagnetic telegraphs. If, however, the units of length, time, and mass are those commonly used in other scientific work, such as the mètre or the centimètre, the second, and the gramme, the units of resistance and of electromotive force will be so small that to express the quantities occurring in practice enormous numbers must be used, and the units of quantity and capacity will be so large that only exceedingly small fractions of them can ever occur in practice. Practical electricians have therefore adopted a set of electrical units deduced by the electromagnetic system from a large unit of length and a small unit of mass.

The unit of length used for this purpose is ten million of mètres, or approximately the length of a quadrant of a meridian of the earth.

The unit of time is, as before, one second.

The unit of mass is 10-11 gramme, or one hundred millionth part of a milligramme.

The electrical units derived from these fundamental units have been named after eminent electrical discoverers. Thus the practical unit of resistance is called the Ohm, and is represented by the resistance-coil issued by the British Association, and described in Art. 340. It is expressed in the electromagnetic system by a velocity of 10,000,000 metres per second.

The practical unit of electromotive force is called the Volt, and is not very different from that of a Daniell's cell. Mr. Latimer Clark has recently invented a very constant cell, whose electromotive force is almost exactly 1.457 Volts.

The practical unit of capacity is called the Farad. The quantity of electricity which flows through one Ohm under the electromotive force of one Volt during one second, is equal to the charge produced in a condenser whose capacity is one Farad by an electromotive force of one Volt.

The use of these names is found to be more convenient in practice than the constant repetition of the words 'electromagnetic units,' with the additional statement of the particular fundamental units on which they are founded.

When very large quantities are to be measured, a large unit is formed by multiplying the original unit by one million, and placing before its name the prefix mega.

In like manner by prefixing micro a small unit is formed, one millionth of the original unit.

The following table gives the values of these practical units in the different systems which have been at various times adopted.

 Fundamental Units. Pratical System. B.A. Report, 1683. Thomson. Weber. Length, Earth's Quadrant, Metre, Centimetre, Millimetre, Time, Second, Second, Second, Second, Mass. 10-11 Gramme. Gramme. Gramme. Milligramme. Resistance Ohm 10⁷ 10⁹ 10¹ Electromotive force Volt 10⁵ 10⁸ 10¹¹ Capacity Farad 10⁻⁷ 10⁻⁹ 10⁻¹⁰ Quantity Farad(charged to a Volt.) 10⁻² 10⁻¹ 10