§ 3. The Absolute System of Electro-Magnetic Units.—The reader should at this point look again at Chapter XIV. § 1, and Chapter XV. § 2. We intend now to explain what are the funda

mental units to which we there referred.

(I.) Unit of current.-If a wire be bent into the arc of a circle, and a pole be placed at its centre, we see by the results of § 2 that the forces due to all the separate elements will act in the same direction on the pole; and, further, since each element is the same distance from the pole, the total force on this will be simply proportional to the current-strength, the pole-strength, the length of the arc, and the inverse square of the radius.

Now we have already defined the units of force, pole-strength, and length. It is, therefore, simplest to define the electromagnetic unit of current to be such as makes the force unity when all the other above quantities are unity. Or

The electro-magnetic unit of current is such that flowing through an arc of unit length (i.e. 1 centimètre), whose radius is unity (i.e. I centimètre), it acts with unit force (i.e. 1 dyne) on a pole of unit strength (see Chapter II. § 7) placed at the centre.

As has been already stated, the practical unit, called the ampère, is one-tenth of the above absolute unit.

(II.) The unit of E.M.F-If the E.M.F. between two points in a circuit be such that unit current flowing for unit time does unit work between these two points, then this E.M.F. is called unit E.M.F. in the electro-magnetic system.

It will be evident to the student that this definition of unit F.M.F. is, in theory at least, simple and clear; for the work can be conceived of as measured by observation of the number of calorimetric heat units given out between the points (as in Chapter XV. § 4); while the current can be measured in the above given absolute units by means of a galvanometer of known

constants.

Such a definition of unit E.M.F. is far better adapted to the requirements of electro-dynamical measurements than is the electrostatical definition given earlier.

As has been already stated, the above given unit is inconveniently small. The practical unit, called the volt, is 108 times this absolute unit.

(III.) The unit of resistance.—Ohm's law defines the unit of resistance as that through which unit E.M.F gives unit current. The practical unit of resistance, called the ohm, is 109 times the absolute unit. Thus, as stated, we have.

§ 4. Summary of Electro-Magnetic Units (see § 3).

Absolute unit of activity is the rate of work when unit current runs

under unit E.M.F., and is I erg per second.

=

Practical unit of activity, the watt, is the rate of work when one ampère works under one volt E.M.F.; it 107 ergs per second. We get watts by multiplying the number of volts by the number of ampères; or, if we are considering only the heat given out in a conductor, by (number of ampères)3 × (number of ohms).

Absolute unit of quantity is that which crosses any section of a conductor in one second when absolute unit of current flows.

Practical unit of quantity, or the coulomb, is the same when one ampère flows.

Absolute unit of capacity is that of a condenser in which the bound charge is absolute unit of quantity when the AV of the plates is absolute unit of E.M.F. or AV.

Practical unit of capacity, or the farad, is that of a condenser in which the bound charge is one coulomb when the AV of the plates is one volt. We may add that the more usual practical unit is the micro-farad (see below).

The megohm, mega-volt, &c., are respectively 1,000,000 times the ohm or volt, &c.

The microhm, micro-volt, &c., are respectively

or volt, &c.

T

of the ohn

Note.-Determination of the units.-The three quantities, C, R, and E, are related to each other by Ohm's law. It may be well to indicate how the absolute units, or any convenient multiple of them, might be determined independently of one another.

Current. When the dimensions and construction of a galvanometer are known, and when the magnetic field H in which it is situated has been determined by the method indicated in Chapter III. §§ 15 and 16, it is possible to measure current in absolute electro-magnetic units by observations of its action upon the galvanometer needle. For such a measurement no knowledge of E

or of R is either directly or indirectly implied.

Resistance. The absolute measurement of the resistance of a conductor is one of the most important experiments in physical science, and one of extreme difficulty. Of the several methods that have been employed to obtain a direct measure of the resistance of a wire, the simplest consists in passing a current through the wire which is contained in a calorimeter similar to that shown in the figure on page 241. Theoretical considerations give for the heat developed per second in the calorimeter the value JHC R. Here C is the current, the value of which can be calculated from the indications of a tangent galvanometer placed in the circuit; H is the heat developed per second, which can be found from the rise of temperature of the liquid by the principles of calorimetry ; and J is the mechanical equivalent of heat (p. 239), which is known from Joule's experiments to be very near to 41,750,000. The only quantity remaining is R, which can therefore be calculated from the other three. Other methods depend upon the laws of electro-magnetic induction given in p. 357; for the details of these the student must refer to more advanced works upon the subject.

Electromotive force. - Having methods of measuring current and resistance absolutely, E. M. F. may be found by the arrangement shown on p. 234. e is the cell whose E. M. F. is required; a tangent galvanometer is placed between R and B, and the current Ae,GB is not required. The current given by the battery P is first adjusted to a convenient strength by the rheostat R, then the contact piece Q is moved along the wire A B until no current flows in G. When this is the case the reasoning of § 13, p. 234, shows that e=aC; a being the resistance of the wire A Q, and C the current in the circuit PR BQAP, which is given by the tangent galvanometer; thus e is determined.

§ 5. The Dimensions of the Derived Units.

When any system of units has been constructed, each derived unit can be expressed in the fundamental units. The manner in which each derived unit involves the fundamental units can be exhibited in a simple form, giving what is called the dimensions of that physical quantity. Thus, in Chapter II., the dimensions of force were given by the relation

F=M.L.T-2.

In any one system of units physical quantities are of different natures if their dimensions. (i.c. the way in which they involve the

fundamental units) are different. This statement may be taken as axiomatic.

But the same physical quantity may have different dimensions in two different systems of units respectively. This is not so easy a matter to understand, but the fact is clear enough, as will be seen from the table given on the next page. Thus the dimensions of current are different according as we regard it from the point of view of electrostatic quantity (see Chapter V. § 1, passing ac: oss a section in unit time, or from the point of view of magnetic actions.

If two physical quantities have different dimensions in the same system, it follows that not only are they different in essential nature, but that in general any alteration of the fundamental units will alter in different ratios the numerical values of the two quantities.

On p. 301 we give a table exhibiting the dimensions of the various physical quantities in the two systems, electro-magnetic and electrostatic, respectively. It would be out of place in this Course to show how the dimensions of each are found, but we give one example.

Example. The dimensions of magnetic pole-strength.-By definition and experiment we have .

force between two equal poles

=

(pole-strength)2
(distance between them)

Using symbols to represent the units of pole-strength, force, and length, we may write .

And therefore

M
=
μ

L

T-1 gives the dimensions required.

With respect to the fact that in the electro-magnetic system the dimensions of R are L T-', or are those of a velocity, the reader is referred to Chapter XXI. § 7 (II.), note.

The ratio-velocity v.- -We must now call attention to the very remarkable fact that when a quantity is expressed both in electrostatic and in electro-magnetic measure, the ratio between the two sets of dimensions always involves simply the dimensions of a velocity; this we have designated by v in the last column of the table on the next page. Direct experiment shows (within the limits of experimental error) that this velocity is a constant.

If C and C' represent the numerical magnitude of the same current measured electro magnetically and electrostatically respectively, and if the same convention be assumed with respect to the other symbols, then by the table on the next page we have

The value of the constant v has been determined from double measurements of the same quantity, the same E.M.F., or the same capacity, respectively; and each investigation agreed in giving to v a value approximately agreeing with the velocity of light, and of the propagation of electro-magnetic induction (see Chapter XX. § 18). It also agrees with the velocity with which two bodies carrying electrostatic charges must move parallel to each other in order that the electrostatic repulsion between them may just balance their electromagnetic attraction (see Chapter XXII. § 15 (e)). It would appear that when rapid motion is given to electrostatic charges of electricity, these give electro-magnetic fields as do currents; but the velocity given to them must be very great in order to render this action at all sensible.

We may then regard the velocity v as one of the great constants in nature, that which expresses the physical constitution of the ether with respect to the propagation of waves in which the vibrations are

transverse.