ever be. Their potential energy will have run down to its lowest minimum, their mutual potential being a negative maximum, and their coefficient of mutual potential M, having its greatest possible value. Two circuits, then, are urged so that their coefficient of mutual potential M shall have the greatest possible value. This justifies Maxwell's Rule (Art. 193), because M represents the number of lines of force mutually intercepted by both circuits. And since in this position each circuit induces as many lines of magnetic force as possible through the other, the coefficient of mutual potential M is also called the coefficient of mutual induction.

NOTE ON MAGNETIC AND ELECTRO-
MAGNETIC UNITS.

321. Magnetic Units.-All magnetic quantities, strength of poles, intensity of magnetisation, etc., are expressed in terms of special units derived from the fundamental units of length, mass, and time, explained in the Note on Fundamental and Derived Units (Art. 254). Most of the following units have been directly explained in the preceding Lesson, or in Lesson XI.; the others follow from them.

Unit Strength of Magnetic Pole.-The unit magnetic pole is one of such a strength, that when placed at a distance of one centimetre (in air) from a similar pole of equal strength, repels it with a force of one dyne (Art. 125). Magnetic Potential.—Magnetic potential being measured by work done in moving a unit magnetic pole against the magnetic forces, the unit of magnetic potential will be measured by the unit of work, the erg.

Unit Difference of Magnetic Potential.-Unit difference of magnetic potential exists between two points when it requires the expenditure of one erg of work to bring a (N.-seeking) unit magnetic pole from one point to the other against the magnetic forces.

Intensity of Magnetic Field is measured by the force it exerts upon a unit magnetic pole: hence,

Unit Intensity of Field is that intensity of field which acts on a unit (N.-seeking) pole with a force of one dync.

322. Electromagnetic Units.-The preceding magnetic units give rise to the following set of electrical units, in which the strength of currents, etc., are expressed in magnetic measure. The relation of this "electromagnetic" set of units to the "electrostatic" set of units of Art. 257 is explained in Art. 365.

Unit Strength of Current.—A current has unit strength when one centimetre length of its circuit bent into an arc of one centimetre radius (so as to be always one centim. away from the magnet-pole) exerts a force of one dyne on a unit magnet-pole placed at the centre (Art. 196). Unit of Quantity of Electricity is that quantity which is conveyed by unit current in one second.

Unit of Difference of Potential (or of Electromotive-force). Potential is work done on a unit of electricity; hence unit difference of potential exists between two points when it requires the expenditure of one erg of work to bring a unit of electricity from one point to the other against the electric force.

Unit of Resistance.-A conductor possesses unit resistance when unit difference of potential between its ends causes a current of unit strength (i.e. one unit of quantity per second) to flow through it.

323. Practical Units.-Several of the above "absolute" units would be inconveniently large and others inconveniently small for practical use. The following are therefore chosen instead, as electromagnetic units :

Electromotive-force.-The Volt, = 108 absolute units (being a little less than the E. M. F. of one Daniell's cell). Resistance.-The Ohm, = 109 absolute units of resistance (and theoretically the resistance represented by the velocity of one earth-quadrant per second). (See Art. 364.) Current.-As a practical unit of current, that furnished by a potential of one volt though one ohm is taken, being ΙΟ of an absolute (electro-magnetic) unit of current, and is known as one Ampère (formerly one "weber "). Quantity.—The Coulomb, = 10-absolute units of quantity of the electromagnetic system.

-1

_9

= 10

(or one one-thousand

Capacity. The Farad,
millionth) of absolute unit of capacity.

Seeing, however, that quantities a million times as great as
some of these, and a million times as small as some, have to be
measured by electricians, the prefixes mega- and micro- are
sometimes used to signify respectively
"" one million" and " one-
millionth part." Thus a megohm is a resistance of one million
ohms, a microfarad a capacity of
of a farad, etc.
The prefix milli- is frequently used for "one-thousandth part ;"
thus a milli-ampère is the thousandth part of one ampère.

1

1.000.000

This system of "practical" units was devised by a committee
of the British Association, who also determined the value of the
"ohm " by experiment, and constructed standard resistance
coils of german-silver, called "B. A. Units " or "ohms."
The "practical" system may be regarded as a system of units
derived not from the fundamental units of centimetre, gramme,
and second, but from a system in which, while the unit of time
remains the second, the units of length and mass are respectively
the earth-quadrant and 10-11
gramme.

324. Dimensions of Magnetic and Electromagnetic Units.
-The fundamental idea of "dimensions" is explained in Art.
258. A little consideration will enable the student to deduce
for himself the following table-

NOTE ON MEASUREMENT OF EARTH'S MAGNETIC FORCE IN ABSOLUTE UNITS.

325a. The intensity of the earth's magnetic force at any place is the force with which a magnet-pole of unit strength is attracted. As explained in Art. 138, it is usual to measure the horizontal component H of this force, and from this and the cosine of the angle of dip to calculate the total force I, as the direct determination of the total force is surrounded with difficulties. To determine H in absolute (or C.G.S.) units, it is necessary to make two observations with a magnet of magnetic moment M; (the magnetic moment being, as mentioned in Art. 313, the product of its length into the strength of one of its poles). In one of these observations the product MH is determined by a M method of oscillations; in the second the quotient is deterH mined by a particular method of deflection. The square root of the quantity obtained by dividing the latter by the former will, of course, give H.

(i.) Determination of MH.-The time t of a complete oscillation to-and-fro of a magnetic bar is

This

where K is the "moment of inertia" of the magnet. formula is, however, only true for very small arcs of vibration. By simple algebra it follows that

4T2K HM = t2

Of these quantities is ascertained by a direct observation of the time of oscillation of the magnet hung by a torsionless fibre ; and K can be either determined experimentally or by one of the following formulæ :—

where w is the mass of the bar in grammes, its length, a

its radius (if round), b its breadth, measured horizontally (if rectangular).

M

(ii.) Determination of The magnet is next caused to deflect a small magnetic needle in the following manner, "broadside on." The magnet is laid horizontally at right angles to the magnetic meridian, and so that its middle point is (magnetically) due south or due north of the small needle, and at a distance from its centre. Lying thus broadside to the small needle its N.-pole will repel, and its S.-pole attract, the N.-pole of the needle, and will exercise contrary actions on the S.-pole of the needle. The total action of the magnet upon the needle will be to deflect the latter through an angle d, whose tangent is directly proportional to and inversely proportional to the cube of the distance r; or

M
H

M

H'

= 73 tan d.

Dividing the former equation by this, and taking the square root, we get,

325b. Seeing that electricians have to deal with quantities requiring in some cases very large numbers, and in other cases very small numbers, to express them, a system of index notation is adopted, in order to obviate the use of long rows of cyphers. In this system the significant figures only of a quantity are put down, the cyphers at the end, or (in the case of a long decimal) at the beginning, being indicated by an index written above. Accordingly, we may write a thousand (= 10 × 10 × 10) as 103, and the quantity 42,000 may be written 42 × 103. British National Debt of £770,000,000 may be written £77 × 107. Fractional quantities will have negative indices when written as exponents. Thus (= 0·01), = I ÷ 10 ÷ 10 = 10-2. And so the decimal o'00028 will be written 28 × 10-5 (being = 28 × 00001). The convenience of this method will be seen by an example or two on electricity. The electrostatic capacity of the earth is 630,000,000 times

The