ELECTRICITY

would cause a deficit at the place from which it is taken and an accumulation at some other place. It is evident that this could not go on indefinitely, for the accumulated quantity would eventually overcome all obstacles and find its way to the place of deficit. Hence it follows that, in order to maintain a continuous current of electricity, it is necessary to provide, first, a continuous difference of potential, and second, a closed conducting-circuit. The first condition we may obtain by any of the several ways of producing electrification; as, for example, contact, induction, chemical action or electromagnetic induction, and the second by making the circuit continuous and of a conducting material. As there are no perfect conductors, any substance we may use will oppose, to a more or less extent, the flow of electricity along it, or, what is the same thing, will require the expenditure of energy in order to maintain a current, and the electrical energy which disappears appears as heat.

To sum up, we have the following quantities ever present in the case of continuous electric

current, connected with each other by simple

laws, which are of the utmost importance:

1. The continuous difference of potential or electromotive force.

2. The resistance of the closed conducting circuit, or that of any one of its parts.

3. The strength of the current which is produced.

4. The energy expended in the circuit, or any of its parts.

R

00000

P

+

A

Cu

Zn

1145

motive force between the points, and briefly written E. M. F. The work done when the unit-charge is carried completely around the circuit is called the electromotive force of the cell, and is equal to the difference of potential between the plates before the wire, R, was added.

$41. STRENGTH OF CURRENT. If the difference of potential of the plates (Fig. 29) is kept constant, the current through the wire, R, is constant, provided the resistance of the wire does not change. Furthermore, the quantity of electricity that passes through any cross-section of the circuit in the same time, as P for example, must be constant; if not, there would be an accumulation of electricity at some point in the circuit. The quantity that passes any section in unit time is called the intensity of the current.

42. RELATION BETWEEN E. M. F., INTENSITY OF CURRENT AND RESISTANCE. Let E be the electromotive force between the points P and P' (Fig. 29), I the strength of current and R the resistance of the conductor between P and P'. The relation between these three quantities is which is known as.

given by the expression I = E

Ohm's law.

R'

$43. ENERGY EXPENDED BY THE CURRENT IN OVERCOMING RESISTANCE. As before, let E be the electromotive force between P and P' (Fig. 29), I the intensity of current, and R the resistance of the wire between these points. By definition, E is the work done on unit-charge in carrying it from P to P'; and as I is the number of units passing between the points in unit-time, EI is the total work done between these points. From Ohm's law, E RI; hence the work per second is EI RI2. This energy, which disappears as electrical energy, appears as heat, which, in most cases, is a loss; but whenever it is desired to produce light or heat by means of the current, this forms the useful part of the energy.

=

The product, EIIR, is a number of workunits; hence if the heat generated is measured, we obtain a value of the mechanical equivalent of heat; for, by the principle of conservation of energy, the amount of energy that appears in the form of heat must be equal to that which disappeared as electrical energy.

5. The field of force about the wire. § 39. CONTINUOUS POTENTIAL DIFFERENCE. Take the case of a simple voltaic cell, as shown in Fig. 29, consisting of two conP' ducting-plates immersed in an immersed in an acid which acts upon one or both of them. Suppose that the wire, B R, is not present, the plates A and B become charged either by contact with the acid or by the conversion of chemical energy into electrical energy, or both. Each plate in contact with the acid is charged to a definite potential, depending upon the material of the plate and the Fig. 29. liquid. (See PRIMARY CELLS, in these Supplements.) In this case the zinc is charged to a higher potential than the copper. If, now, the plates be connected by the wire, R, we have but to consider two plates, as the charged bodies A and B of Fig. 10; the liquid corresponds to the connecting wire in the same figure, and the wire, R, is the return-path, the whole making up a complete circuit, which may be taken as a typical circuit. The liquid is necessarily a conductor, and conducts in a manner peculiar to liquids; it is called an electrolyte. The following table gives the specific resistance (See ELECTROLYSIS, in these Supplements.) and relative conductivity of the more common $40. ELECTROMOTIVE FORCE. The work substances; also the resistance of a wire or coldone by the electric forces when unit-charge is umn of the substance one meter long and one propelled from P to P' is called the electro-square millimeter in cross-section.

I

$44. CONDUCTIVITY. If R is the resistance of a conducting wire, then R ductivity.

is defined as its con

It is often necessary to refer to the resistance and conductivity of a substance, rather than a definite portion of it. The resistance of a substance is termed its specific resistance, and is defined as the resistance of a conductor of the material having unit-length and unit cross-section; i.e., a unit cube. The conductivity is the reciprocal of the specific resistance.

THE

$47a. DIRECTIONS AND INTENSITY OF ELECTROMAGNETIC FIELD. The direction of the force at any point in an electro-magnetic field is defined as the direction in which a north-seeking magnet-pole tends to move when placed at that point. The direction of the force can readily be found when the direction of the current is known, by remembering the following simple analogy: Suppose an ordinary watch to be pierced by a hole through its center, and perpendicular to its face; imagine the wire to pass through this hole so that the current will enter the face of the watch, and leave at the back. The direction in which the north-seeking pole would move around the wire is the same as that in which the hands of the watch move.

The intensity of the electro-magnetic field at any point is defined as the force in dynes exerted upon unit positive pole when placed at the point. Unit pole is a small magnet-pole of such a strength that it will repel a like pole with unit force (one dyne) when placed at unit distance (1 c. m.) from it. $48. REPRESENTATION OF THE ELECTROMAGNETIC FIELD BY LINES OF FORCE. Fig. 34 shows

a

1147

ds

of the current producing the field, and the length of
the conductor, if very short as compared with the
distance of the point from the con-
ductor, and inversely proportional
to the square of the distance of
the element of current from the
point.

In Fig. 35 let I be the intensity
of current flowing in the circle,
and P a point at the center; ds is

Fig. 35.

a small length of the conductor, and d its distance
from the point. Then the intensity of the magnetic
KIds
where K is a constant de-
field at P is f=
d2
pending upon the unit current used. If P is at
the center of the circle, the intensity at this point
will be

§ 50. UNIT CURRENT. The magnetic field about a conductor in which a current is flowing enables us to select a definite current to use as a unit or standard current.

If the conductor, Fig. 35, is a circle of unitradius, and the current flowing is such that each unit length of the conductor exerts unit force on unit magnet-pole placed at the center, the current is said to be of unit strength. If the force at the center due to unit length of the conductor is unity, that due to the whole circumference is 27. If the current is not unity, but I, and the radius, any radius, r, the electromagnetic intensity at the Hence K in the pre

center is f=

or

r

2arl 2πΙ r2 ceding article is unity, if the above unit of current is chosen.

851. GALVANOMETERS. The action of the electromagnetic field upon a magnet, or the mutual action between two

such fields, serve as a
basis for current meas-
urement.

N

Let ab (Fig. 36) be a circle bearing a current I; SN is a small magnet suspended at the center of the circle, and short as compared with the diameter of the circle. Imagine the the lines of force in the field about a circular ring plane of the circle as bearing a current. The plane of the paper is per- vertical and placed to coincide with the vertical pendicular to the plane of the ring, and passes plane through the magnet when there is no curthrough its center; the current passes down at Arent through the circle, or coil as it is usually and up at B.

Fig. 34.

$49. INTENSITY OF THE FIELD AT THE CENTER OF A CIRCLE. The intensity of the magnetic field at any point is directly proportional to the strength

called.

FIG. 36.

The magnet is suspended from the point O by a light fiber which is without torsion as nearly as possible, and hangs nearly north and south when

Let NS (Fig. 37) be before deflection and Hm

s'

N

FIG. 37.

N

the position of the needle
N'S' that after. The in-
tensity of the magnetic
field at any point due
to the earth's magnet-
ism is expressed in the
same manner as the
field due to a current;
i.e., by the force ex-
erted upon unit-pole
placed at the point.
Furthermore, it is only
that component of the
earth's field lying in
the horizontal plane
which can act to bring

the needle back to its initial position when de-
flected. If the pole-strength of the needle is not
unity, but m, then the force acting on each pole of
the needle is Hm when H is the horizontal compo-
nent of the magnetic field at the point before the
current flows in the coil. The force f, due to the cur-
rent, is practically perpendicular to the plane of the
coil in the small region occupied by the magnet,
27mI
and has for its value f= , where m is the pole-
strength of the needle, I the current, and r the
radius of the coil. Consider, first, the force Hm.
It is evident that only that component of it per-
pendicular to the needle can rotate it about the
point of suspension. The value of this compo-
nent is Hm cosa. Similarly, only the component of
the force f, which is perpendicular to the needle
is to be taken into account; this component is
2mI

r

r

cos ẞ; now, when the needle is acted upon by both forces, it will come to rest at the point where these components are equal; at this point we have 2πmI

r

cos Hm cos a, but the angle a is equal to 0, the deflection, and is the complement of 0; hence cosa sin and cos cos 0; hence we may write, 2mI -tan 0. If the cos 0=Hm sin@, or I = H coil have n turns close together, and n is not large, then the value of the current is I= H- -tan 0. It

r

r

2π

r 271

will be noticed that H is a force which can be meas-
ured, and is practically constant during any one
r is made up of
measurement and the quantity 2πn
constants; hence the current is proportional to
the tangent of the angle of deflection, and the in-
strument is called a tangent galvanometer. Such
an instrument is useful only to determine the ab-
solute value of the current; i. e., measure it directly
with the unit we have defined. The chief objec-
tion to its use is the variability of H. For practi-

cal purposes other forms are used, the principal types of which are,

1. Galvanometers similar to the tangent galvanometer, in that they have a fixed coil and movable magnet.

2. Those in which the magnet is fixed in posi tion, and the coil moves.

3. Electrodynamometers, instruments in which both fields are produced by the current, one of the coils being suspended or movable.

$ 52. SUSPENDED NEEDLE GALVANOMETERS. Fig. 38 shows the construction of an ordinary

Fig. 38.

mirror or reflecting galvanometer. There are two small magnets, N and S, connected by a light aluminium rod, to the center of which is attached the mirror, M. The magnets, connecting-rod and mirror constitute the system, or suspension as it is sometimes called. Each of the magnets is built up of several small bits of magnetized steel wire, one fourth to three eighths of an inch long. The poles of the two magnets are reversed, as shown in the figure, and the upper magnet is the stronger; hence the force tending to set the system north and south is the difference between the strength of the two needles. If the needles were of the same strength, the system would remain in any position. Such a system is called an astatic system. N'S' is a magnet placed on the case of the instrument, and above the system. (The case is not shown.) This magnet can be moved up and down, and rotated about a vertical axis. In the position shown, N'S' creates a magnetic field at the upper needle which is opposite to that of the earth at the same point; hence if N'S' is lowered to a certain point, it neutralizes the earth's field, and the system becomes as nearly

ELECTRICITY