The electro-motive force of a battery is a quantity such, that when it is multiplied by the strength of the current, the product is the energy produced by the battery in a given time (such as an hour). It is proportional to the number of cells.

Let M, then, denote the electro-motive force of one cell, n the number of cells; also, let E be the energy developed per lb. of zinc consumed, as stated in Article 394; then

Mny = Enz.

..(2.)

So that

M = .03744 E = for Daniell's battery, 41014;
for Smee's battery, 26013.

In these values of M, it is to be borne in mind, that the unit of force is one pound weight, and the unit of time an hour. In Pro

1 32.2

fessor Thomson's papers, the unit of force is of the weight of

a grain, and the unit of time a second.

The heat produced in a given time by a given current in the same circuit is proportional to the square of the strength of the That quantity of heat, then, is expressed by

current.

Where R is a quantity called the resistance of the circuit, being the heat developed in it in an unit of time by a current of unit strength.

The resistance of a circuit is the sum of the resistances of the various parts of which it consists, comprehending the plates and liquid of the cells, and the conductor which completes the circuit. The resistances of conductors made of a given substance are directly as their lengths and inversely as their sectional areas, or directly as the squares of their lengths and inversely as their weights. Let / be the length of any one conductor in a circuit, in feet, whether solid or liquid; w its weight, in lbs.; then

where e is a co-efficient depending on the material, and called the specific resistance of that material. Professor Thomson gives values of in which the unit of force is

1

32.2

of a grain weight, the unit

of mass, that of a grain, and the unit of time one second: to reduce these to values in which the unit of force is one pound weight, the unit of mass, that of a pound, and the unit of time one hour, they are to be multiplied by

3600

32.2 X 49000000

The following are examples of the results of that reduction for temperatures of 50° Fahrenheit:—

Copper wire,
Mercury,............................

from 176 to 128.

?= 10,356.

When the circuit produces no chemical decomposition out of the cells, no magnetic induction, and no mechanical or other external work, the whole of the energy developed by the chemical action in the cells takes the form of heat in different parts of the circuit. This fact is expressed by the following equation:

one of the consequences of which is the following:

..(6.)

.(7.)

or, the strength of the current is directly as the electro-motive force and inversely as the resistance of the circuit; being the celebrated principle known as "Ohm's Law."

Another consequence shows the rapidity of chemical action in a given circuit, viz. :—

M2 n2
ER

.(8.)

397. Efficiency of Electro-magnetic Engines.—Equations 1, 2, 3, 4, and 5 of Article 396 are applicable to all electro-chemical circuits whatsoever. Equations 6, 7, and 8 are applicable only to an ille battery, as it may be called, in which all the energy is spent in producing heat in the materials of the circuit.

An electric circuit may move mechanism against resistance, and so perform mechanical work, in three ways.

Ì. By the mutual attractions and repulsions of currents, or of parts of one current. Currents in the same direction attract, and currents in contrary directions repel each other. This method has been used in philosophical apparatus only.

II. By the attractions and repulsions between currents and permanent magnets. A magnet placed with its south pole towards the

eye of the spectator attracts currents whose direction is that of right-handed revolution relatively to its axis, and repels those whose direction is that of left-handed revolution.

III. By the attractions and repulsions between temporary and permanent magnets. A conductor coiled round a soft iron bar, when a current is sent through it, magnetizes the bar in that direction which makes it attract the current, according to the principle stated above under head II.; when the current ceases the magnetism ceases; when the current is reversed the direction of the magnetism is reversed. Opposite poles of magnets attract, similar poles repel each other; so that by periodically reversing the temporary magnetism of a soft iron bar, it may be made to take a reciprocating motion towards and from a permanent magnet.

IV. By the mutual attractions of temporary magnets.

The efficiency of the engine in all those cases is governed by two principles: 1. The performance of external work by an electric circuit produces a counteractive force, opposing the electromotive force, whose magnitude is equal to the external work performed in an unit of time divided by the strength of the current.

Let U be the external work performed in an hour by the engine. This gives rise to a certain counteractive force, which causes the current to be of less strength than that which the battery produces when idle. Let y be the strength of the current in the idle circuit, as given by equation 6 of Article 396; and the strength when the work U is performed per hour. Then the counteractive force is, U ÷ 7'

and the strength of current' is the same as if the electromotive force, instead of being M n, were M n that is to say,

U

This principle might be deduced as a consequence from the law of the conservation of energy; for multiplying equation 1 by 'R, and transposing, we find,

which expresses, that the useful work of the engine is the excess of the whole energy developed in the battery Mn, above the energy wasted in producing heat Ry'2.

2. A second principle is, that the attractions and repulsions produced by a given circuit and apparatus arranged in a given way are proportional to the square of the strength of the current (a law discovered by Mr. Joule); so that we may make

where A is a factor depending on the apparatus used. Hence equation 2 becomes

Hence are deduced the following expressions:-
For the rapidity of the chemical action,

.(5.)

From which it appears that the efficiency of the engine approxi

mates towards unity as the factor A increases; but at the same time the absolute work performed diminishes without limit.

398. Rotating Disc Engine. This machine, the simplest of all electro-magnetic engines, but hitherto used in the lecture room only, is the result of a disBcovery of Arago's. In fig. 174, N and S are the north and south poles of a permanent magnet, so shaped as to approach very near to the two faces of a copper

disc D, near its lower edge; that disc turns on an axis A, whose bearings (not shown in the figure) must rest on insulating supports. The lower edge of the disc between the poles of the magnet dips

into a cup M, containing mercury. C and Z are conducting wires, connecting respectively the axis of the disc and the mercury in the cup with the electrodes of a galvanic battery. By the arrangement shown in the figure, an electric current is made to pass from the positive electrode to the axis of the disc; thence through the disc to the mercury, and thence to the negative electrode of the battery. The action of the poles of the magnet on the disc is shown by the diagram, fig. 175. S is the magnet, with the south pole exposed to view; the arrow head on the circle shows the direction of the revolving current to which the magnet is equivalent. A B and A E are two portions of the current in the disc, from the axis to the mercury. According to the principle that currents in the same direction attract each other, and currents in opposite directions repel each other, the magnet attracts A B and repels A E, and so keeps up a continuous rotation of the disc in the direction BE. The direction of rotation can be reversed by reversing the current; that is, by connecting A with Z and M with C.

399. Rotating Bar Engine.—This machine, the invention of Mr. Webster, is shown in fig. 176. NS, NS,

are two semicircular permanent magnets fixed within a frame of brass or other diamagnetic material, and having two gaps between their pairs of contiguous poles, which are similar, as indicated by the letters. M is a mercury-cup of non-conducting material on a pedestal; it is divided into two parts by a diametral non-conducting partition, in the plane of the permanent magnets, as shown in fig. 177. In the centre of the cup stands a

M

Fig. 176.

pivot, on which rotates the horizontal soft iron bar A B; the two arms of that bar are encircled by the two portions of a long coil of conducting wire. The two ends of that coil dip into the two halves of the mercury cup, which halves are connected with the electrodes of a battery by the wires C Z. The ends of the soft iron bar pass between the poles of the permanent magnet, so as to come very near them, but not to touch them. To produce rotation in the direction indicated by the arrow, round the bar A B is so arranged that when the end A is moving from SS to N N, and the end B from N N to SS, A is a south

pole, and B a north pole.

is

{

Fig. 177.

the coil

(repelled attracted {by S S, and instant that the ends of the bar

Then
B