On account of its small resistance, the tangent compass is well adapted for currents of low tension, but in which a considerable quantity of

Fig. 538.

electricity is set in motion. For currents which can overcome great resistance, but have only a small quantity of electricity, the multiplier is best fitted.

715. Ohm's law. For a knowledge of the conditions which regulate the action of the voltaic current, science is indebted to the late Professor Ohm.

His results were at first deduced from theoretical considerations, but by his own researches as well as by those of Fechner, Pouillet, Daniell, De la Rive, Wheatstone, and others, they have received the fullest confirmation, and their great theoretical and practical importance has been fully established.

i. The force or cause by which electricity is set in motion in the voltaic circuit is called the electromotive force. The quantity of electricity which in any unit of time flows through a section of the circuit is called the intensity of the current. Ohm found that this intensity is the same in all parts of one and the same circuit however heterogeneous they were; and also that it is proportional to the electromotive force.

It has further been found that when the same current is passed respectively through a short and through a long wire of the same material, its action on the magnetic needle is less in the latter case than in the former. Ohm accordingly supposed that in the latter case there

was a greater resistance to the passage of the current than in the former; and he proved that 'the resistance is inversely proportional to the intensity of the current.'

On these principles Ohm founded the celebrated law which bears his name, that

The intensity of the current is equal to the electromotive force divided by the resistance.

Which is expressed by the simple formula

where I is the intensity of the current, E the electromotive force, and R the resistance.

ii. The resistance of a conductor depends on three properties; its comductivity, which is a constant, determined for each conductor; its section ; and its length. The resistance is obviously inversely proportional to the conductivity, that is, the less the conducting power the greater the resistance. This has been experimentally shown, and it has also been proved that the resistance is inversely as the section and directly as the length of a conductor. If then is the conductivity, w the section, and a the length of a conductor, we have

к

that is, the intensity of a current is inversely proportional to the length of the conductor and directly proportional to its section and conductivity.

iii. In a voltaic battery composed of different elements, the intensity of the current is equal to the sum of the electromotive forces of all the elements divided by the sum of the resistances. Usually, however, s battery is composed of elements of the same kind, each having the same electromotive force and the same resistance.

In an ordinary element there are essentially two resistances to be considered: 1. That offered by the liquid conductor between the two plates, which is frequently called the internal or essential resistance; and 2, That offered by the interpolar conductor which connects the two plates outside the liquid; this conductor may consist either wholly of metal, or partly of metal and partly of liquids to be decomposed; it is the external or non-essential resistance. Calling the former R and the latter T, Ohm's formula becomes

iv. If any number, n, of similar elements are joined together, there is n times the electromotive force, but at the same time n times the

internal resistance, and the formula becomes

nE nR+r

If the resistance

in the interpolar, r, is very small, which is the case, for instance, when it is a short thick copper wire, it may be neglected in comparison with the internal resistance, and then we have

=

I — "E_E
nk R

that is, a battery consisting of several elements produces in this case no greater effect than a single element. If, however, the external resistancer is very great, which is the case where the current has to pass through a long thin wire, or through a liquid, the intensity is within certain limits very nearly proportional to the number of elements.

v. If the plates of an element be made m times as large, there is no increase in the electromotive force, for this depends on the nature of the metals and of the liquid, but the resistance is m times as small, for the section is m times larger, the expression becomes then

Hence, an increase in the size of the plate, or, what is the same thing, a decrease in the internal resistance, does not increase the intensity to an indefinite extent; for ultimately the resistance of the element R vanishes in comparison with the resistance r, and the intensity always approximates to the value I=E.

r

In a thermoelectric pile, which consists of very short metallic conductors, the internal resistance R is very small. It may hence be neglected, and Ohm's formula becomes

E
I=

that is, the intensity is inversely as the length of the connecting wire. vi. Ohm's law enables us to arrange a battery so as to obtain the greatest effect in any given case. For instance, with a battery of six Bunsen's elements there are the following four ways of arranging them: 1. In a single series (fig. 539), in which the zinc Z of one element is united with the copper C of the second; the zinc of this with the copper of the third, and so on; 2. Arranged in a system of three double elements, each element being formed by joining two of the former (fig. 540); 3. In a system of two elements, each of which consists of three of the original elements joined, so as to form one of triple the surface (fig. 541); 4. Lastly, of one large element, all the zincs and all the coppers being joined, so as to form a larger pair of six times the surface (fig. 542).

With a series of twelve elements there may be six different combinations, and so on for a larger number.

Now let us suppose that in the particular case of a battery of six elements the internal resistance R of each element is 3 and the external resistance r= 12. Then in the first case where there are six elements we have the value

If they were united so as to form three elements, each of double the surface, as in the second case (fig. 540), the electromotive force then would be, the electromotive force E in each element; there would als be a resistance R in each element, but this would only be half as great, for the section of the plate is now double; hence the intensity in this case would be

hence this change would lessen the intensity.

If with the same elements the resistance in the connecting wire were only r=2, we should have the values in the two cases respectively

The result in this case is, therefore, more favourable. If the resistancer were 9 the intensity would be the same in both cases. Hence, by altering the size of the plates or the arrangement favourable or unfavourable results are obtained according to the relation between R and r.

In any given combination the maximum effect is obtained when the total resistance in the elements is equal to the resistance of the interpolar. Suppose that in a given case n elements are arranged so as to form a battery of s couples, each consisting of t cells, n = st. Denoting the resistance of a single element by r, the total resistance is

r's

t

ing to the above law the maximum effect is obtained when

Now accord

= 1, where

r82

But t=" hence = 1, or

n

If in a given case we have 8 elements, each offering a resistance 15 and an interpolar with the resistance 40, we get s= 4.3. But this is an impossible arrangement, for it is not a whole number, and the nearest whole number must be taken. This is 4, and it will be found on making a calculation analogous to that above, that when arranged so as to form 4 elements, each of double surface, the greatest effect is obtained.