the surface of the copper there is another electromotive force and a consequent sudden drop in potential bringing us again to the starting point at C.

If in the diagram AE is laid off equal to BC, then EZ is the difference of potential which represents the total electromotive force of the cell, for it is the difference between the two oppositely directed electromotive forces AZ and BC, and the fall of potential from A to B is equal to that from E to C. It is therefore clear that

the total fall of potential around the circuit, including that which takes place within the acid as well as that in the outside circuit, is equal to the total electromotive force of the cell.

It will be noticed that CD, the difference of potential between the two poles, is equal to the fall of potential in the external circuit, and is therefore less than the total electromotive force of the cell whenever there is any current flowing, for there is then a fall of potential within the cell itself from A to B.

642. Ohm's Law Applied to Part of a Circuit.—It has been seen (§605) that in any whole circuit

where E is the electromotive force in the circuit and R is a constant known as the resistance of the circuit. Similarly in any part of a circuit such as that between A and B (Fig. 361) the current I is proportional to the difference of potential between those points and may be written

Р

or Ir P

where P is the drop in potential between A and B and r is the resistance of that part of the circuit.

When there is no source of electromotive force, such as a battery cell, in a given portion of a circuit, the difference of potential between its ends in volts is equal to the product of the current in amperes by the resistance of that portion in ohms.

When an electromotive force E is included in any part of the circuit considered, the difference of potential between the ends of that part may be written

E

A

For Ir measures the drop in potential in the direction of the current, but when E is with the current it lifts the potential, as seen in the preceding paragraph. The sign of E is therefore opposite to that of Ir in that case.

643. Resistances in Series.-In a complete circuit made up of several conductors, having resistances r r1 r2, and including an electromotive force E, as in figure 361,

1

FIG. 361.

the successive steps in potential may be written thus

from A to B resistance= =r P=Ir

from B to C resistance=r, P1=Ir1

from C to A resistance=r2 P2=Ir2-E

1

2

0000

(5)

(6)

(7)

The sum P+P1+P, is evidently the total change of potential around the circuit from A around to A again, but this must be zero for it ends at the same potential as it began. Therefore adding 5, 6, and 7 we have

Ir+Ir,+Ir2-E=0

but this may be written

1

E

and by Ohm's Law

1

hence R=r+r1+r2; i.e., the resistance of several conductors connected in series is the sum of their separate resistances.

644. Combining Resistances in Parallel.—Let three conductors having resistances r1, T2, T3 be joined in parallel in a battery circuit as shown in figure 362. It was shown by Faraday that the sum of the currents in the branches is equal to the total current I before it divides,

r1

B

FIG. 362.

But the drop in potential from A to
B must be the same along either

branch. Letting P represent this drop we have

But if R is the effective resistance of the three branches combined

The reciprocal of the resistance of a conductor is called its conductance, hence the sum of the conductances of several conductors joined in parallel is the conductance of the combination.

If the three resistances above considered are equal the combination will have one-third the resistance of one alone. 645. Galvanometer Shunt.-It frequently happens that a current is to be measured by a galvanometer adapted to smaller currents. In such a case a wire S of suitable resistance, called a shunt (Fig. 363), may be connected across from one galvanometer terminal to the other. The current then divides between the shunt and the galvanometer. If the resistance of the galvanometer is just 9 times that of the shunt used, the current will divide in the ratio 1:9, so that one-tenth of it flows through the galvanometer and nine-tenths through the shunt.

000000

S

FIG. 363.

646. Resistance of Wires and Specific Resistance.-The resistance of a wire or of any conductor of uniform cross-section increases with its length and is inversely proportional to its cross section. The results of the last two articles show that this is so. For if the cross-section of a conductor is doubled it is equivalent to two of the original conductors side by side in parallel, and hence by $644 the resistance is one-half as much as before.

The resistance of a cylindrical conductor of a given substance one centimeter long and one square centimeter in cross section is called the specific resistance of that substance, or its resistivity.

When a wire of length l and cross-section s is made of a substance having resistivity p, its resistance R is given by the formula

ρι R

S

647. Resistivities. The curves given in figure 364 show the specific resistances of certain pure metals and alloys and also the variation of the resistances with temperature.

་

If the curves for the pure metals are produced it will be found that they intersect the base line in the region of the absolute zero (-273°). The experiments of Dewar and Fleming on the resistances of metals at low temperatures point to the conclusion that the electrical resistances of the metals approach zero as they approach the absolute zero of temperature.

The coefficients of increase in resistance increase somewhat with the temperature, as is shown by the upward curvatures of the lines, so that the proportional increase in resistance from 0° to 100° C. is in most cases larger than the expansion of gases. for that range. The following table shows some specific resistances at 0° C. in millionths of an ohm with the corresponding increase in resistance per ohm when the temperature is raised from 0° to 100° C. (From Dewar and Fleming.)

It will be noticed how these increments of resistance with temperature compare with the increase in volume of air as it expands from unit volume at 0° C. to 1.3667 at 100° C.

The resistance of carbon decreases with rise of temperature instead of increasing, so that the filament of an incandescent lamp may have only one-third as much resistance when hot as when cold.

The resistivities of alloys cannot in general be calculated from those of their constituents, but are often much greater than would be expected. The temperature coefficients of German silver, platinoid, and manganin are much less than those of pure metals; for this reason as well as for their large specific resistances these substances have been used extensively in making resistance coils.