758. Conjugate Branches.-Wheatstone's bridge may be otherwise described as consisting of six branches connecting four points, two and two, in every possible way, the four points being A, B, C, J in Fig. 488. A battery is inserted in the branch which connects any two of these points, A and B, and a galvanometer is inserted in the branch which connects the other two, C and J. These two branches may be called opposite, and in like manner A C is opposite to B J, and BC to AJ. The condition of no current going through the galvanometer is expressed in § 756 as a proportion. Multiplying extremes and means, and writing the names of the branches for the resistances in them, the condition is

AC. BJ BC. AJ,

where each member of the equation is the product of the resistances in opposite branches. When this condition is fulfilled, the remaining pair of opposite branches A B and CJ are conjugate, that is to say, a battery in one produces no current in the other. The symmetry of the relations shows that the battery may change places with the galvanometer.

759. Conjugate Branches when there are Several Batteries.-When there are batteries in more branches than one, the current in any branch will be the algebraical sum of the currents due to the several batteries considered separately. Hence when there is equality between the two products of opposite resistances, as in last section, the current in either of the two remaining branches will be independent of the electro-motive force of the battery in the other; and these two branches are still said to be conjugate. In estimating the resistance of any branch which contains a battery, the resistance of the battery must of course be included.

Thus far we have not discussed the effect of change of resistance in one of two conjugate branches. The introduction of additional resistance into any branch can affect the current in the rest only by altering the difference of potentials between the ends of this branch; and the same remark applies to the introduction of a source of electro-motive force into any branch. Two changes, one of resistance, and the other of electro-motive force, in a branch, will have the same effect on the rest of the circuit, if they have the same effect on the difference of potentials of the ends of this branch. Hence if the current in one of two branches be independent of the electromotive force in the other, it must also be independent of the resistance in the other.

CONJUGATE BRANCHES.

739

As this reasoning may appear doubtful to some of our readers, we subjoin a formal investigation leading to the same result.

B

60. Investigation of Condition of Conjugateness.-Let A, B, C, J (Fig. 488 or Fig. 491), be four points connected, two and two, by six branches. Let the resistance in the branch connecting A and B be denoted by A B or BA, and the electro-motive force in it (positive if tending from A to B) by ab. Let the current in this branch (positive if from A to B) be denoted by y, and the currents in the branches BC, CA, by a, 6. Then the current in the branch AJ (positive if from A to J) will be ẞ-y; for this current together with y carries off from A the supply brought by 6. Similarly, the currents in BJ, CJ, will be y − a, a − ß. Then, since the sum of the falls of potential in travelling round a circuit must equal the sum of the rises, we have, for the circuit J BC, the equation

B

y-a

a

Fig. 491.--Theory of Conjugate
Branches.

a.BC+(a - ß СJ+(a− y) JB=bc+cj+jb.

Similarly, for the circuits JC A, JA B, we have

B. CA + (b − y) A J + 'ß − a) JC = ca+aj+jc

C

These three equations are sufficient to determine the three currents a, ß, y, in terms of the electro-motive forces and resistances. Multiplying the equations in order, by 1, l,m (l and m being multipliers to be afterwards determined), and adding; the coefficients of a, ß, Y, will be

and the second member of the equation will be

dc+l.ca+m. ab + (l − m) aj + (m − 1) bj + (1 − 1) cj.

To find the value of a, we must equate the coefficients of ẞ and γ to zero, and then divide the second member by the coefficient of a. The electro-motive force aj appears only in the term (l–m) aj, and the resistance AJ only in the terms (l–m) AJ and (m-1) AJ. Hence the equality of l to m is the condition alike of the disappearance of

aj and of AJ. Putting lm, and equating the coefficients of ẞ and y to zero, we have

whence

CJ

BJ

1=

=

CA+CJAB+BJ'

CJ.AB=BJ.CA,

which is, accordingly, the condition of conjugateness. That is to say, if the product of one pair of opposite resistances be equal to the product of another pair, the remaining pair of branches will be so related that the current in each is independent of the electro-motive force and resistance in the other.

761. Thomson's Method of Measuring the Resistance of a Galvanometer.—The resistance of a galvanometer can be measured without the use of another galvanometer, by the following method, due to Sir Wm. Thomson.

In a system of six branches joining four points, let a battery and a contact key respectively be in one pair of opposite branches. Then, if the products of the resistances in opposite branches be equal for the four remaining pairs, we know by § 758 that no current will pass through the branch containing the key, and hence making or breaking contact with the key will be nugatory; hence the galvanometer will not have its deflection altered by making or breaking contact with the key. The experiment is to be conducted by altering the resistance in one of the branches until the key has no effect on the galvanometer. The resistance of the galvanometer is then calculated from the equality of the products of opposite pairs.

This method was suggested by the following.

762. Mance's Method of Finding the Resistance of a Battery.In this method a galvanometer and a contact key are in a pair of opposite branches, and the battery is in one of the four remaining branches, while the other three contain known resistances. The observation is made by varying one of these resistances till the galvanometer is not affected by the key. The branches containing the key and the galvanometer are then conjugate (and the resistance of the battery can be calculated), if the putting down of the, key does not alter the electro-motive force of the battery. This condition is seldom fulfilled.

In this method, as well as in that described in the preceding section, the galvanometer does not stand at zero, but shows a steady deflection, which is unaltered by opening or closing the branch containing the key.

CHAPTER LVII.

RELATIONS BETWEEN ELECTRICITY AND HEAT.

763. Heating of Wires.-The heating of a wire by the passage of a current may conveniently be exhibited by the aid of the apparatus represented in Fig. 492. Two uprights mounted on a stand are furnished, at different

is done with a battery of suitable power, the wire is first seen to droop in consequence of expansion, then to redden, and finally to melt, becoming inflamed if the metal is sufficiently combustible.

If a file is attached to one of the terminals of a battery, and the other terminal is drawn along the file, a rapid succession of sparks will be obtained; and if the battery be sufficiently powerful, globules of incandescent metal will be scattered about with brilliant effect.

764. Joule's Law. The energy of a current is equal to the product of the quantity of electricity that passes and the electro-motive force that drives it. As the numerical measure of a current is the quantity of electricity which passes in unit time, it follows that the energy of a current C lasting for a time t, is ECt, E denoting the electro-motive

force. But again, by Ohm's law, E is equal to the product of the current C and the whole resistance R. The expression for the energy therefore becomes

and this energy is all transformed into heat in the circuit, subject to a small correction for the Peltier and Thomson effects which will be described in a later section (§ 771). It has accordingly been found, first by Joule, and afterwards by Lenz, Becquerel, and others, that the formula C2Rt represents the quantity of heat generated by a current under ordinary circumstances. The experiments have usually been conducted by passing a current through a spiral of wire immersed in water or alcohol, and observing the elevation of temperature of the liquid.

• 2,

This law of Joule's, like that of Ohm, may be applied to any part of a circuit, as well as to the circuit considered as a whole; that is to say, if the circuit consists of parts whose resistances are r1, r1⁄2 the quantities of heat generated in them are respectively C2 r1t, C2r, t, . . ., and are therefore proportional to the resistances 71, 72 r2t, of the respective parts, since C and t are necessarily the same for all.

765. Relation of Heat in Circuit to Chemical Action in Battery.The energy of a current, and consequently the heat developed in the circuit, is the exact equivalent of the potential energy of chemical affinity which runs down in the cells of the battery. This fact, first verified approximately by Joule, has been more accurately confirmed by the experiments of Favre, who introduced into the muffle of his mercurial calorimeter, already described and figured in § 509, a small voltaic cell with its poles connected by a fine wire. He found that the consumption of 33 grammes of zinc in the cell corresponded to a generation of heat amounting to 18,796 gramme-degrees. But the chemical action in the cell is complex. The 33 grammes of zinc unite with 8 grammes of oxygen, and in so doing generate 42,451 grammedegrees. The combination of these 41 grammes of oxide of zinc with 40 grammes of sulphuric acid, produces 10,456 gramme-degrees, making in all 52,907. But an equivalent of water undergoes decomposition, and this absorbs 34,463, which must be subtracted from the above sum, leaving 18,444 gramme-degrees as the balance of heat generated in the whole complex action. The heat actually observed in the experiment agrees almost precisely with this calculated amount. (Compare § 734.)