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sufficiently explained by the diagrams and by what has preceded in § 5. (Read § 5 again.) We will take several cases.

I. Two equal branches.-Here we have for the equivalent resistance

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Hence, if R be the resistance in the rest of the circuit and in the battery together, and if E be the E.M.F.

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III. Two unequal branches.-Here we have

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IV. N unequal branches.—In this most general case we have

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Example in divided circuits.—'Two wires, ADB and AEC B, of uniform section, &c., and of equal resistance, connect the points A and B. A third wire, AFC, of equal resistance, connects the point A with the middle point C of one wire. When a current flows

from A to B, find what fraction of it passes through the wire A F C.'

In the figure we represent the arrangement intended, and we have called the resistances of ADB and AFC each I, while those of AEC and of CB are

FIG. iv.

each We will first find the equivalent resistance of the paths from A through C to B, and so find the fraction of the current that passes through ADB; and next we will find what fraction of the remainder takes the route

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AFC. Now the resistance of ADB = 1, or

And the equivalent resistance

6

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routes D and C are in the ratio of 6: 5 respectively.

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current passes by D, and passes from A through C.

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II

will be distributed between the branches AEC and AFC

in the inverse ratio of their resistances; that is, in the direct ratio of 2: I. I 6

Hence of will pass by the route A FC; or the answer required is

3 II

2

II

§ 10. 'Shunts. The case of two unequal branches has a very important application. It is often necessary (i.) to protect a sensitive galvanometer against a strong current by allowing only a part of the current to pass through the galvanometer, and yet (ii.) to measure the current. This is done by connecting the terminals of the galvanometer by a wire of greater or less resistance, and thus leaving a greater or smaller fraction of the current to pass through the galvanometer. If this fraction be known, and its magnitude be measured, we can easily calculate the total current. We shall see, however, that the introduction of this 'short cut,' or 'shunt,' has the effect of increasing the total current by decreasing the total resistance; and hence the fraction measured is a certain fraction of the new total current, not of the original current. Let R be the resistance of the battery and rest of the circuit combined; let G be the resistance of the galvanometer which here takes the place of ri in § 9, case (III.); let S be the resistance of the shunt which here takes the place of r; and let R' be the equivalent resistance of G and S. This R' is of course less than either G or S. Let C be the total current; Co the current passing through the galvanometer; and C, the current through the shunt. the original current passing before the shunt was used. have.

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Let C, be

Then we

We will now proceed to consider several points.

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(ii.) If G be negligible as compared with R, then will C = C„, whatever shunt we use. This follows from the above formulæ, since we there may neglect G, and therefore a fortiori may neglect R', which is less than G.

(iii.) If G, and G, be so great that R may be neglected, then we

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find C = n C ̧; and as a consequence we should have passing through the galvanometer a current = C = 11. n C1 = C。; thus

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exposing the galvanometer to a current as strong as the whole original current.

By having in the circuit an adjustable resistance, we can introduce a compensating resistance, and so maintain C equal to C..

Note. To keep C = Co, we must have the total resistance of circuit the same; GS or the compensating resistance must be such that G+R=

+R+r;

G+S

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SII. Fall of Potential through the Circuit.-The following view of a circuit, a view indicated by the figure to Chapter XI. § 7, is convenient, and, though not accurate in detail, it can for general purposes be substituted for a more accurate view, and can lead to no error in such considerations as follow.

In the figure referred to, it is supposed that in the cell the E.M.F. occurs as an abrupt rise in potential; in fact, that the energy of the cell is spent in, as it were, pumping the electricity straight up from the lower to the higher level. Thus, the line CZ

represents, in the graphic method, the total E.M.F. E of the circuit. The potential falls round the circuit from the higher to the lower level again; the total fall being E.

Ohm's law, as expressed by the formulæ given in § 2, means that this fall proceeds proportionally to the resistance. Hence, if R' be the resistance of any portion of the circuit, and if R be the total resistance of the whole circuit, then the fall of potential between the beginning and end of R' will be E. We deduce the following.

R'
R

(a) In the case of negligible external resistance the whole fall of potential will (approx.) take place in the cell itself. We find (approx.) no AV between the beginning and the end of the external circuit; or the poles of the battery will have (approx.) the same potential.

(b) In the case of negligible internal resistance the whole fall of potential will (approx.) take place externally to the cell. We shall find (approx.) between the beginning and the end of the external circuit a AV equal to the whole E.M.F. of the battery ; or the poles of the battery will have (approx.) the same AV as if the circuit were broken.

(c) Where internal resistance equals external, we shall have the fall of potential equally divided. The poles of the battery will exhibit a AV equal to

I

E. 2

(d) Where there is a multiple arc between two points A and B, and we wish to find the AV E between A and B, we have merely to calculate the equivalent resistance R'. We then have

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Note.—All the above cases are readily exhibited graphically (see § 7).

§ 12. Kirchhoff's Two Laws. By a careful application of Ohm's law, and of the principle that when the V of a point is constant there must be as much electricity flowing away from it as flows to it in each second, it is possible to investigate the distribution of current and potential in very complicated cases; cases where we have any number of cells connected by a net-work of conductors in any way whatever.

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