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observations; the battery should therefore be one of constant

E. M. F. and the observations should be taken in quick succession

so as to avoid change in the current due to the increase of the

resistance of the circuit caused by the heating effect of the current.

311. Resistance Coils. The known resistances used as de-

scribed in the preceding paragraph are usually in the form of coils.

These resistance coils, especially those used as standards of resist-

ance, are made with great care and accuracy and embody many

refinements. They range from .001 of an ohm to 10,000 ohms.

A section of one is shown in Fig. 129. From the ebonite lid there

extends downwards a hollow metal cylinder which has an insulat-

ing covering of shellac-coated silk. Around this cylinder is wrapped

the coil proper which is of silk-insulated manganin wire (Par. 289).

For reasons which are explained later (Par. 315), the wire is

doubled upon itself at its middle point and the winding is begun

at this loop. The ends of the coil are attached to heavy copper

VOLTAIC ELECTRICITY.

235

terminals bent downward as shown. The coil is connected up in

the circuit by inserting these turned-down ends into mercury cups

which in turn are connected to the lead wires. The whole is pro-

Fig. 129.

tected by a brass case which is perforated by many small openings.

The object of the interior metal cylinder is to conduct away heat

developed in the wire and at the same time to afford a large sur-

face for radiation. The object of the openings is to allow the

enclosed coil to cool off more rapidly and also to permit the tem-

perature to be kept down by submerging the entire coil in oil.

The plug in the center of the lid is to permit the insertion of a

thermometer for reading the temperature of the coil so that the

proper correction for temperature may be applied.

312. Drop in Divided Circuit. The usual way of measuring

ordinary resistances is by means of the Wheatstone bridge, a piece

of apparatus whose principle will be understood from the following

explanation. Consider a divided circuit of two branches and let

A (Fig. 130) be the point of high potential. The current at B

divides into two parts inversely proportional to the resistances of

the two branches, i. e., the greater part goes along the branch of

least resistance, the lesser part along the branch of greater resist-

ance. There is a continuous drop of potential along each branch

of the circuit from B to D, in other words, the drop of potential

over the two branches is exactly the same. Suppose following the

right hand branch we reach a point M at which we have passed

over one-half of the resistance in that branch; the difference of po-

tential between M and D is only one-half of that between B and D.

236

ELEMENTS OF ELECTRICITY.

Fig. 130.

Similarly, following the left hand branch and reaching a point

N at which we have passed over one-half of the resistance in that

A branch, the difference of potential between

I JV and D is only one-half of that between B

and D. Hence, the points M and N are at

the same potential. This can be shown by

connecting between these points a sensitive

galvanometer G. (Galvanometers are de-

scribed in Chapter 30. For the present it is

sufficient for us to know that a galvanometer

(more strictly a galvanoscope) is an instru-

ment which indicates by the movement of

its needle that a current is flowing in the

circuit of which it forms a part, and by the

direction of the motion of the needle indi-

cates the direction of the current.) Should

there be a difference of potential between M

and N, a current would be produced and

would be revealed by a deflection of the gal-

vanometer needle, but the needle will be found to remain at rest.

The foregoing illustration is based on the supposition that the

resistance of BM and of B N are each one-half of the resistance of

the respective branches, but the prin-

ciple is equally true for I/nth, that is,

if the resistance of BM be I /nth of

that of the right hand branch and the

resistance of B N be I/nth of that of

the left hand branch, the points M and

N will be at the same potential and

there will be no flow of current between

them if they be connected through a

galvanometer.

313. Principle of the Wheatstone

Bridge. Let us now consider a divided

circuit of two branches (Fig. 131),

each branch subdivided into two parts

as shown, and suppose that in the left

hand branch we know the resistance

Fig. 131.

of A and of R and further can vary that of R at pleasure, and

that in the right hand branch we know the resistance of the

VOLTAIC ELECTRICITY 237

portion B but do not know that of the remainder X and wish to

determine it. Of the total resistance of the right hand branch,

X is some definite fraction, say I /nth. Since R may be varied at

pleasure, it can be adjusted so that it is 1/wth of the total resist-

ance in the left hand branch. When such a state of affairs is

reached, the points M and N will, from what has been shown

above, be at the same potential and the galvanometer connected

between M and N will reveal no current. The system is now

said to be "balanced."

Since X is I/rath of the total resistance in the right hand branch,

B is n 1/wths, and since R has been made 1/wth of that in the

left hand branch, A is n 1/wths.

Hence A : B : : R : X

Whence X

or, when the system has been

brought to a balance, the resistance in X is equal to the product of

the resistances in the adjacent arms divided by that of the opposite arm.

314. A Second Demonstration. The same thing can be readily

shown by applying the principle of drop directly. Call the current

in the left hand branch I', that in the right hand branch I", and

the resistance in the four arms A, B, R, and X, respectively. The

drop from S to N is equal to the current times the resistance or

FA; that from S to M is equal to I"B. But M and N being at

the same potential these drops are equal. Similarly, the drop

from N to T, or I'R, is equal to the drop from M to T, or I"X.

We then have the two equations,

(I) FA = T'E

(II) I'R =I"X

Dividing (II) by (I) and striking out common factors

R X

A = B

Whence as above

BR

The foregoing is the principle upon which the Wheatstone

bridge is constructed.

238

ELEMENT^ OF ELECTRICITY.

"DT> T>

The expression X = r- can be written X = -j R, whence it is

seen that if B and A be so selected that B/A is some multiple or

submultiple of ten, calculations will be simplified since all that

will then be necessary will be to point off decimal places or add

zeros to the value of the known resistance R.

315. Arrangement of Resistances. In the actual apparatus

the resistance in the arms A, B, and R is usually varied by re-

moving or changing the position of certain plugs. For example,

Fig. 132.

the arm A, a portion of which is represented in Fig. 132, consists

of a heavy brass bar DE secured to the ebonite plate FF and cut

entirely through at regular intervals by tapering openings into

which fit the corresponding ebonite-handled brass plugs A, B, C.

The separate sections into which the bar is divided are connected

beneath the plate FF by the resistance coils G, H, K. These are

wound as described in Par. 311 so as to avoid self-induction. For

the present we may explain this by stating that when a circuit

through a coil of wire is completed there is produced through

induction an opposing E. M. F. which causes the current to lag

and prevents it from rising to its full strength at once. When a

coil is made by winding it from a loop at its middle point, each

turn of the coil carrying a current is paralleled by an equal turn

in which the current flows in the opposite direction and the

inductive effects of the two turns exactly neutralize each other.

These coils have a resistance of 1 ohm, 10 ohms, 100 ohms, etc.,

VOLTAIC ELECTRICITY.

239

and therefore bear to each other the ratio of 1 : 10 : 100, etc.

With the plugs in position the current passes from DtoE through

the bar and coils, the combined resistance of which is so small as

to be negligible. With the plug B removed, the current must

follow the path D M H N E, that is, the resistance of the coil

H has been introduced into the circuit.

In the arm R the arrangement is similar but there is a much

greater number of coils whose resistances are in ohms 1, 2, 3, 4,

10, 20, 30, 40, 100, 200, 300, 400, 1000, 2000, 3000, 4000, etc., thus

enabling any combination from 1 to 11110 to be obtained. This

arm is usually called the "rheostat" and is consequently desig-

nated in diagrams by letter JR.

316. Evolution in Form. The theory of the Wheatstone

bridge is best explained as above from a diagrammatic diamond-

(5)

shaped figure as in (1), Fig. 133. The commercial form of this

apparatus bears no superficial resemblance to the figure but has

been evolved directly from it as the following will show.

1st step. The galvanometer need not be placed in the diamond

but may be connected outside as shown in (2).

2d step. A and B need not make an angle with each other

but may be flattened down as shown in (3).

3d step. R being the arm which carries the greatest number

of resistance coils should, relatively to A and B, be elongated as

shown in (4).

4th step. Finally, for the sake of compactness, the arm R

may be folded back upon itself as shown in (5).

Other minor changes consist in the arrangement of the terminals

to facilitate connections, and in the insertion in the battery and

240

ELEMENTS OF ELECTRICITY.

galvanometer circuits of keys permanently attached to the instru-

ment. Sometimes a galvanometer is included in the case. The

final result is an instrument of which Fig. 134 shows a form made

by the Leeds & Northrup Co.

Fig. 134.

The various circuits between the keys and other parts of the

bridge are inside the case but are usually indicated by white lines

marked on the cover.

317. Connections for a Measurement. Whatever be the form

of the bridge it is well- to bear in mind the following: first, the

current enters (or leaves) at the junction of A and B and leaves

(or enters) at the junction of R and X', second, the galvanometer

is connected between the junction of A and R and that of B and X.

(It should however be observed that it may readily be shown that

the battery and the galvanometer may be interchanged, the

resistance of their respective leads altered at will, and the E. M. F.

of the battery varied, all this without affecting the balance.)

Finally, in the factor by which R is to be multiplied, the resistance

of B, the arm connected to X, is the numerator and that of A,

the arm opposite X, is the denominator.

318. Operation of Measurement. To measure the resistance,

say of a wire X, the apparatus is brushed free from dust, and

plugs brightened, being especially careful to remove all grease

VOLTAIC ELECTRICITY.

241

or oil so as to insure perfect contacts. Connections are then made

as shown in Fig. 135. A plug is removed from coil of the same

resistance in both A and B, their ratio therefore being unity.

Various plugs are then removed from R until with both battery

Fig. 135.

and galvanometer keys closed the apparatus is as nearly balanced

as possible. At this point the sum of the unplugged resistances

in -R is as near the unknown resistance X as it is possible to get

with the ratio of unity in B/A.

319. Bracketing. The plugs in R are not removed at hap-

hazard but preferably the resistance should be arrived at by a

system of "bracketing." For example, the first plug to be removed

should be selected so that the resistance thrown in is certainly

greater or less than the one to be measured. Suppose it to be less.

The battery key K is closed and then the galvanometer key H.

Suppose the galvanometer needle to be deflected to the left.

Replace the plug and remove a second one so as to throw in a

resistance certainly greater than that to be measured. Upon

closing the keys, if the needle is now deflected to the right the

unknown resistance lies between the two. Replace the plug and

remove a third which will throw in a resistance as near half way

of the interval between the first two as possible. If upon closing

the keys the needle is deflected to the left, the third resistance is

too small, if to the right it is too great. Proceed in this way

keeping the unknown resistance between limits and halving the

interval at each successive attempt.

'242 ELEMENTS OF ELECTRICITY.

With a little experience the bracketing can be materially

shortened by observing the amount of swing produced in the

needle by the trial resistances. This decreases rapidly as the

correct resistance is approached and indicates which of two is the

nearer.

320. Order of Closing Keys. The order in which the battery

and galvanometer keys are closed is not a matter of indifference.

It is essential that the battery key be closed first. For consider

Fig. 135. The coils in R are wound so as to avoid self-induction

but this object may not be completely attained and with a number

of coils unplugged the inductance may not be negligible. Again,

if the resistance X be that of a coil, especially if it be wrapped

around an iron core, its self-induction will be large. Finally, if X

be a cable it may have considerable capacity as a condenser. In

any of these cases, when the battery key K is closed the current

will not rise at once to its full strength in the branch affected.

Suppose the bridge to be balanced accurately and the galvanom-

eter key closed first; when K is closed the current in one branch

or the other not rising at once to its full strength, M and N will

be momentarily at different potentials and there will be an in-

stantaneous rush of current through G causing a deflection of the

needle and incorrectly indicating a lack of balance. On the other

hand, if K be closed first there will still be this retardation but its

effect will disappear in a fraction of a second, M and N will reach

the same potential and when H is closed there will be no deflection

of the galvanometer needle.

There may be used 'ajspecial key which by making successive

contacts as it is pressed down will insure the proper sequence of

closing.

To avoid violent swings of the needle, the galvanometer key at

first should be given a mere tap.

321. Proper Ratio to Use. The first determination gives the

resistance of X to the nearest unit or ohm. If it be desired to

measure it to the first, second, or third place of decimals, the plugs

in A and B must be so adjusted that the ratio B/A is .1, .01, or

.001 and the corresponding decimal places are pointed off in the

final reading of R. If the resistance to be measured be large, the

ratio B/A must be 10, or 100, or 1000.

It will be noted that some of the ratios can be obtained by several

VOLTAIC ELECTRICITY.

243

different combinations, thus }, %%%, !#> all gi ye the ratio unity.

It can be shown that other things being equal, the greatest sensi-

bility is obtained when the resistances in the four arms of the

bridge are as nearly equal as possible. For example, if the resist-

ance to be measured is about 100 ohms and this is to be measured

to the nearest unit, the ratio should be {%%, or if to the nearest

tenth then T VA-

The instrumental sensibility depends directly upon the sensi-

tiveness of the galvanometer, or its ability to indicate very minute

currents when the bridge is nearly balanced.

322. Bridge with Reversible Ratios. There is sometimes used,

instead of the bridge described above, a variation by which a coil

is saved in each of the arms A and B, making six instead of eight,

and yet the same ratios are preserved.

(2)

Fig. 136.

Its arrangement is shown in (2) in Fig. 136 and is as if the arms

A and B of (1) had been separated at S and each rotated outward

from the center. These outer ends (2) are then connected by a

heavy wire with S which must now be regarded as the junction

of A and B and, according to Par. 317, is the point at which the

battery current enters. The inner ends of A and B are connected

to the R and the X arms by movable plugs. With the plugs in

the positions shown by the small circles in (2), A is connected to

R and B to X. If these plugs be shifted to the positions marked

by the crosses, A becomes connected to X and B to R, in other

words (see Par. 317), A and B interchange.

244

ELEMENTS OF ELECTRICITY.

The A arm contains the coils 1, 10, 100, the B arm 10, 100,

1000. The smallest B/A ratio obtainable with the plugs in the

first position is T W or .1. If it be desired to use a smaller one, shift

the two plugs, A becomes B and B becomes A, and the ratios

T y and joVo become available.

323. The Dial Bridge. In the bridges described above, resist-

ances are thrown in the rheostat by removing plugs. There are

other forms, such as the dial bridge and the decade bridge, in

which resistances are introduced by inserting plugs. The connec-

tions of a dial bridge are shown diagrammatically in Fig. 137.

coooooioooocp

Fig. 137.

The A and B arms are like those of the ordinary bridge but the

rheostat is composed of dials, usually four, which are marked

units, tens, hundreds, and thousands, respectively. Each dial

consists of a heavy center piece of brass surrounded by ten key-

stone shaped pieces, these being numbered 0, 1, 2, etc., to 9.

Between the successive keystone pieces, except numbers 9 and 0,

are resistance coils, those at each dial being all of the same

resistance. Thus, at the unit dial each coil has a resistance of one

ohm; at the ten dial each has a resistance of ten ohms, and so on.

The current entering by A goes to the center of the first dial, then

through the plug to the corresponding keystone piece, thence

through the coils in series to the keystone and thence to the

second dial, etc. The diagram represents a resistance plugged in

of 5135 ohms.

This form is more expensive than the first but has a number of

advantages, among them, the smaller number of plugs to be

handled and consequent smaller number of contacts (four as com-

pared to fifteen or more) and the much less danger of error in

reading off resistances.

VOLTAIC ELECTRICITY.

245

324. Resistances that may be Measured by Bridge. The

bridge is not suited to the measurement of very high or of very low

resistances. Theory requires that with the plugs inserted the

resistances in the arms should be zero while, as a matter of fact,

they have a resistance which may affect the fourth place of deci-

mals. The resistance in the contacts of the plugs themselves may

affect the third place. Therefore, in measuring very small resist-

ances these neglected resistances may cause a considerable error,

and in the case of a very large resistance any error in the balance

is multiplied a hundred or a thousandfold by applying the ratio

B/A. In general, the measurements should lie between .01 and

100,000 ohms.

325. The Slide Wire Bridge. A simplified form of bridge, used

especially in the measurement of low resistances, is the so-called

slide wire bridge. This consists (Fig. 138) of a wire WW of uniform

Fig. 138.

cross-section stretched between heavy copper terminals and above

a graduated scale. Since this scale is usually a meter subdivided

into millimeters, the instrument is often called a "meter bridge."

Connections are made as shown in the figure, R being a standard

resistance coil (Par. 311) whose resistance is preferably as near as

possible to that of X, the resistance to be measured. The terminal

P of the galvanometer is slid backwards and forwards along the

wire WW until balance is attained, at which point, if A and B

be the resistances of the corresponding portions of the wire, we

have, as in any other bridge, X = BR/A. Since the wire is of

uniform cross-section, the resistance of the portions is directly

proportional to their lengths, hence in the above expression the

lengths of A and B, which may be read directly from the printed

scale, can be and are used instead of the actual resistances, which

last need not be known at all.

246

ELEMENTS OF ELECTRICITY.

326. Measurement of High Resistance. The principle of the

measurement of high resistance is simple. We measure accurately

the current driven through the resistance by a known E. M. F.,

whence, by Ohm's law, the resistance is obtained at once. For

example, to measure the resistance of the rubber insulation of a

reel of submarine cable, the entire cable, except the two free ends,

is submerged in a tank of water (Fig. 139). To one of fcne ends of

Fig. 139.

the cable core is attached a terminal of a battery. The other

terminal is connected to G, a very delicate current-measuring

instrument (a reflecting galvanometer, Par. 378), and the circuit

is completed by a wire extending from G and dipping into the water

in the tank. The E. M. F. of the battery is measured by the

instrument V, and the resulting current by G, whence R follows

from Ohm's law. Reflection will show that should the total

length of the cable be n yards, the average resistance per yard is

n times the total resistance. In actually carrying out this measure-

ment, many refinements and precautions are observed, not

necessary to mention here.

327. Measurement of Resistance of Electrolytes. The re-

sistance of an electrolyte can not be measured by the means

described above. We have seen (Par. 215) that the passage of a

current through an electrolyte produces chemical decomposition;

the current used in balancing a bridge would therefore bring about

this electrolysis. If gas be released at either anode or cathode, the

resistance which we are trying to measure would be very greatly

increased. Also the products of electrolysis will still set up a back

E.M.F. which by cutting down the current through the electrolyte

would lessen the drop in the corresponding branch and render value-

less observations based on movements of the galvanometer needle.

VOLTAIC ELECTRICITY. 247

We may, however, make these measurements by employing a

rapidly alternating current, that is, a current which many times

a second reverses its direction of flow. In this case, a galvanometer

can not be used to indicate a balance but in its stead a telephone

receiver is employed, taking the place of G in Fig. 138. So long as

an alternating current flows through the receiver a buzzing sound

is produced, but when the bridge is balanced the sound dies out.

Explanation of these facts will be given later.

328. Measurement of Internal Resistance of Cells. In meas-

uring the internal resistance of a cell the same difficulties are

encountered as in the case of electrolytes and in addition the cur-

rent produced by the cell itself prevents the use of the bridge.

There are, however, several methods by which this internal

resistance may be measured. The simplest is by using the instru-

ments for measuring E. M. F. and current, which instruments

will be described in Chapter 34. We first measure the E. M. F.

of the cell when no current is flowing. We then cause a moderate

current to flow from the cell, measure this current and the external

or useful volts (Par. 305). The difference between the E. M. F.

of the cell and the useful volts is the lost volts or Ir, and knowing

I we determine r.

248 ELEMENTS OF ELECTRICITY.

CHAPTER 27.

THE POTENTIOMETER.

329. Measurement of Electro -Motive Force of Cells. The

simplest and usual way of measuring the electro-motive force of

a cell is by means of a voltmeter, an instrument described in Chap-

ter 34. It will be shown, however, that in order to obtain a read-

ing from the voltmeter, there must be a flow of current through

the instrument. It is true that this current is so small that for

all ordinary cases it is entirely negligible, but if there be a current

there will also be lost volts (Par. 305) and since a voltmeter reads

only the useful volts, its indications are always some slight

amount less than the true E. M. F. Therefore, to obtain strictly

accurate results, the E. M. F. of a cell should be measured when

no current is flowing. This may be done with an electrometer,

as explained in Chapter 11, but preferably by a potentiometer, an

instrument which we shall now describe.

330. Preliminary Arrangement of a Potentiometer. Let us

suppose that we start with one or two cells giving us ajxmstant

D

Fig. 140.

E. M. F. of about two volts, seven or eight feet of rather thin wire

of uniform cross-section, and a graduated paper scale. Provided

E. M. F. and the observations should be taken in quick succession

so as to avoid change in the current due to the increase of the

resistance of the circuit caused by the heating effect of the current.

311. Resistance Coils. The known resistances used as de-

scribed in the preceding paragraph are usually in the form of coils.

These resistance coils, especially those used as standards of resist-

ance, are made with great care and accuracy and embody many

refinements. They range from .001 of an ohm to 10,000 ohms.

A section of one is shown in Fig. 129. From the ebonite lid there

extends downwards a hollow metal cylinder which has an insulat-

ing covering of shellac-coated silk. Around this cylinder is wrapped

the coil proper which is of silk-insulated manganin wire (Par. 289).

For reasons which are explained later (Par. 315), the wire is

doubled upon itself at its middle point and the winding is begun

at this loop. The ends of the coil are attached to heavy copper

VOLTAIC ELECTRICITY.

235

terminals bent downward as shown. The coil is connected up in

the circuit by inserting these turned-down ends into mercury cups

which in turn are connected to the lead wires. The whole is pro-

Fig. 129.

tected by a brass case which is perforated by many small openings.

The object of the interior metal cylinder is to conduct away heat

developed in the wire and at the same time to afford a large sur-

face for radiation. The object of the openings is to allow the

enclosed coil to cool off more rapidly and also to permit the tem-

perature to be kept down by submerging the entire coil in oil.

The plug in the center of the lid is to permit the insertion of a

thermometer for reading the temperature of the coil so that the

proper correction for temperature may be applied.

312. Drop in Divided Circuit. The usual way of measuring

ordinary resistances is by means of the Wheatstone bridge, a piece

of apparatus whose principle will be understood from the following

explanation. Consider a divided circuit of two branches and let

A (Fig. 130) be the point of high potential. The current at B

divides into two parts inversely proportional to the resistances of

the two branches, i. e., the greater part goes along the branch of

least resistance, the lesser part along the branch of greater resist-

ance. There is a continuous drop of potential along each branch

of the circuit from B to D, in other words, the drop of potential

over the two branches is exactly the same. Suppose following the

right hand branch we reach a point M at which we have passed

over one-half of the resistance in that branch; the difference of po-

tential between M and D is only one-half of that between B and D.

236

ELEMENTS OF ELECTRICITY.

Fig. 130.

Similarly, following the left hand branch and reaching a point

N at which we have passed over one-half of the resistance in that

A branch, the difference of potential between

I JV and D is only one-half of that between B

and D. Hence, the points M and N are at

the same potential. This can be shown by

connecting between these points a sensitive

galvanometer G. (Galvanometers are de-

scribed in Chapter 30. For the present it is

sufficient for us to know that a galvanometer

(more strictly a galvanoscope) is an instru-

ment which indicates by the movement of

its needle that a current is flowing in the

circuit of which it forms a part, and by the

direction of the motion of the needle indi-

cates the direction of the current.) Should

there be a difference of potential between M

and N, a current would be produced and

would be revealed by a deflection of the gal-

vanometer needle, but the needle will be found to remain at rest.

The foregoing illustration is based on the supposition that the

resistance of BM and of B N are each one-half of the resistance of

the respective branches, but the prin-

ciple is equally true for I/nth, that is,

if the resistance of BM be I /nth of

that of the right hand branch and the

resistance of B N be I/nth of that of

the left hand branch, the points M and

N will be at the same potential and

there will be no flow of current between

them if they be connected through a

galvanometer.

313. Principle of the Wheatstone

Bridge. Let us now consider a divided

circuit of two branches (Fig. 131),

each branch subdivided into two parts

as shown, and suppose that in the left

hand branch we know the resistance

Fig. 131.

of A and of R and further can vary that of R at pleasure, and

that in the right hand branch we know the resistance of the

VOLTAIC ELECTRICITY 237

portion B but do not know that of the remainder X and wish to

determine it. Of the total resistance of the right hand branch,

X is some definite fraction, say I /nth. Since R may be varied at

pleasure, it can be adjusted so that it is 1/wth of the total resist-

ance in the left hand branch. When such a state of affairs is

reached, the points M and N will, from what has been shown

above, be at the same potential and the galvanometer connected

between M and N will reveal no current. The system is now

said to be "balanced."

Since X is I/rath of the total resistance in the right hand branch,

B is n 1/wths, and since R has been made 1/wth of that in the

left hand branch, A is n 1/wths.

Hence A : B : : R : X

Whence X

or, when the system has been

brought to a balance, the resistance in X is equal to the product of

the resistances in the adjacent arms divided by that of the opposite arm.

314. A Second Demonstration. The same thing can be readily

shown by applying the principle of drop directly. Call the current

in the left hand branch I', that in the right hand branch I", and

the resistance in the four arms A, B, R, and X, respectively. The

drop from S to N is equal to the current times the resistance or

FA; that from S to M is equal to I"B. But M and N being at

the same potential these drops are equal. Similarly, the drop

from N to T, or I'R, is equal to the drop from M to T, or I"X.

We then have the two equations,

(I) FA = T'E

(II) I'R =I"X

Dividing (II) by (I) and striking out common factors

R X

A = B

Whence as above

BR

The foregoing is the principle upon which the Wheatstone

bridge is constructed.

238

ELEMENT^ OF ELECTRICITY.

"DT> T>

The expression X = r- can be written X = -j R, whence it is

seen that if B and A be so selected that B/A is some multiple or

submultiple of ten, calculations will be simplified since all that

will then be necessary will be to point off decimal places or add

zeros to the value of the known resistance R.

315. Arrangement of Resistances. In the actual apparatus

the resistance in the arms A, B, and R is usually varied by re-

moving or changing the position of certain plugs. For example,

Fig. 132.

the arm A, a portion of which is represented in Fig. 132, consists

of a heavy brass bar DE secured to the ebonite plate FF and cut

entirely through at regular intervals by tapering openings into

which fit the corresponding ebonite-handled brass plugs A, B, C.

The separate sections into which the bar is divided are connected

beneath the plate FF by the resistance coils G, H, K. These are

wound as described in Par. 311 so as to avoid self-induction. For

the present we may explain this by stating that when a circuit

through a coil of wire is completed there is produced through

induction an opposing E. M. F. which causes the current to lag

and prevents it from rising to its full strength at once. When a

coil is made by winding it from a loop at its middle point, each

turn of the coil carrying a current is paralleled by an equal turn

in which the current flows in the opposite direction and the

inductive effects of the two turns exactly neutralize each other.

These coils have a resistance of 1 ohm, 10 ohms, 100 ohms, etc.,

VOLTAIC ELECTRICITY.

239

and therefore bear to each other the ratio of 1 : 10 : 100, etc.

With the plugs in position the current passes from DtoE through

the bar and coils, the combined resistance of which is so small as

to be negligible. With the plug B removed, the current must

follow the path D M H N E, that is, the resistance of the coil

H has been introduced into the circuit.

In the arm R the arrangement is similar but there is a much

greater number of coils whose resistances are in ohms 1, 2, 3, 4,

10, 20, 30, 40, 100, 200, 300, 400, 1000, 2000, 3000, 4000, etc., thus

enabling any combination from 1 to 11110 to be obtained. This

arm is usually called the "rheostat" and is consequently desig-

nated in diagrams by letter JR.

316. Evolution in Form. The theory of the Wheatstone

bridge is best explained as above from a diagrammatic diamond-

(5)

shaped figure as in (1), Fig. 133. The commercial form of this

apparatus bears no superficial resemblance to the figure but has

been evolved directly from it as the following will show.

1st step. The galvanometer need not be placed in the diamond

but may be connected outside as shown in (2).

2d step. A and B need not make an angle with each other

but may be flattened down as shown in (3).

3d step. R being the arm which carries the greatest number

of resistance coils should, relatively to A and B, be elongated as

shown in (4).

4th step. Finally, for the sake of compactness, the arm R

may be folded back upon itself as shown in (5).

Other minor changes consist in the arrangement of the terminals

to facilitate connections, and in the insertion in the battery and

240

ELEMENTS OF ELECTRICITY.

galvanometer circuits of keys permanently attached to the instru-

ment. Sometimes a galvanometer is included in the case. The

final result is an instrument of which Fig. 134 shows a form made

by the Leeds & Northrup Co.

Fig. 134.

The various circuits between the keys and other parts of the

bridge are inside the case but are usually indicated by white lines

marked on the cover.

317. Connections for a Measurement. Whatever be the form

of the bridge it is well- to bear in mind the following: first, the

current enters (or leaves) at the junction of A and B and leaves

(or enters) at the junction of R and X', second, the galvanometer

is connected between the junction of A and R and that of B and X.

(It should however be observed that it may readily be shown that

the battery and the galvanometer may be interchanged, the

resistance of their respective leads altered at will, and the E. M. F.

of the battery varied, all this without affecting the balance.)

Finally, in the factor by which R is to be multiplied, the resistance

of B, the arm connected to X, is the numerator and that of A,

the arm opposite X, is the denominator.

318. Operation of Measurement. To measure the resistance,

say of a wire X, the apparatus is brushed free from dust, and

plugs brightened, being especially careful to remove all grease

VOLTAIC ELECTRICITY.

241

or oil so as to insure perfect contacts. Connections are then made

as shown in Fig. 135. A plug is removed from coil of the same

resistance in both A and B, their ratio therefore being unity.

Various plugs are then removed from R until with both battery

Fig. 135.

and galvanometer keys closed the apparatus is as nearly balanced

as possible. At this point the sum of the unplugged resistances

in -R is as near the unknown resistance X as it is possible to get

with the ratio of unity in B/A.

319. Bracketing. The plugs in R are not removed at hap-

hazard but preferably the resistance should be arrived at by a

system of "bracketing." For example, the first plug to be removed

should be selected so that the resistance thrown in is certainly

greater or less than the one to be measured. Suppose it to be less.

The battery key K is closed and then the galvanometer key H.

Suppose the galvanometer needle to be deflected to the left.

Replace the plug and remove a second one so as to throw in a

resistance certainly greater than that to be measured. Upon

closing the keys, if the needle is now deflected to the right the

unknown resistance lies between the two. Replace the plug and

remove a third which will throw in a resistance as near half way

of the interval between the first two as possible. If upon closing

the keys the needle is deflected to the left, the third resistance is

too small, if to the right it is too great. Proceed in this way

keeping the unknown resistance between limits and halving the

interval at each successive attempt.

'242 ELEMENTS OF ELECTRICITY.

With a little experience the bracketing can be materially

shortened by observing the amount of swing produced in the

needle by the trial resistances. This decreases rapidly as the

correct resistance is approached and indicates which of two is the

nearer.

320. Order of Closing Keys. The order in which the battery

and galvanometer keys are closed is not a matter of indifference.

It is essential that the battery key be closed first. For consider

Fig. 135. The coils in R are wound so as to avoid self-induction

but this object may not be completely attained and with a number

of coils unplugged the inductance may not be negligible. Again,

if the resistance X be that of a coil, especially if it be wrapped

around an iron core, its self-induction will be large. Finally, if X

be a cable it may have considerable capacity as a condenser. In

any of these cases, when the battery key K is closed the current

will not rise at once to its full strength in the branch affected.

Suppose the bridge to be balanced accurately and the galvanom-

eter key closed first; when K is closed the current in one branch

or the other not rising at once to its full strength, M and N will

be momentarily at different potentials and there will be an in-

stantaneous rush of current through G causing a deflection of the

needle and incorrectly indicating a lack of balance. On the other

hand, if K be closed first there will still be this retardation but its

effect will disappear in a fraction of a second, M and N will reach

the same potential and when H is closed there will be no deflection

of the galvanometer needle.

There may be used 'ajspecial key which by making successive

contacts as it is pressed down will insure the proper sequence of

closing.

To avoid violent swings of the needle, the galvanometer key at

first should be given a mere tap.

321. Proper Ratio to Use. The first determination gives the

resistance of X to the nearest unit or ohm. If it be desired to

measure it to the first, second, or third place of decimals, the plugs

in A and B must be so adjusted that the ratio B/A is .1, .01, or

.001 and the corresponding decimal places are pointed off in the

final reading of R. If the resistance to be measured be large, the

ratio B/A must be 10, or 100, or 1000.

It will be noted that some of the ratios can be obtained by several

VOLTAIC ELECTRICITY.

243

different combinations, thus }, %%%, !#> all gi ye the ratio unity.

It can be shown that other things being equal, the greatest sensi-

bility is obtained when the resistances in the four arms of the

bridge are as nearly equal as possible. For example, if the resist-

ance to be measured is about 100 ohms and this is to be measured

to the nearest unit, the ratio should be {%%, or if to the nearest

tenth then T VA-

The instrumental sensibility depends directly upon the sensi-

tiveness of the galvanometer, or its ability to indicate very minute

currents when the bridge is nearly balanced.

322. Bridge with Reversible Ratios. There is sometimes used,

instead of the bridge described above, a variation by which a coil

is saved in each of the arms A and B, making six instead of eight,

and yet the same ratios are preserved.

(2)

Fig. 136.

Its arrangement is shown in (2) in Fig. 136 and is as if the arms

A and B of (1) had been separated at S and each rotated outward

from the center. These outer ends (2) are then connected by a

heavy wire with S which must now be regarded as the junction

of A and B and, according to Par. 317, is the point at which the

battery current enters. The inner ends of A and B are connected

to the R and the X arms by movable plugs. With the plugs in

the positions shown by the small circles in (2), A is connected to

R and B to X. If these plugs be shifted to the positions marked

by the crosses, A becomes connected to X and B to R, in other

words (see Par. 317), A and B interchange.

244

ELEMENTS OF ELECTRICITY.

The A arm contains the coils 1, 10, 100, the B arm 10, 100,

1000. The smallest B/A ratio obtainable with the plugs in the

first position is T W or .1. If it be desired to use a smaller one, shift

the two plugs, A becomes B and B becomes A, and the ratios

T y and joVo become available.

323. The Dial Bridge. In the bridges described above, resist-

ances are thrown in the rheostat by removing plugs. There are

other forms, such as the dial bridge and the decade bridge, in

which resistances are introduced by inserting plugs. The connec-

tions of a dial bridge are shown diagrammatically in Fig. 137.

coooooioooocp

Fig. 137.

The A and B arms are like those of the ordinary bridge but the

rheostat is composed of dials, usually four, which are marked

units, tens, hundreds, and thousands, respectively. Each dial

consists of a heavy center piece of brass surrounded by ten key-

stone shaped pieces, these being numbered 0, 1, 2, etc., to 9.

Between the successive keystone pieces, except numbers 9 and 0,

are resistance coils, those at each dial being all of the same

resistance. Thus, at the unit dial each coil has a resistance of one

ohm; at the ten dial each has a resistance of ten ohms, and so on.

The current entering by A goes to the center of the first dial, then

through the plug to the corresponding keystone piece, thence

through the coils in series to the keystone and thence to the

second dial, etc. The diagram represents a resistance plugged in

of 5135 ohms.

This form is more expensive than the first but has a number of

advantages, among them, the smaller number of plugs to be

handled and consequent smaller number of contacts (four as com-

pared to fifteen or more) and the much less danger of error in

reading off resistances.

VOLTAIC ELECTRICITY.

245

324. Resistances that may be Measured by Bridge. The

bridge is not suited to the measurement of very high or of very low

resistances. Theory requires that with the plugs inserted the

resistances in the arms should be zero while, as a matter of fact,

they have a resistance which may affect the fourth place of deci-

mals. The resistance in the contacts of the plugs themselves may

affect the third place. Therefore, in measuring very small resist-

ances these neglected resistances may cause a considerable error,

and in the case of a very large resistance any error in the balance

is multiplied a hundred or a thousandfold by applying the ratio

B/A. In general, the measurements should lie between .01 and

100,000 ohms.

325. The Slide Wire Bridge. A simplified form of bridge, used

especially in the measurement of low resistances, is the so-called

slide wire bridge. This consists (Fig. 138) of a wire WW of uniform

Fig. 138.

cross-section stretched between heavy copper terminals and above

a graduated scale. Since this scale is usually a meter subdivided

into millimeters, the instrument is often called a "meter bridge."

Connections are made as shown in the figure, R being a standard

resistance coil (Par. 311) whose resistance is preferably as near as

possible to that of X, the resistance to be measured. The terminal

P of the galvanometer is slid backwards and forwards along the

wire WW until balance is attained, at which point, if A and B

be the resistances of the corresponding portions of the wire, we

have, as in any other bridge, X = BR/A. Since the wire is of

uniform cross-section, the resistance of the portions is directly

proportional to their lengths, hence in the above expression the

lengths of A and B, which may be read directly from the printed

scale, can be and are used instead of the actual resistances, which

last need not be known at all.

246

ELEMENTS OF ELECTRICITY.

326. Measurement of High Resistance. The principle of the

measurement of high resistance is simple. We measure accurately

the current driven through the resistance by a known E. M. F.,

whence, by Ohm's law, the resistance is obtained at once. For

example, to measure the resistance of the rubber insulation of a

reel of submarine cable, the entire cable, except the two free ends,

is submerged in a tank of water (Fig. 139). To one of fcne ends of

Fig. 139.

the cable core is attached a terminal of a battery. The other

terminal is connected to G, a very delicate current-measuring

instrument (a reflecting galvanometer, Par. 378), and the circuit

is completed by a wire extending from G and dipping into the water

in the tank. The E. M. F. of the battery is measured by the

instrument V, and the resulting current by G, whence R follows

from Ohm's law. Reflection will show that should the total

length of the cable be n yards, the average resistance per yard is

n times the total resistance. In actually carrying out this measure-

ment, many refinements and precautions are observed, not

necessary to mention here.

327. Measurement of Resistance of Electrolytes. The re-

sistance of an electrolyte can not be measured by the means

described above. We have seen (Par. 215) that the passage of a

current through an electrolyte produces chemical decomposition;

the current used in balancing a bridge would therefore bring about

this electrolysis. If gas be released at either anode or cathode, the

resistance which we are trying to measure would be very greatly

increased. Also the products of electrolysis will still set up a back

E.M.F. which by cutting down the current through the electrolyte

would lessen the drop in the corresponding branch and render value-

less observations based on movements of the galvanometer needle.

VOLTAIC ELECTRICITY. 247

We may, however, make these measurements by employing a

rapidly alternating current, that is, a current which many times

a second reverses its direction of flow. In this case, a galvanometer

can not be used to indicate a balance but in its stead a telephone

receiver is employed, taking the place of G in Fig. 138. So long as

an alternating current flows through the receiver a buzzing sound

is produced, but when the bridge is balanced the sound dies out.

Explanation of these facts will be given later.

328. Measurement of Internal Resistance of Cells. In meas-

uring the internal resistance of a cell the same difficulties are

encountered as in the case of electrolytes and in addition the cur-

rent produced by the cell itself prevents the use of the bridge.

There are, however, several methods by which this internal

resistance may be measured. The simplest is by using the instru-

ments for measuring E. M. F. and current, which instruments

will be described in Chapter 34. We first measure the E. M. F.

of the cell when no current is flowing. We then cause a moderate

current to flow from the cell, measure this current and the external

or useful volts (Par. 305). The difference between the E. M. F.

of the cell and the useful volts is the lost volts or Ir, and knowing

I we determine r.

248 ELEMENTS OF ELECTRICITY.

CHAPTER 27.

THE POTENTIOMETER.

329. Measurement of Electro -Motive Force of Cells. The

simplest and usual way of measuring the electro-motive force of

a cell is by means of a voltmeter, an instrument described in Chap-

ter 34. It will be shown, however, that in order to obtain a read-

ing from the voltmeter, there must be a flow of current through

the instrument. It is true that this current is so small that for

all ordinary cases it is entirely negligible, but if there be a current

there will also be lost volts (Par. 305) and since a voltmeter reads

only the useful volts, its indications are always some slight

amount less than the true E. M. F. Therefore, to obtain strictly

accurate results, the E. M. F. of a cell should be measured when

no current is flowing. This may be done with an electrometer,

as explained in Chapter 11, but preferably by a potentiometer, an

instrument which we shall now describe.

330. Preliminary Arrangement of a Potentiometer. Let us

suppose that we start with one or two cells giving us ajxmstant

D

Fig. 140.

E. M. F. of about two volts, seven or eight feet of rather thin wire

of uniform cross-section, and a graduated paper scale. Provided

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46