A Treatise on Electricity and Magnetism/Volume 1/Part 2/Chapter 11

by James Clerk Maxwell

CHAPTER XI.

THE MEASUREMENT OF ELECTRIC RESISTANCE.

335.] In the present state of electrical science, the determination of the electric resistance of a conductor may be considered as the cardinal operation in electricity, in the same sense that the determination of weight is the cardinal operation in chemistry.

The reason of this is that the determination in absolute measure of other electrical magnitudes, such as quantities of electricity, electromotive forces, currents, &c., requires in each case a complicated series of operations, involving generally observations of time, measurements of distances, and determinations of moments of inertia, and these operations, or at least some of them, must be repeated for every new determination, because it is impossible to preserve a unit of electricity, or of electromotive force, or of current, in an unchangeable state, so as to be available for direct comparison.

But when the electric resistance of a properly shaped conductor of a properly chosen material has been once determined, it is found that it always remains the same for the same temperature, so that the conductor may be used as a standard of resistance, with which that of other conductors can be compared, and the comparison of two resistances is an operation which admits of extreme accuracy.

When the unit of electrical resistance has been fixed on, material copies of this unit, in the form of 'Resistance Coils,' are prepared for the use of electricians, so that in every part of the world electrical resistances may be expressed in terms of the same unit. These unit resistance coils are at present the only examples of material electric standards which can be preserved, copied, and used for the purpose of measurement. Measures of electrical capacity, which are also of great importance, are still defective, on account of the disturbing influence of electric absorption.

336.] The unit of resistance may be an entirely arbitrary one, as in the case of Jacobi's Etalon, which was a certain copper wire of 22.4932 grammes weight, 7.61975 metres length, and 0.667 millimetres diameter. Copies of this have been made by Leyser of Leipsig, and are to be found in different places.

According to another method the unit may be defined as the resistance of a portion of a definite substance of definite dimensions. Thus, Siemens' unit is defined as the resistance of a column of mercury of one metre long, and one square millimetre section, at the temperature 0°C.

337.] Finally, the unit may be defined with reference to the electrostatic or the electromagnetic system of units. In practice the electromagnetic system is used in all telegraphic operations, and therefore the only systematic units actually in use are those of this system.

In the electromagnetic system, as we shall shew at the proper place, a resistance is a quantity homogeneous with a velocity, and may therefore be expressed as a velocity. See Art. 628.

338.] The first actual measurements on this system were made by Weber, who employed as his unit one millimetre per second. Sir W. Thomson afterwards used one foot per second as a unit, but a large number of electricians have now agreed to use the unit of the British Association, which professes to represent a resistance which, expressed as a velocity, is ten millions of metres per second. The magnitude of this unit is more convenient than that of Weber's unit, which is too small. It is sometimes referred to as the B.A. unit, but in order to connect it with the name of the discoverer of the laws of resistance, it is called the Ohm.

339.] To recollect its value in absolute measure it is useful to know that ten millions of metres is professedly the distance from the pole to the equator, measured along the meridian of Paris. A body, therefore, which in one second travels along a meridian from the pole to the equator would have a velocity which, on the electromagnetic system, is professedly represented by an Ohm.

I say professedly, because, if more accurate researches should prove that the Ohm, as constructed from the British Association's material standards, is not really represented by this velocity, electricians would not alter their standards, but would apply a correction. In the same way the metre is professedly one ten-millionth of a certain quadrantal arc, but though this is found not to be exactly true, the length of the metre has not been altered, but the dimensions of the earth are expressed by a less simple number.

According to the system of the British Association, the absolute value of the unit is originally chosen so as to represent as nearly as possible a quantity derived from the electromagnetic absolute system.

340.] When a material unit representing this abstract quantity has been made, other standards are constructed by copying this unit, a process capable of extreme accuracy—of much greater accuracy than, for instance, the copying of foot-rules from a standard foot.

These copies, made of the most permanent materials, are distributed over all parts of the world, so that it is not likely that any difficulty will be found in obtaining copies of them if the original standards should be lost.

But such units as that of Siemens can without very great labour be reconstructed with considerable accuracy, so that as the relation of the Ohm to Siemens unit is known, the Ohm can be reproduced even without having a standard to copy, though the labour is much greater and the accuracy much less than by the method of copying.

Fig. 27.

Finally, the Ohm may be reproduced by the electromagnetic method by which it was originally determined. This method, which is considerably more laborious than the determination of a foot from the seconds pendulum, is probably inferior in accuracy to that last mentioned. On the other hand, the determination of the electromagnetic unit in terms of the Ohm with an amount of accuracy corresponding to the progress of electrical science, is a most important physical research and well worthy of being repeated.

The actual resistance coils constructed to represent the Ohm were made of an alloy of two parts of silver and one of platinum in the form of wires from .5 millimetres to .8 millimetres diameter, and from one to two metres in length. These wires were soldered to stout copper electrodes. The wire itself was covered with two layers of silk, imbedded in solid paraffin, and enclosed in a thin brass case, so that it can be easily brought to a temperature at which its resistance is accurately one Ohm. This temperature is marked on the insulating support of the coil. (See Fig. 27.)

On the Forms of Resistance Coils.

341.] A Resistance Coil is a conductor capable of being easily placed in the voltaic circuit, so as to introduce into the circuit a known resistance.

The electrodes or ends of the coil must be such that no appreciable error may arise from the mode of making the connexions. For resistances of considerable magnitude it is sufficient that the electrodes should be made of stout copper wire or rod well amalgamated with mercury at the ends, and that the ends should be made to press on flat amalgamated copper surfaces placed in mercury cups.

For very great resistances it is sufficient that the electrodes should be thick pieces of brass, and that the connexions should be made by inserting a wedge of brass or copper into the interval between them. This method is found very convenient.

The resistance coil itself consists of a wire well covered with silk, the ends of which are soldered permanently to the electrodes.

The coil must be so arranged that its temperature may be easily observed. For this purpose the wire is coiled on a tube and covered with another tube, so that it may be placed in a vessel of water, and that the water may have access to the inside and the outside of the coil.

To avoid the electromagnetic effects of the current in the coil the wire is first doubled back on itself and then coiled on the tube, so that at every part of the coil there are equal and opposite currents in the adjacent parts of the wire.

When it is desired to keep two coils at the same temperature the wires are sometimes placed side by side and coiled up together. This method is especially useful when it is more important to secure equality of resistance than to know the absolute value of the resistance, as in the case of the equal arms of Wheatstone's Bridge, (Art. 347).

When measurements of resistance were first attempted, a resistance coil, consisting of an uncovered wire coiled in a spiral groove round a cylinder of insulating material, was much used. It was called a Rheostat. The accuracy with which it was found possible to compare resistances was soon found to be inconsistent with the use of any instrument in which the contacts are not more perfect than can be obtained in the rheostat. The rheostat, however, is still used for adjusting the resistance where accurate measurement is not required.

Resistance coils are generally made of those metals whose resistance is greatest and which vary least with temperature. German silver fulfils these conditions very well, but some specimens are found to change their properties during the lapse of years. Hence for standard coils, several pure metals, and also an alloy of platinum and silver, have been employed, and the relative resistance of these during several years has been found constant up to the limits of modern accuracy.

342.] For very great resistances, such as several millions of Ohms, the wire must be either very long or very thin, and the construction of the coil is expensive and difficult. Hence tellurium and selenium have been proposed as materials for constructing standards of great resistance. A very ingenious and easy method of construction has been lately proposed by Phillips^[1]. On a piece of ebonite or ground glass a fine pencil-line is drawn. The ends of this filament of plumbago are connected to metallic electrodes, and the whole is then covered with insulating varnish. If it should be found that the resistance of such a pencil-line remains constant, this will be the best method of obtaining a resistance of several millions of Ohms.

343.] There are various arrangements by which resistance coils may be easily introduced into a circuit.

For instance, a series of coils of which the resistances are 1, 2, 4, 8, 16, &c., arranged according to the powers of 2, may be placed in a box in series.

Fig. 28.

The electrodes consist of stout brass plates, so arranged on the outside of the box that by inserting a brass plug or wedge between two of them as a shunt, the resistance of the corresponding coil may be put out of the circuit. This arrangement was introduced by Siemens.

Each interval between the electrodes is marked with the resistance of the corresponding coil, so that if we wish to make the resistance box equal to 107 we express 107 in the binary scale as 64 + 32 + 8 + 2 + 1 or 1101011. We then take the plugs out of the holes corresponding to 64, 32, 8, 2 and 1, and leave the plugs in 16 and 4.

This method, founded on the binary scale, is that in which the smallest number of separate coils is needed, and it is also that which can be most readily tested. For if we have another coil equal to 1 we can test the equality of 1 and 1’, then that of 1 + 1’ and 2, then that of 1 + 1’ + 2 and 4, and so on.

The only disadvantage of the arrangement is that it requires a familiarity with the binary scale of notation, which is not generally possessed by those accustomed to express every number in the decimal scale.

344.] A box of resistance coils may be arranged in a different way for the purpose of measuring conductivities instead of resistances.

Fig. 29.

The coils are placed so that one end of each is connected with a long thick piece of metal which forms one electrode of the box, and the other end is connected with a stout piece of brass plate as in the former case.

The other electrode of the box is a long brass plate, such that by inserting brass plugs between it and the electrodes of the coils it may be connected to the first electrode through any given set of coils. The conductivity of the box is then the sum of the conductivities of the coils.

In the figure, in which the resistances of the coils are 1, 2, 4, &c., and the plugs are inserted at 2 and 8, the conductivity of the box is ${\frac {1}{2}}+{\frac {1}{8}}={\frac {5}{8}}$, and the resistance of the box is therefore ${\frac {8}{5}}$ or 1.6.

This method of combining resistance coils for the measurement of fractional resistances was introduced by Sir W. Thomson under the name of the method of multiple arcs. See Art. 276.

On the Comparison of Resistances.

345.] If $E$ is the electromotive force of a battery, and $R$ the resistance of the battery and its connexions, including the galvanometer used in measuring the current, and if the strength of the current is $I$ when the battery connexions are closed, and $I_{1},\,I_{2}$ when additional resistances $r_{1},\,r_{2}$ are introduced into the circuit, then, by Ohm's Law,

$E=IR=I_{1}(R+r_{1})=I_{2}(R+r_{2}).$

Eliminating $E,$ the electromotive force of the battery, and $R$ the resistance of the battery and its connexions, we get Ohm's formula

${\frac {r_{1}}{r_{2}}}={\frac {(I-I_{1})I_{2}}{(I-I_{2})I_{1}}}.$

This method requires a measurement of the ratios of $I,\,I_{1}$ and $I_{2},$ and this implies a galvanometer graduated for absolute measurements.

If the resistances $r_{1}$ and $r_{2}$ are equal, then $I_{1}$ and $I_{2}$ are equal, and we can test the equality of currents by a galvanometer which is not capable of determining their ratios.

But this is rather to be taken as an example of a faulty method than as a practical method of determining resistance. The electromotive force $E$ cannot be maintained rigorously constant, and the internal resistance of the battery is also exceedingly variable, so that any methods in which these are assumed to be even for a short time constant are not to be depended on.

346.] The comparison of resistances can be made with extreme

Fig. 30.

accuracy by either of two methods, in which the result is in dependent of variations of $R$ and $E$.

The first of these methods depends on the use of the differential galvanometer, an instrument in which there are two coils, the currents in which are independent of each other, so that when the currents are made to flow in opposite directions they act in opposite directions on the needle, and when the ratio of these currents is that of $m$ to $n$ they have no resultant effect on the galvanometer needle.

Let $I_{1},\,I_{2}$ be the currents through the two coils of the galvanometer, then the deflexion of the needle may be written

$\delta =mI_{1}-nI_{2}.$

Now let the battery current $I$ be divided between the coils of the galvanometer, and let resistances $A$ and $B$ be introduced into the first and second coils respectively. Let the remainder of the resistance of their coils and their connexions be $\alpha$ and $\beta$ respectively, and let the resistance of the battery and its connexions between $C$ and $D$ be $r$, and its electromotive force $E.$

Then we find, by Ohm's Law, for the difference of potentials between $C$ and $D,$

$C-D=I_{1}(A+\alpha )=I_{2}(B+\beta )=E-Ir,$

and since

$I_{1}+I_{2}=I,$

$I_{1}=E{\frac {B+\beta }{D}},\qquad I_{2}=E{\frac {A+\alpha }{D}},\qquad I=E{\frac {A+\alpha +B+\beta }{D}},$

where

$D=(A+\alpha )(B+\beta )+r(A+\alpha +B+\beta ).$

The deflexion of the galvanometer needle is therefore

$\delta ={\frac {E}{D}}\{m(B+\beta )-n(A+\alpha )\},$

and if there is no observable deflexion, then we know that the quantity enclosed in brackets cannot differ from zero by more than a certain small quantity, depending on the power of the battery, the suitableness of the arrangement, the delicacy of the galvanometer, and the accuracy of the observer.

Suppose that $B$ has been adjusted so that there is no apparent deflexion.

Now let another conductor $A'$ be substituted for $A$, and let $A'$ be adjusted till there is no apparent deflexion. Then evidently to a first approximation $A'=A.$

To ascertain the degree of accuracy of this estimate, let the altered quantities in the second observation be accented, then

$m(B+\beta )-n(A+\alpha )={\frac {D}{E}}\delta ,$

$m(B+\beta )-n(A'+\alpha )={\frac {D'}{E'}}\delta '.$

Hence

$n(A'-A)={\frac {D}{E}}\delta -{\frac {D'}{E'}}\delta '.$

If $\delta$ and $\delta '$, instead of being both apparently zero, had been only observed to be equal, then, unless we also could assert that $E=E'$, the right-hand side of the equation might not be zero. In fact, the method would be a mere modification of that already described.

The merit of the method consists in the fact that the thing observed is the absence of any deflexion, or in other words, the method is a Null method, one in which the non-existence of a force is asserted from an observation in which the force, if it had been different from zero by more than a certain small amount, would have produced an observable effect.

Null methods are of great value where they can be employed, but they can only be employed where we can cause two equal and opposite quantities of the same kind to enter into the experiment together.

In the case before us both $\delta$ and $\delta '$ are quantities too small to be observed, and therefore any change in the value of $E$ will not affect the accuracy of the result.

The actual degree of accuracy of this method might be ascertained by taking a number of observations in each of which $A'$ is separately adjusted, and comparing the result of each observation with the mean of the whole series.

But by putting $A'$ out of adjustment by a known quantity, as, for instance, by inserting at $A$ or at $B$ an additional resistance equal to a hundredth part of $A$ or of $B$, and then observing the resulting deviation of the galvanometer needle, we can estimate the number of degrees corresponding to an error of one per cent. To find the actual degree of precision we must estimate the smallest deflexion which could not escape observation, and compare it with the deflexion due to an error of one per cent.

^[2]If the comparison is to be made between $A$ and $B$, and if the positions of $A$ and $B$ are exchanged, then the second equation becomes

$m(A+\beta )-(B+\alpha )={\frac {D'}{E}}\delta ',$

whence

$(m+n)(B-A)={\frac {D}{E}}\delta -{\frac {D'}{E'}}\delta '.$

If $m$ and $n,\,A\,{\mbox{and}}\,B,\,\alpha \,{\mbox{and}}\,\beta \,$ are approximately equal, then

$B-A={\frac {1}{2nE}}(A+\alpha )(A+\alpha +2r)(\delta -\delta ').$

Here $\delta -\delta '$ may be taken to be the smallest observable deflexion of the galvanometer.

If the galvanometer wire be made longer and thinner, retaining the same total mass, then $n$ will vary as the length of the wire and $\alpha$ as the square of the length. Hence there will be a minimum value of ${\frac {(A+\alpha )(A+\alpha +2r)}{n}}$ when

$\alpha ={\frac {1}{3}}(a+r)\left\{2{\sqrt {1-{\frac {3}{4}}{\frac {r^{2}}{(A+r)^{2}}}}}-1\right\}.$

If we suppose $r$, the battery resistance, small compared with $A$, this gives

$\alpha ={\frac {1}{3}}A;$

or, the resistance of each coil of the galvanometer should be one-third of the resistance to be measured.

We then find

$B-A={\frac {8}{9}}{\frac {A^{2}}{nE}}(\delta -\delta ').$

If we allow the current to flow through one only of the coils of the galvanometer, and if the deflexion thereby produced is $\Delta$ (supposing the deflexion strictly proportional to the deflecting force), then

$\Delta ={\frac {mE}{A+\alpha +r}}={\frac {3}{4}}{\frac {nE}{A}}\;{\mbox{if}}\;r=0\;{\mbox{and}}\;a={\frac {1}{3}}A.$

Hence

${\frac {B-A}{A}}={\frac {2}{3}}{\frac {\delta -\delta '}{\Delta }}.$

In the differential galvanometer two currents are made to produce equal and opposite effects on the suspended needle. The force with which either current acts on the needle depends not only on the strength of the current, but on the position of the windings of the wire with respect to the needle. Hence, unless the coil is very carefully wound, the ratio of $m$ to $n$ may change when the position of the needle is changed, and therefore it is necessary to determine this ratio by proper methods during each course of experiments if any alteration of the position of the needle is suspected.

The other null method, in which Wheatstone's Bridge is used, requires only an ordinary galvanometer, and the observed zero deflexion of the needle is due, not to the opposing action of two currents, but to the non-existence of a current in the wire. Hence we have not merely a null deflexion, but a null current as the phenomenon observed, and no errors can arise from want of regularity or change of any kind in the coils of the galvanometer. The galvanometer is only required to be sensitive enough to detect the existence and direction of a current, without in any way determining its value or comparing its value with that of another current.

347.] Wheatstone's Bridge consists essentially of six conductors connecting

Fig. 31.

four points. An electromotive force $E$ is made to act between two of the points by means of a voltaic battery introduced between $B$ and $C$. The current between the other two points $O$ and $A$ is measured by a galvanometer.

Under certain circumstances this current becomes zero. The conductors $BC$ and $OA$ are then said to be conjugate to each other, which implies a certain relation between the resistances of the other four conductors, and this relation is made use of in measuring resistances.

If the current in $OA$ is zero, the potential at $O$ must be equal to that at $A$. Now when we know the potentials at $B$ and $C$ we can determine those at $O$ and $A$ by the rule given at Art. 274, provided there is no current in $OA$,

$O={\frac {B\gamma +C\beta }{\beta +\gamma }},\qquad A={\frac {Bb+Cc}{b+c}},$

whence the condition is

$b\beta =c\gamma ,$

where $b,\,c,\,\beta ,\,\gamma$ are the resistances in $CA,\,AB,\,BO\,{\mbox{and}}\,OC$ respectively.

To determine the degree of accuracy attainable by this method we must ascertain the strength of the current in $OA$ when this condition is not fulfilled exactly.

Let $A,\,B,\,C,$ and $O$ be the four points. Let the currents along $BC,\,CA\,{\mbox{and}}\,AB$ be $x,\,y\,{\mbox{and}}\,z,$ and the resistances of these conductors $a,\,b\,{\mbox{and}}\,c.$ Let the currents along $OA,\,OB\,{\mbox{and}}\,OC$ be $\xi ,\,\eta ,\,\zeta$ and the resistances $\alpha ,\,\beta \,{\mbox{and}}\,\gamma .$ Let an electromotive force $E$ act along $BC.$ Required the current $\xi$ along $OA.$

Let the potentials at the points $A,\,B,\,C\,{\mbox{and}}\,O$ be denoted by the symbols $A,\,B,\,C\,{\mbox{and}}\,0.$ The equations of conduction are

${\begin{array}{ll}ax=B-C+E,\qquad &\alpha \xi =O-A,\\by=C-A,&\beta \eta =O-B,\\cz=A-B,&\gamma \zeta =O-C;\end{array}}$

with the equations of continuity

${\begin{array}{l}\xi +y-z=0,\\\eta +z-x=0,\\\zeta +x-y=0.\end{array}}$

By considering the system as made up of three circuits $OBC,\,OCA\,{\mbox{and}}\,OAB$ in which the currents are $x,\,y,\,z$ respectively, and applying Kirchhoff's rule to each cycle, we eliminate the values of the potentials $O,\,A,\,B,\,C,$ and the currents $\xi ,\,\eta ,\,\zeta ,$ and obtain the following equations for $x,\,y\,{\mbox{and}}\,z,$

${\begin{array}{llll}(a+\beta +\gamma )x&-\gamma y&-\beta z&=E,\\{}-\gamma x&{}+(b+\gamma +\alpha )y&-\alpha z&=0,\\{}-\beta x&{}-\alpha y&{}+(c+\alpha +\beta )z&=0.\end{array}}$

Hence, if we put

$D=\left|{\begin{array}{ccc}a+\beta +\gamma &-\gamma &-\beta \\-\gamma &b+\gamma +\alpha &-\alpha \\-\beta &-\alpha &c+\alpha +\beta \end{array}}\right|,$

we find

$\xi ={\frac {E}{D}}(b\beta -c\gamma ),$

and

$x={\frac {E}{D}}\{(b+\gamma )(c+\beta )+\alpha (b+c+\beta +\gamma )\}.$

348.] The value of $D$ may be expressed in the symmetrical form, $D=\alpha bc+bc(\beta +\gamma )+ca(\gamma +\alpha )+ab(\alpha +\beta )+(a+b+c)(\beta \gamma +\gamma \alpha +\alpha \beta )$ or, since we suppose the battery in the conductor $a$ and the galvanometer in $\alpha$, we may put $B$ the battery resistance for $a$ and $G$ the galvanometer resistance for $\alpha$. We then find

$D={\begin{array}{rl}BG(b+c+\beta +\gamma )+{}&B(b+\gamma )(c+\beta )\\&\;{}+G(b+c)(\beta +\gamma )+bc(\beta +\gamma )+\beta \gamma (b+c).\end{array}}$

If the electromotive force $E$ were made to act along $OA,$ the resistance of $OA$ being still $\alpha$, and if the galvanometer were placed in $BC$. the resistance of $BC$ being still $a$, then the value of $D$ would remain the same, and the current in $BC$ due to the electromotive force $E$ acting along $OA$ would be equal to the current in $OA$ due to the electromotive force $E$ acting in $BC$.

But if we simply disconnect the battery and the galvanometer, and without altering their respective resistances connect the battery to $O$ and $A$ and the galvanometer to $B$ and $C$, then in the value of $D$ we must exchange the values of $B$ and $G$. If $D'$ be the value of $D$ after this exchange, we find $${\begin{aligned}D'-D&=(G-B)\{(b+c)(\beta +\gamma )-(b+\gamma )(\beta +c)\},\\&=(B-G)\{(b-\beta )(c-\gamma \}.\end{aligned}}$$

Let us suppose that the resistance of the galvanometer is greater than that of the battery.

Let us also suppose that in its original position the galvanometer connects the junction of the two conductors of least resistance $\beta$, $\gamma$ with the junction of the two conductors of greatest resistance $b$, $c$, or, in other words, we shall suppose that if the quantities $b$, $c$, $\gamma$, $\beta$ are arranged in order of magnitude, $b$ and $c$ stand together, and $\gamma$ and $\beta$ stand together. Hence the quantities $b-\beta$ and $c-\gamma$ are of the same sign, so that their product is positive, and therefore $D'-D$ is of the same sign as $B-G$.

If therefore the galvanometer is made to connect the junction of the two greatest resistances with that of the two least, and if the galvanometer resistance is greater than that of the battery, then the value of $D$ will be less, and the value of the deflexion of the galvanometer greater, than if the connexions are exchanged.

The rule therefore for obtaining the greatest galvanometer deflexion in a given system is as follows:

Of the two resistances, that of the battery and that of the galvanometer, connect the greater resistance so as to join the two greatest to the two least of the four other resistances.

349.] We shall suppose that we have to determine the ratio of the resistances of the conductors $AB$ and $AC$, and that this is to be done by finding a point $O$ on the conductor $BOC$, such that when the points $A$ and $O$ are connected by a wire, in the course of which a galvanometer is inserted, no sensible deflexion of the galvanometer needle occurs when the battery is made to act between $B$ and $C$.

The conductor $BOC$ may be supposed to be a wire of uniform resistance divided into equal parts, so that the ratio of the resistances of $BO$ and $OC$ may be read off at once.

Instead of the whole conductor being a uniform wire, we may make the part near $O$ of such a wire, and the parts on each side may be coils of any form, the resistance of which is accurately known.

We shall now use a different notation instead of the symmetrical notation with which we commenced.

Let the whole resistance of $BAC$ be $R$.

Let $c=mR$ and $b=(1-m)R$.

Let the whole resistance of $BOC$ be $S$.

Let $\beta =nS$ and $\gamma =(1-n)S$.

The value of $n$ is read off directly, and that of $m$ is deduced from it when there is no sensible deviation of the galvanometer.

Let the resistance of the battery and its connexions be $B$, and that of the galvanometer and its connexions $G$.

We find as before

${\begin{aligned}D=G\{BR+BS+RS\}&+m(1-m)R^{2}(B+S)+n(1-n)S^{2}(B+R)\\&+(m+n-2mn)BRS,\end{aligned}}$

and if $\xi$ is the current in the galvanometer wire

$\xi ={\frac {ERS}{D}}(n-m)$.

In order to obtain the most accurate results we must make the deviation of the needle as great as possible compared with the value of $(n-m)$. This may be done by properly choosing the dimensions of the galvanometer and the standard resistance wire.

It will be shewn, when we come to Galvanometry, Art. 716, that when the form of a galvanometer wire is changed while its mass remains constant, the deviation of the needle for unit current is proportional to the length, but the resistance increases as the square of the length. Hence the maximum deflexion is shewn to occur when the resistance of the galvanometer wire is equal to the constant resistance of the rest of the circuit.

In the present case, if $\delta$ is the deviation,

$\delta =C{\sqrt {G}}\xi$,

where $C$ is some constant, and $G$ is the galvanometer resistance which varies as the square of the length of the wire. Hence we find that in the value of $D$, when $\delta$ is a maximum, the part involving $G$ must be made equal to the rest of the expression.

If we also put $m=n$, as is the case if we have made a correct observation, we find the best value of $G$ to be

$G=n(1-n)(R+S)$.

This result is easily obtained by considering the resistance from $A$ to $O$ through the system, remembering that $BC$, being conjugate to $AO$, has no effect on this resistance.

In the same way we should find that if the total area of the acting surfaces of the battery is given, the most advantageous arrangement of the battery is when

$B={\frac {RS}{R+S}}.$

Finally, we shall determine the value of $S$ such that a given change in the value of $n$ may produce the greatest galvanometer deflexion. By differentiating the expression for $\xi$ we find

$S^{2}={\frac {BR}{B+R}}\left(R+{\frac {G}{n(1-n)}}\right).$

If we have a great many determinations of resistance to make in which the actual resistance has nearly the same value, then it may be worth while to prepare a galvanometer and a battery for this purpose. In this case we find that the best arrangement is

$S=R,\qquad B={\frac {1}{2}}R,\qquad G=2n(1-n)R,$

and if $n={\frac {1}{2}},\;G={\frac {1}{2}}R$.

On the Use of Wheatstone's Bridge.

350.] We have already explained the general theory of Wheatstone's Bridge, we shall now consider some of its applications.

Fig. 32.

The comparison which can be effected with the greatest exactness is that of two equal resistances.

Let us suppose that $\beta$ is a standard resistance coil, and that we wish to adjust $\gamma$ to be equal in resistance to $\beta$.

Two other coils, $b$ and $c$, are prepared which are equal or nearly equal to each other, and the four coils are placed with their electrodes in mercury cups so that the current of the battery is divided between two branches, one consisting of $\beta$ and $\gamma$ and the other of $b$ and $c$. The coils $b$ and $c$ are connected by a wire $PR$, as uniform in its resistance as possible, and furnished with a scale of equal parts.

The galvanometer wire connects the junction of $\beta$ and $\gamma$ with a point $Q$ of the wire $PR$, and the point of contact at $Q$ is made to vary till on closing first the battery circuit and then the galvanometer circuit, no deflexion of the galvanometer needle is observed.

The coils $\beta$ and $\gamma$ are then made to change places, and a new position is found for $Q$. If this new position is the same as the old one, then we know that the exchange of $\beta$ and $\gamma$ has produced no change in the proportions of the resistances, and therefore $\gamma$ is rightly adjusted. If $Q$ has to be moved, the direction and amount of the change will indicate the nature and amount of the alteration of the length of the wire of $\gamma$, which will make its resistance equal to that of $\beta$.

If the resistances of the coils $b$ and $c$, each including part of the wire $PR$ up to its zero reading, are equal to that of $b$ and $c$ divisions of the wire respectively, then, if $x$ is the scale reading of $Q$ in the first case, and $y$ that in the second,

${\frac {c+x}{b-x}}={\frac {\beta }{\gamma }},\qquad {\frac {c+y}{b-y}}={\frac {\gamma }{\beta }},$

whence

${\frac {\gamma ^{2}}{\beta ^{2}}}=1+{\frac {(b+c)(y-x)}{(c+x)(b-y)}}$

Since $b-y$ is nearly equal to $c+x$, and both are great with respect to $x$ or $y$, we may write this

${\frac {\gamma ^{2}}{\beta ^{2}}}=1+4{\frac {y-x}{b+c}},$

and

$\gamma =\beta \left(1+2{\frac {y-x}{b+c}}\right).$

When $\gamma$ is adjusted as well as we can, we substitute for $b$ and $c$ other coils of (say) ten times greater resistance.

The remaining difference between $\beta$ and $\gamma$ will now produce a ten times greater difference in the position of $Q$ than with the original coils $b$ and $c$, and in this way we can continually increase the accuracy of the comparison.

The adjustment by means of the wire with sliding contact piece is more quickly made than by means of a resistance box, and it is capable of continuous variation.

The battery must never be introduced instead of the galvanometer into the wire with a sliding contact, for the passage of a powerful current at the point of contact would injure the surface of the wire. Hence this arrangement is adapted for the case in which the resistance of the galvanometer is greater than that of the battery.

On the Measurement of Small Resistances.

351.] When a short and thick conductor is introduced into a circuit its resistance is so small compared with the resistance occasioned by unavoidable faults in the connexions, such as want of contact or imperfect soldering, that no correct value of the

Fig. 33.

resistance can be deduced from experiments made in the way described above.

The object of such experiments is generally to determine the specific resistance of the substance, and it is resorted to in cases when the substance cannot be obtained in the form of a long thin wire, or when the resistance to transverse as well as to longitudinal conduction has to be measured.

Fig. 34.

Sir W. Thomson,^[3] has described a method applicable to such cases, which we may take as an example of a system of nine conductors.

The most important part of the method consists in measuring the resistance, not of the whole length of the conductor, but of the part between two marks on the conductor at some little distance from its ends.

The resistance which we wish to measure is that experienced by a current whose intensity is uniform in any section of the conductor, and which flows in a direction parallel to its axis. Now close to the extremities, when the current is introduced by means of electrodes, either soldered, amalgamated, or simply pressed to the ends of the conductor, there is generally a want of uniformity in the distribution of the current in the conductor. At a short distance from the extremities the current becomes sensibly uniform. The student may examine for himself the investigation and the diagrams of Art. 193, where a current is introduced into a strip of metal with parallel sides through one of the sides, but soon becomes itself parallel to the sides.

The resistance of the conductors between certain marks $S,\,S'$ and $TT'$ is to be compared.

The conductors are placed in series, and with connexions as perfectly conducting as possible, in a battery circuit of small resistance. A wire $SVT$ is made to touch the conductors at $S$ and $T$, and $S'V'T'$ is another wire touching them at $S'$ and $T'$.

The galvanometer wire connects the points $V$ and $V'$ of these wires.

The wires $SVT$ and $S'V'T'$ are of resistance so great that the resistance due to imperfect connexion at $S,\,T,\,S'$ or $T'$ may be neglected in comparison with the resistance of the wire, and $V,\,V'$ are taken so that the resistance in the branches of either wire leading to the two conductors are nearly in the ratio of the resistances of the two conductors.

Calling	$H$ and $F$ the resistances of the conductors $SS'$ and $TT'$.
"	$A$ and $C$ those of the branches $SV$ and $VT$.
"	$P$ and $R$ those of the branches $S'V'$ and $V'T'$.
"	$Q$ that of the connecting piece $S'T'$.
"	$B$ that of the battery and its connexions.
"	$G$ that of the galvanometer and its connexions.

The symmetry of the system may be understood from the skeleton diagram. Fig. 33.

The condition that $B$ the battery and $G$ the galvanometer may be conjugate conductors is, in this case,

${\frac {F}{C}}-{\frac {H}{A}}+\left({\frac {R}{C}}-{\frac {P}{A}}\right){\frac {Q}{P+Q+R}}=0.$

Now the resistance of the connector $Q$ is as small as we can make it. If it were zero this equation would be reduced to

${\frac {F}{C}}={\frac {H}{A}},$

and the ratio of the resistances of the conductors to be compared would be that of $C$ to $A$, as in Wheatstone's Bridge in the ordinary form.

In the present case the value of $Q$ is small compared with $P$ or with $R$, so that if we assume the points $V,\,V'$ so that the ratio of $R$ to $C$ is nearly equal to that of $P$ to $A$, the last term of the equation will vanish, and we shall have

$F:H::C:A.$

The success of this method depends in some degree on the perfection of the contact between the wires and the tested conductors at $SS',\,T'$ and $T$. In the following method, employed by Messrs. Matthiessen and Hockin^[4], this condition is dispensed with.

Fig. 35.

352.] The conductors to be tested are arranged in the manner already described, with the connexions as well made as possible, and it is required to compare the resistance between the marks $SS'$ on the first conductor with the resistance between the marks $T'T$ on the second.

Two conducting points or sharp edges are fixed in a piece of insulating material so that the distance between them can be accurately measured. This apparatus is laid on the conductor to be tested, and the points of contact with the conductor are then at a known distance $SS'$. Each of these contact pieces is connected with a mercury cup, into which one electrode of the galvanometer may be plunged.

The rest of the apparatus is arranged, as in Wheatstone's Bridge, with resistance coils or boxes $A$ and $C$, and a wire $PR$ with a sliding contact piece $Q$, to which the other electrode of the galvanometer is connected.

Now let the galvanometer be connected to $S$ and $Q$, and let $A_{1}$ and $C_{1}$ be so arranged, and the position of $Q$ so determined, that there is no current in the galvanometer wire.

Then we know that

${\frac {XS}{SY}}={\frac {A_{1}+PQ}{C_{1}+QR}}$

where $XS,\,PQ,$ &c. stand for the resistances in these conductors.

From this we get

${\frac {XS}{XY}}={\frac {A_{1}+PQ_{1}}{A_{1}+C_{1}+PR}}.$

Now let the electrode of the galvanometer be connected to $S'$, and let resistance be transferred from $C$ to $A$ (by carrying resistance coils from one side to the other) till electric equilibrium of the galvanometer wire can be obtained by placing $Q$ at some point of the wire, say $Q_{2}$. Let the values of $C$ and $A$ be now $C_{2}$ and $A_{2}$, and let

$A_{2}+C_{2}+PR=A_{1}+C_{1}+PR=R.$

Then we have, as before,

${\frac {XS'}{XY}}={\frac {A_{2}+PQ_{2}}{R}}.$

Whence

${\frac {SS'}{XY}}={\frac {A_{2}-A_{1}+Q_{1}Q_{2}}{R}}.$

In the same way, placing the apparatus on the second conductor at $TT'$ and again transferring resistance, we get, when the electrode is in $T'$,

${\frac {XT'}{XY}}={\frac {A_{3}+PQ_{3}}{R}},$

and when it is in $T$,

${\frac {XT}{XY}}={\frac {A_{4}+PQ_{4}}{R}}.$

Whence

${\frac {T'T}{XY}}={\frac {A_{4}-A_{3}+Q_{3}Q_{4}}{R}}.$

We can now deduce the ratio of the resistances $SS'$ and $T'T$, for

${\frac {SS'}{T'T}}={\frac {A_{2}-A_{1}+Q_{1}Q_{2}}{A_{4}-A_{3}+Q_{3}Q_{4}}}.$

When great accuracy is not required we may dispense with the resistance coils $A$ and $C$, and we then find

${\frac {SS'}{T'T}}={\frac {Q_{1}Q_{2}}{Q_{3}Q_{4}}}$.

The readings of the position of $Q$ on a wire of a metre in length cannot be depended on to less than a tenth of a millimetre, and the resistance of the wire may vary considerably in different parts owing to inequality of temperature, friction, &c. Hence, when great accuracy is required, coils of considerable resistance are introduced at $A$ and $C$, and the ratios of the resistances of these coils can be determined more accurately than the ratio of the resistances of the parts into which the wire is divided at $Q$.

It will be observed that in this method the accuracy of the determination depends in no degree on the perfection of the contacts at SS' or T'T.

This method may be called the differential method of using Wheatstone's Bridge, since it depends on the comparison of observations separately made.

An essential condition of accuracy in this method is that the resistance of the connexions should continue the same during the course of the four observations required to complete the determination. Hence the series of observations ought always to be repeated in order to detect any change in the resistances.

On the Comparison of Great Resistances.

353.] When the resistances to be measured are very great, the comparison of the potentials at different points of the system may be made by means of a delicate electrometer, such as the Quadrant Electrometer described in Art. 219.

If the conductors whose resistance is to be measured are placed in series, and the same current passed through them by means of a battery of great electromotive force, the difference of the potentials at the extremities of each conductor will be proportional to the resistance of that conductor. Hence, by connecting the electrodes of the electrometer with the extremities, first of one conductor and then of the other, the ratio of their resistances may be determined.

This is the most direct method of determining resistances. It involves the use of an electrometer whose readings may be depended on, and we must also have some guarantee that the current remains constant during the experiment.

Four conductors of great resistance may also be arranged as in Wheatstone's Bridge, and the bridge itself may consist of the electrodes of an electrometer instead of those of a galvanometer. The advantage of this method is that no permanent current is required to produce the deviation of the electrometer, whereas the galvanometer cannot be deflected unless a current passes through the wire.

354.] When the resistance of a conductor is so great that the current which can be sent through it by any available electromotive force is too small to be directly measured by a galvanometer, a condenser may be used in order to accumulate the electricity for a certain time, and then, by discharging the condenser through a galvanometer, the quantity accumulated may be estimated. This is Messrs. Bright and Clark's method of testing the joints of submarine cables.

355.] But the simplest method of measuring the resistance of such a conductor is to charge a condenser of great capacity and to connect its two surfaces with the electrodes of an electrometer and also with the extremities of the conductor. If $E$ is the diffference of potentials as shewn by the electrometer, $S$ the capacity of the condenser, and $Q$ the charge on either surface, $R$ the resistance of the conductor and $x$ the current in it, then, by the theory of condensers,

$Q=SE,$

By Ohm's Law,

$E=Rx$,

and by the definition of a current,

$x=-{\frac {dQ}{dt}}$.

Hence

$Q=RS{\frac {dQ}{dt}}$,

and

$Q=Q_{0}e^{-{\frac {t}{RS}}}$,

where $Q_{0}$ is the charge at first when $t=0$.

Similarly

$E=E_{0}e^{-{\frac {t}{RS}}}$

where $E_{0}$ is the original reading of the electrometer, and $E$ the same after a time $t$. From this we find

$R={\frac {t}{s\{\log _{e}E_{0}-\log _{e}E\}}}$,

which gives $R$ in absolute measure. In this expression a knowledge of the value of the unit of the electrometer scale is not required.

If $S$, the capacity of the condenser, is given in electrostatic measure as a certain number of metres, then $R$ is also given in electrostatic measure as the reciprocal of a velocity.

If $S$ is given in electromagnetic measure its dimensions are ${\frac {T^{2}}{L}}$ and $R$ is a velocity.

Since the condenser itself is not a perfect insulator it is necessary to make two experiments. In the first we determine the resistance of the condenser itself, $R_{0}$, and in the second, that of the condenser when the conductor is made to connect its surfaces. Let this be $R'$. Then the resistance, $R$, of the conductor is given by the equation

${\frac {1}{R}}={\frac {1}{R'}}-{\frac {1}{R_{0}}}$.

This method has been employed by MM. Siemens.

Thomson's^[5] Method for the Determination of the Resistance of the Galvanometer.

356.] An arrangement similar to Wheatstone's Bridge has been employed with advantage by Sir W. Thomson in determining the

Fig. 36.

resistance of the galvanometer when in actual use. It was suggested to Sir W. Thomson by Mance's Method. See Art. 357. Let the battery be placed, as before, between $B$ and $C$ in the figure of Article 347, but let the galvanometer be placed in $CA$ instead of in $OA$. If $b\beta -c\gamma$ is zero, then the conductor $OA$ is conjugate to $BC$, and, as there is no current produced in $OA$ by the battery in $BC$, the strength of the current in any other conductor is independent of the resistance in $OA$. Hence, if the galvanometer is placed in $CA$ its deflexion will remain the same whether the resistance of $OA$ is small or great. We therefore observe whether the deflexion of the galvanometer remains the same when $O$ and $A$ are joined by a conductor of small resistance, as when this connexion is broken, and if, by properly adjusting the resistances of the conductors, we obtain this result, we know that the resistance of the galvanometer is

$b={\frac {c\gamma }{\beta }}$

where $c,\,\gamma$, and $\beta$ are resistance coils of known resistance.

It will be observed that though this is not a null method, in the sense of there being no current in the galvanometer, it is so in the sense of the fact observed being the negative one, that the deflexion of the galvanometer is not changed when a certain contact is made. An observation of this kind is of greater value than an observation of the equality of two different deflexions of the same galvanometer, for in the latter case there is time for alteration in the strength of the battery or the sensitiveness of the galvanometer, whereas when the deflexion remains constant, in spite of certain changes which we can repeat at pleasure, we are sure that the current is quite independent of these changes.

The determination of the resistance of the coil of a galvanometer can easily be effected in the ordinary way of using Wheatstone's Bridge by placing another galvanometer in $OA$. By the method now described the galvanometer itself is employed to measure its own resistance.

Mance's^[6] Method of determining the Resistance of the Battery.

357.] The measurement of the resistance of a battery when in action is of a much higher order of difficulty, since the resistance of the battery is found to change considerably for some time after the strength of the current through it is changed. In many of the methods commonly used to measure the resistance of a battery such alterations of the strength of the current through it occur in the course of the operations, and therefore the results are rendered doubtful.

In Mance's method, which is free from this objection, the battery is placed in $BC$ and the galvanometer in $CA$. The connexion between $O$ and $B$ is then alternately made and broken.

If the deflexion of the galvanometer remains unaltered, we know that $OB$ is conjugate to $CA$, whence $c\gamma =a\alpha$, and $a$, the resistance of the battery, is obtained in terms of known resistances $c,\,\gamma ,\,\alpha$.

When the condition $c\gamma =a\alpha$ is fulfilled, then the current through the galvanometer is

$y={\frac {E\alpha }{b\alpha +C(b+\alpha +\gamma )}},$

and this is independent of the resistance $\beta$ between $O$ and $B$. To test the sensibility of the method let us suppose that the condition $c\gamma =a\alpha$ is nearly, but not accurately, fulfilled, and that $y_{0}$ is the

Fig. 37.

current through the galvanometer when $O$ and $B$ are connected by a conductor of no sensible resistance, and $y_{1}$ the current when $O$ and $B$ are completely disconnected.

To find these values we must make $\beta$ equal to $0$ and to $\infty$ in the general formula for $y$, and compare the results.

In this way we find

${\frac {y_{0}-y_{1}}{y}}={\frac {\alpha }{\gamma }}{\frac {c\gamma -a\alpha }{(c+\alpha )(\alpha +\gamma )}}$,

where $y_{0}$ and $y_{0}$ are supposed to be so nearly equal that we may, when their difference is not in question, put either of them equal to $y$, the value of the current when the adjustment is perfect.

The resistance, $c$, of the conductor $AB$ should be equal to $a$, that of the battery, $\alpha$ and $\gamma$, should be equal and as small as possible, and $b$ should be equal to $a+y$.

Since a galvanometer is most sensitive when its deflexion is small, we should bring the needle nearly to zero by means of fixed magnets before making contact between $O$ and $B$.

In this method of measuring the resistance of the battery, the current in the battery is not in any way interfered with during the operation, so that we may ascertain its resistance for any given strength of current, so as to determine how the strength of current effects the resistance.

If $y$ is the current in the galvanometer, the actual current through the battery is $x_{0}$ with the key down and $x_{1}$ with the key up, where

$x_{0}=y\left(1+{\frac {b}{\alpha +\gamma }}\right),\qquad x_{1}=y\left(1+{\frac {b}{\gamma }}+{\frac {\alpha c}{\gamma (\alpha +c)}}\right),$

the resistance of the battery is

$a={\frac {c\gamma }{\alpha }}$,

and the electromotive force of the battery is

$E=y\left(b+c+{\frac {c}{\alpha }}(b+\gamma )\right)$.

The method of Art. 356 for finding the resistance of the galvanometer differs from this only in making and breaking contact between $O$ and $A$ instead of between $O$ and $B$, and by exchanging $\alpha$ and $\beta$ we obtain for this case

${\frac {y_{0}-y_{1}}{y}}={\frac {\beta }{\gamma }}{\frac {c\gamma -b\beta }{(c+\beta )(\beta +\gamma )}}$.

On the Comparison of Electromotive Forces.

358.] The following method of comparing the electromotive forces of voltaic and thermoelectric arrangements, when no current passes through them, requires only a set of resistance coils and a constant battery.

Let the electromotive force $E$ of the battery be greater than that of either of the electromotors to be compared, then, if a sufficient

Fig. 38.

resistance, $R_{1}$, be interposed between the points $A_{1},\,B_{1}$ of the primary circuit $EB_{1}A_{1}E$, the electromotive force from $B_{1}$ to $A_{1}$ may be made equal to that of the electromotor $E_{1}$. If the electrodes of this electromotor are now connected with the points $A_{1},\,B_{1}$ no current will flow through the electromotor. By placing a galvanometer $G_{l}$ in the circuit of the electromotor $E_{1}$, and adjusting the resistance between $A_{1}$ and $B_{1}$, till the galvanometer $G_{1}$ indicates no current, we obtain the equation

$E_{1}=R_{1}C$,

where $R_{1}$ is the resistance between $A_{1}$ and $B_{1}$. and $C$ is the strength of the current in the primary circuit.

In the same way, by taking a second electromotor $E_{2}$ and placing its electrodes at $A_{2}$ and $B_{2}$, so that no current is indicated by the galvanometer $G_{2}$,

$E_{2}=R_{2}C$,

where $R_{2}$ is the resistance between $A_{2}$ and $B_{2}$. If the observations of the galvanometers $G_{l}$ and $G_{2}$ are simultaneous, the value of $C$, the current in the primary circuit, is the same in both equations, and we find

$E_{1}:E_{2}::R_{1}:R_{2}$.

In this way the electromotive force of two electromotors may be compared. The absolute electromotive force of an electromotor may be measured either electrostatically by means of the electrometer, or electromagnetically by means of an absolute galvanometer.

This method, in which, at the time of the comparison, there is no current through either of the electromotors, is a modification of Poggendorff's method, and is due to Mr. Latimer Clark, who has deduced the following values of electromotive forces:

					Concentrated solution of		Volts.
Daniell	I.	Amalgamated Zinc	HSO4 +	4 aq.	Cu SO 4	Copper	= 1.079
	II.	"	HSO4 +	12 aq.	Cu SO4	Copper	= 0.978
	III.	"	HSO4 +	12 aq.	Cu NO6	Copper	= 1.00
Bunsen	I.	"	"	"	H NO6	Carbon	= 1.964
	II.	"	"	"	sp. g. 1.38	Carbon	= 1.888
Grove		"	HSO4 +	4 aq.	H NO6	Platinum	= 1.956
A Volt is an electromotive force equal to 100,000,000 units of the centimetre-gramme-second system.

1. Phil. Mag., July, 1870.
2. This investigation is taken from Weber's treatise on Galvanometry. Göttingen Transactions, x. p. 65.
3. Proc. R. S., June 6, 1861.
4. Laboratory. Matthiessen and Hockin on Alloys.
5. Proc. R. S,, Jan. 19, 1871.
6. Proc. R. S., Jan. 19, 1871