THE MEASUREMENT OF ELECTRIC RESISTANCE

Excerpts

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  The remaining difference between and 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 MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC 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 MEASUREMENT OF ELECTRIC RESISTANCE

  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 affects the resistance.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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:

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  In the electromagnetic system a resistance is a quantity homogeneous with a velocity, and may therefore be expressed as a velocity.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  The coils and 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 and has produced no change in the proportions of the resistances, and therefore is rightly adjusted.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  By the method now described the galvanometer itself is employed to measure its own resistance.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  This method, as has been pointed out by Professor Oliver Lodge, is not free from error on account of the variation of the E.M.F. of the battery, as the current through it is diminished or increased by raising or depressing the key.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  Let the electromotive force E of the battery be greater than that of either of the electromotors to be compared, then, if a sufficient resistance, R_1, be interposed between the points A_1, B_1 of the primary circuit EB_1A_1E, 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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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 MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  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.

Equations

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  E = IR = I_1( R + r_1 ) = I_2( R + r_2 )

  The battery's electromotive force equals the current times the total circuit resistance, and so it also equals each current times the resistance with the extra resistance added.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \frac{r_1}{r_2} = \frac{(I-I_1)I_2}{(I-I_2)I_1}

  The ratio of two unknown resistances is found from the ratios of the measured currents, with the battery's own resistance and emf eliminated.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \delta = mI_1 - nI_2

  The deflexion of the differential galvanometer needle is the difference between the two coil currents, each weighted by its coil constant.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  C-D = I_1(A+\alpha) = I_2(B+\beta) = E-Ir

  The potential difference between C and D equals the current in each galvanometer branch times that branch's total resistance, and also equals the emf less the drop across the battery's internal resistance.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  I_1 + I_2 = I

  The battery current divides between the two galvanometer coils so that the two branch currents add up to the total current.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  I_1 = E\,\frac {B + \beta}{D}

  The current through the first coil equals the emf times the second branch's total resistance divided by the common denominator D.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  D is the common denominator for the branch currents, built from the two coil branch resistances and the battery resistance.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The galvanometer deflexion is the emf over D times the difference between the two weighted branch resistances; zero deflexion means that bracket is nearly zero.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The difference between the two substituted resistances is set by the two observed deflexions, each corrected by its own D and emf.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  After exchanging the resistances A and B, the difference between them is set by the two observed deflexions, with the sum of the coil constants as a factor.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  When the coil constants and resistances are nearly equal, the difference B minus A is approximately given by the smallest observable deflexion difference.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The galvanometer coil resistance alpha that minimises the sensitivity-error factor is given by this expression in terms of A and the battery resistance r.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \alpha = \tfrac{1}{3} A

  When the battery resistance r is small compared with A, each galvanometer coil should have one-third of the resistance to be measured.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  With the optimal galvanometer coil resistance, the difference B minus A is approximately 8/9 times A squared over nE, times the difference of the smallest observable deflexions.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \Delta = \frac{mE}{A + \alpha + r} = \frac{3}{4}\frac{nE}{A}

  The deflexion from one coil alone is mE over the total circuit resistance; with r zero and alpha one-third of A, this equals three-quarters of nE over A.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The fractional error in the comparison of A and B is two-thirds of the smallest observable deflexion difference divided by the single-coil deflexion.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  O = \frac{B \gamma + C \beta}{ \beta + \gamma}

  With no current in OA, the potential at O is the average of the potentials at B and C, weighted by the opposite resistances beta and gamma.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  A = \frac{Bb + Cc}{b + c}

  With no current in OA, the potential at A is the average of the potentials at B and C, weighted by the opposite resistances c and b.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  b \beta = c \gamma

  The bridge is balanced, with no current in OA, when the products of opposite arm resistances are equal, so the two conductors are conjugate.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  x_1 = y \left(1 + \frac{b}{\alpha + \gamma} \right)

  With the key up, the actual current through the battery equals the galvanometer current times one plus b over alpha plus gamma.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \frac{c + x}{b - x} = \frac{\beta}{\gamma}

  With the coils in their first position, the bridge balances when the ratio of the two arm resistances equals the ratio of the standard coil to the coil being adjusted.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \frac{c + y}{b - y} = \frac{\gamma}{\beta}

  With the coils beta and gamma interchanged, the bridge balances at the second scale reading y, with the ratio of arm resistances now equal to gamma over beta.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  Combining the two balance conditions gives the ratio of the squares of the two resistances in terms of the two scale readings and the arm resistances.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  When b and c are nearly equal and large compared with x and y, the ratio of the squares of the resistances is approximately one plus four times the scale difference over b plus c.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  To first order, the resistance to be adjusted equals the standard resistance times one plus twice the scale difference over b plus c.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  c=\sqrt{a\alpha}

  The best value of the bridge arm c, given the battery and galvanometer resistances, is the square root of their product, as shown by Heaviside.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  b=\sqrt{a\gamma\frac{\alpha + \gamma}{a + \gamma}}

  The best value of the bridge arm b, given the battery, galvanometer, and measured resistances, is the square root of a gamma times the ratio of alpha plus gamma to a plus gamma.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \beta = \sqrt{\alpha \gamma \frac{a + \gamma }{\alpha + \gamma}}

  The best value of the standard coil beta, given the battery, galvanometer, and measured resistances, is the square root of alpha gamma times the ratio of a plus gamma to alpha plus gamma.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  When the galvanometer in CA gives the same deflexion whether or not O and A are joined, the galvanometer resistance b equals c times gamma over beta.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  y = \frac{E\alpha}{b\alpha + c(b+\alpha+\gamma)}

  When the bridge condition is satisfied, the current through the galvanometer equals the battery emf times alpha divided by a combination of the arm resistances, and does not depend on beta.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \frac{y_0-y_1}{y} = \frac{\alpha}{\gamma}\frac{c\gamma-a\alpha}{(c+\alpha )(\alpha+\gamma)}

  The sensitivity of Mance's method: the relative change in galvanometer current between the two switch positions is proportional to the departure of c gamma from a alpha.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  c\gamma=a\alpha

  The condition under which the galvanometer deflexion is unchanged when O and B are connected or disconnected: the product of c and gamma equals the product of a and alpha.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  x_0 = y \left(1 + \frac{b}{\gamma} + \frac{\alpha c}{\gamma (\alpha + c)} \right)

  With the key down, the actual current through the battery equals the galvanometer current times a sum of terms in the bridge resistances.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The resistance of the battery equals c times gamma divided by alpha.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The electromotive force of the battery equals the galvanometer current times a combination of the bridge resistances.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \frac{y_0 - y_1}{y} = \frac{\beta}{\gamma} \frac{c \gamma - b \beta}{(c + \beta)(\beta + \gamma)}

  For the galvanometer-resistance arrangement, obtained from the battery case by exchanging alpha and beta, the relative change in galvanometer current has the same form with b beta in place of a alpha.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  E_1 = R_1C

  When no current flows through the electromotor E1, its emf equals the resistance R1 between A1 and B1 times the primary-circuit current C.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  E_2 = R_2C

  When no current flows through the electromotor E2, its emf equals the resistance R2 between A2 and B2 times the primary-circuit current C.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  E_1 : E_2 :: R_1 : R_2

  Two electromotive forces are in the same ratio as the resistances that balance them against the same primary current.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  E = IR = I_1( R + r_1 ) = I_2( R + r_2 )\text{.}

  By Ohm's law, the battery's electromotive force equals the current times the total circuit resistance in each of the three states: no added resistance, and with r_1 or r_2 added.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \frac{r_1}{r_2} = \frac{(I-I_1)I_2}{(I-I_2)I_1}\text{.}

  The ratio of two resistances can be found from the ratio of the currents measured with each one in the circuit, without knowing E or R.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \delta = mI_1 - nI_2\text{.}

  The deflexion of the differential galvanometer needle is the difference between the two coil currents, each weighted by its coil constant.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The needle deflexion in the differential galvanometer arrangement is proportional to the bracketed difference of the coil-resistance combinations, so zero deflexion means that bracket vanishes.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  D = (A + \alpha)(B + \beta) + r(A + \alpha + B + \beta)\text{.}

  D is defined as the combined resistance quantity of the differential galvanometer circuit, built from the two coil branches and the battery resistance r.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  When the resistances and coil constants are approximately equal, the difference B - A is approximately given by the smallest observable deflexion difference times a factor depending on the circuit.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \alpha = \tfrac{1}{3} A\text{;}

  For a battery of negligible resistance, the galvanometer coil resistance should be one-third of the resistance being measured.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The fractional difference between the two resistances is two-thirds of the smallest observable deflexion divided by the deflexion produced by one coil alone.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  \Delta = \frac{mE}{A + \alpha + r} = \frac{3}{4}\frac{nE}{A}\text{ if }r = 0\text{ and }\alpha = \frac{1}{3} A\text{.}

  The deflexion produced by one galvanometer coil alone is mE over the total resistance, which reduces to three-quarters of nE/A when r is zero and alpha is one-third of A.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  b \beta = c \gamma\text{,}

  The bridge gives no current in the galvanometer branch when the product of the opposite arm resistances b and beta equals the product of c and gamma.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  O = \frac{B \gamma + C \beta}{ \beta + \gamma}\text{,}

  The potential at O is the weighted average of the potentials at B and C, weighted by the opposite resistances.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The current along OA in the bridge is proportional to the departure of the bridge from balance, b beta minus c gamma.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  D = abc + bc(\beta+\gamma)+ca(\gamma+\alpha)+ab(\alpha+\beta)+(a+b+c)(\beta\gamma+\gamma\alpha+\alpha\beta)

  The bridge determinant D is written in a symmetrical form in the six conductor resistances.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  G = n(1 - n)(R + S)\text{.}

  The best galvanometer resistance for a bridge with m equal to n is n(1 - n) times the sum of the total bridge resistance R and the total resistance S of BOC.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  For a battery of given total electrode area, the most advantageous battery resistance is RS over R plus S.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The value of S that makes a given change in n produce the greatest galvanometer deflexion is found from this relation, obtained by differentiating the current expression.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

  G = 2n(1-n)R\text{,}

  For many determinations of nearly equal resistance, the best galvanometer resistance is 2n(1-n) times R, with S equal to R and B equal to half of R.

• THE MEASUREMENT OF ELECTRIC RESISTANCE

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

  The deviation of the galvanometer needle is proportional to the square root of the galvanometer resistance times the current in its wire.

Problems

No exercises in this chapter.