METHODS OF MAINTAINING AN ELECTRIC CURRENT

Excerpts

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  The electrification is here produced between the surfaces of two different substances, such as glass and amalgam or ebonite and fur.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  The electromotive force of the machine is the excess of the potential of the comb above that of the rubber. The most convenient test of the electromotive force of an electrical machine is the length of the sparks which it will give.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  I do not propose it as a useful form of machine, but as an example of the method by which the contrivance called in heat-engines a regenerator may be applied to an electrical machine to prevent loss of work.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  A great number of the experiments by which Coulomb established the fundamental laws of electricity were made by measuring the force between two small spheres charged with electricity, one of which was fixed while the other was held in equilibrium by two forces, the electrical action between the spheres, and the torsional elasticity of a glass fibre or metal wire.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Besides this, by connecting the guard-ring with a metal case surrounding the back of the attracted disk and all its suspending apparatus, the electrification of the back of the disk is rendered impossible, for it is part of the inner surface of a closed hollow conductor all at the same potential.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  In the ordinary frictional electrical machine the work done in overcoming friction is far greater than that done in increasing the electrification. Hence any arrangement by which the electrification may be produced entirely by mechanical work against the electrical forces is of scientific importance if not of practical value.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  In order to determine large differences of potential in absolute measure we may employ the attracted disk electrometer, and compare the attraction with the effect of a weight.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  If the conductor is not large compared with the electrometer, K’ will be comparable with K, and unless we can ascertain the values of K and K’ the second term of the expression will have a doubtful value.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Since the sphere is not electrified it will be at the potential of the air at the place.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  This method of using an auxiliary electrification besides the electrification to be measured is called the Heterostatic method of electrometry, in opposition to the Idiostatic method in which the whole effect is produced by the electrification to be measured.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  If there be a hollow nearly surrounded by the conductor, then the potential at any point of the air in this hollow will be very nearly that of the conductor.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  If by any means we can cause a succession of small bodies to detach themselves from the end of the electrode, the potential of the electrode will approximate to that of the surrounding air. This may be done by causing shot, filings, sand, or water to drop out of a funnel or pipe connected with the electrode. The point at which the potential is measured is that at which the stream ceases to be continuous and breaks into separate parts or drops.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  To ascertain the potential of a charged conductor of finite size we may connect the conductor with one electrode of the electrometer, while the other is connected to earth or to a body of constant potential. The electrometer reading will give the potential of the conductor after the division of its electricity between it and the part of the electrometer with which it is put in contact.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  But if we can make the potential of the electrode of the electrometer very nearly equal to that of the body before making contact, then the uncertainty of the values of K and K’ will be of little consequence.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Now let the sphere thus discharged be carried to the end of the wire and made to touch it. Since the sphere is not electrified it will be at the potential of the air at the place. If the electrode wire is at the same potential it will not be affected by the contact, but if the electrode is at a different potential it will by contact with the sphere be made nearer to that of the air than it was before. By a succession of such operations, the sphere being alternately discharged and made to touch the electrode, the potential of the electrode of the electrometer will continually approach that of the air at the given point.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  To measure the potential of a conductor without touching it, we may measure the potential of the air at any point in the neighbourhood of the conductor, and calculate that of the conductor from the result. If there be a hollow nearly surrounded by the conductor, then the potential at any point of the air in this hollow will be very nearly that of the conductor.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  In this way it has been ascertained by Sir W. Thomson that if two hollow conductors, one of copper and the other of zinc, are in metallic contact, then the potential of the air in the hollow surrounded by zinc is positive with reference to that of the air in the hollow surrounded by copper.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Thus the rotation of the machine carries the positive electrification of the surface of the glass from the rubber to the comb, and the negative electric wind of the comb either neutralizes the positively electrified surface, or is carried round with it to the rubber, so that there is a continual current of positive electricity kept up from the rubber to the comb, or, what is the same thing, of negative electricity from the comb to the rubber, or, since the mode of expressing the fact is indifferent, we may, if we please, describe it as consisting of a positive current in the one direction combined with a negative current in the other the arithmetical sum of these two imaginary currents being the actual current observed.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  In order that the machine may work to the best advantage this slipping back of the electricity must be prevented.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  It was by means of the revolving doubler that Volta succeeded in developing from the electrification of the pile an electrification capable of affecting his electrometer.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  On the other hand, the quantity pU - qV continually increases, so that, however little pU may exceed or fall short of qV at first, the difference will be increased in a geometrical ratio in each revolution till the electromotive forces become so great that the insulation of the apparatus is overcome.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  This conductor C’, by which the carrier is enabled to be connected to earth without a spark, answers to the contrivance called a regenerator in heat-engines. We shall therefore call it a Regenerator.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  In all electrometers it is of the greatest importance to know what force we are measuring. The force acting on the suspended sphere is due partly to the direct action of the fixed sphere, but partly also to the electrification, if any, of the sides of the case.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  When the distance is too small a small change of absolute distance makes a great change in the force, since the force varies inversely as the square of the distance, so that any error in the absolute distance introduces a large error in the result unless the distance is large compared with the limits of error of the micrometer screw.

Equations

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  EE_1 aa_1\sin\theta\left\{\frac{1}{r^3}-\frac{1}{b^3}\right\}=M(\theta-\phi)

  With the spherical case large compared with the spheres' distances, the total moment from the charges and their image balances the torsional moment of the fibre.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  z = -QU

  The charge on the carrier, while it is in contact with earth so that its potential is zero, is minus Q times the potential of the inductor it was in.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  (C'+c')U+C'V'=aV

  Charge on the carrier while it is within the inductor C, with its capacity C' + c' and coefficient of induction -C'.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  C'V'=aV

  Condition under which the carrier's potential is reduced to zero at the regenerator, so it can touch the earth spring without a spark.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  C'V'=aV\text{,\quad and\quad}CV = a'V'

  The two conditions the regenerators must satisfy so that the carrier can touch the earth spring without a spark at either regenerator.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Fa \cos \tfrac{1}{2} \theta = M ( \theta - \phi )

  Balance of the torsion arm: the moment of the electric force about the axis of torsion equals the torsional moment of the fibre acting through the twist theta minus phi.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  M = \frac{4\pi^2I}{T^2}

  The moment of torsion equals four pi squared times the moment of inertia divided by the square of the period of a double vibration.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  =\frac{EE_1}{r^2}

  The repulsion between the two small spheres, neglecting induction on them, equals the product of their charges divided by the square of the distance between them.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  \frac{EE_1aa_1 \sin \theta}{ r^3}

  The moment of that repulsion about the vertical axis through the centre of motion.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  = EE_1 \frac{aa_1\sin\theta}{b^3 \left\{ 1 - 2 \dfrac{aa_1}{b^2} \cos \theta + \dfrac{a^2{a_1}^2}{b^4} \right\}^\frac{3}{2}}

  The moment about the axis of the attraction between the suspended sphere and the image of the fixed sphere in the spherical case, which is the second line of the displayed calculation.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Wg &= \frac{V^2A}{8 \pi D^2}

  The weight W times gravity g balances the attraction between the disks, which is V squared times the area A over 8 pi D squared.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V = D\, \sqrt{\frac{8 \pi gW}{A}}

  The potential difference between the disks in terms of the distance, the weight and the disk area.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  A = \tfrac{1}{2} \pi (R^2 + R'^2)

  For a circular suspended disk of radius R in a guard-ring aperture of radius R', the effective area is half pi times the sum of the squares of the two radii.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V = 4D\, \sqrt{ \frac{gW}{R^2 + R'^2}}

  The potential difference between the disks for a circular suspended disk inside a guard-ring.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  \alpha = B \frac{log_e2}{\pi}

  Defines alpha, a correction length, as the breadth B of the annular gap times log base e of 2 over pi.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  \alpha = 0.220635 (R' - R)

  Numerical value of the correction length alpha for the guard-ring gap, to the stated precision.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Q &= V \left\{ \frac{R^2 + R'^2}{8D} - \frac{R'^2 - R^2}{8D} \frac{\alpha}{D + \alpha} \right\}

  The charge on the suspended disk when the guard-ring surface lies in the same plane as the disk, for potential difference V between the disks.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Q = V \left\{ \frac{R^2 + R'^2}{8D} - \frac{R'^2 - R^2}{8D} \frac{\alpha}{D + \alpha} + \frac{R+R'}{D}(D'-D) \log_e \frac{4 \pi(R+R')}{D'-D} \right\}

  The charge on the suspended disk when the guard-ring surface is not exactly in the plane of the disk, with an extra term for the height difference z.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  A = \tfrac{1}{2} \pi \left\{R^2 + R'^2 -(R'^2 - R^2) \frac{\alpha}{D + \alpha} + 8 (R + R')(D' - D) \log_e \frac{4 \pi (R + R')}{D' - D} \right\}

  The corrected effective area of the suspended disk, to be used in the attraction formula when the guard-ring is not exactly in the plane of the disk.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V - V' = (D - D') \sqrt{ \frac{8 \pi g W}{A}}

  The difference of two potentials is found from the difference of the two disk distances at which the suspended disk is in its sighted position, so that the uncertain zero of D is avoided.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  \overline{V} = \frac{KV + K'V'}{K + K'}\text{.}

  After a conductor of capacity K at potential V is put in contact with an electrometer part of capacity K' at potential V', their common potential is the capacity-weighted mean (KV + K'V')/(K + K').

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V = \overline{V} + \frac{K'}{K} (\overline{V} - V')\text{.}

  The original potential of the conductor can be recovered from the measured common potential, the capacities, and the electrometer's original potential.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Q = V'a\text{,}

  The charge on the sphere equals its radius times the potential the electrometer reads for the sphere after it is earthed, insulated and carried into the room.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V + V' = 0\text{,}

  The potential of the air at the sphere's centre is equal in magnitude and opposite in sign to the sphere's measured potential, so the two sum to zero.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  -Va = Q

  The charge on the sphere, placed with its centre at a point of potential V, equals minus the potential times the sphere's radius.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  p^2 = \xp\dfrac {Q}{B}

  Defines p as the square root of the coefficient of induction Q divided by the capacity B.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  q^2 = \xp\dfrac {Q'}{A}

  Defines q as the square root of the coefficient of induction Q' divided by the capacity A.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  U_{n + 1} &= U_n - \frac {Q'}{A} V_n

  The potential of inductor A after n+1 half revolutions equals its potential after n half revolutions minus Q'/A times the potential of C.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V_{n + 1} &= V_n - \frac {Q}{B} U_n

  The potential of inductor C after n+1 half revolutions equals its previous potential minus Q/B times the potential of A.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  pU_{n + 1} + qV_{n + 1} &= \left(pU_n + qV_n \right) \left( 1 - pq \right) = \left(pU_0 + qV_0 \right) \left( 1 - pq \right)^{n + 1}

  The combination pU + qV is multiplied by (1 - pq) at each half revolution, so after n+1 half revolutions it equals its initial value times (1 - pq) to the power n+1.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  pU_{n + 1} - qV_{n + 1} &= \left( pU_n - qV_n \right) \left( 1 + pq \right) = \left( pU_0 - qV_0 \right) \left( 1 + pq \right)^{n + 1}

  The combination pU - qV is multiplied by (1 + pq) at each half revolution, so it grows geometrically with the number of revolutions.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  U_n &= U_0 \left\{ ( 1 - pq )^n + ( 1 + pq )^n \right\} + \frac{q}{p} V_0 \left\{ ( 1 - pq )^n - ( 1 + pq )^n \right\}

  Closed form for the potential of inductor A after n half revolutions, in terms of the initial potentials.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V_n &= \frac{p}{q} U_0 \left\{ ( 1 - pq )^n - ( 1 + pq )^n \right\} + V_0 \left\{ ( 1 - pq )^n + ( 1 + pq )^n \right\}

  Closed form for the potential of inductor C after n half revolutions, in terms of the initial potentials.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  C'V'=aV\text{,}

  Condition that the carrier, leaving the regenerator C, is brought to zero potential so it can touch the earth spring without a spark.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  (C'+c')U+C'V'=aV\text{.}

  The charge on the carrier when it is in regenerator C equals aV, where U is the carrier's potential.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  -\frac{A'V'+BV}{A'+a'}\text{,}

  The potential the carrier would have at the middle of receiver A' if it kept its charge, found from the charges and capacities involved.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  C'V'=aV\text{,\quad and\quad}CV = a'V'\text{.}

  The two conditions that the regenerators must satisfy so that the carrier leaves each receiver at zero potential without sparking.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Fa \cos \tfrac{1}{2} \theta = M ( \theta - \phi )\text{.}

  The electric force between the spheres, times the lever arm, balances the torsional moment of the fibre, which is proportional to the twist angle theta minus phi.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  M = \frac{4\pi^2I}{T^2}\text{.}

  The moment of torsion equals four pi squared times the moment of inertia of the arm divided by the square of the period of its double vibration.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  EE_1 aa_1\sin\theta\left\{\frac{1}{r^3}-\frac{1}{b^3}\right\}=M(\theta-\phi)\text{.}

  With the spherical case included, the total moment tending to turn the torsion-arm equals the torsional moment M times the twist angle theta minus phi.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V = 4D\, \sqrt{ \frac{gW}{R^2 + R'^2}}.

  For a circular suspended disk inside a guard-ring, the potential difference between the disks equals 4D times the square root of gW over R squared plus R' squared.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Wg &= \frac{V^2A}{8 \pi D^2}\text{,}

  The attraction between the two disks, of area A at distance D and potential difference V, equals the weight W times gravity g that balances it.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V &= D\, \sqrt{\frac{8 \pi gW}{A}}\text{.}

  The potential difference between the disks equals D times the square root of 8 pi g W over A.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  V - V' = (D - D') \sqrt{ \frac{8 \pi g W}{A}}\text{.}

  The difference of two potentials equals the difference of the corresponding disk distances times the same constant factor, so only differences of readings are needed.

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  v = (D - D') \sqrt{ \frac{8 \pi gW}{A}}\text{.}

  The electromotive force v of the battery equals the difference of the micrometer readings times the constant factor sqrt(8 pi g W / A).

• METHODS OF MAINTAINING AN ELECTRIC CURRENT

  Q &= V \left\{ \frac{R^2 + R'^2}{8D} - \frac{R'^2 - R^2}{8D} \frac{\alpha}{D + \alpha} \right\}\text{,}

  The quantity of electricity on the suspended disk, with the guard-ring present, in terms of the potential difference V and the disk dimensions.

Problems

No exercises in this chapter.