ELECTRO-MAGNETIC INDUCTION IN WATER

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ELECTRO-MAGNETIC INDUCTION INWATERH. D. Einhorn Dr.Ing. & E. NewberyPublished online: 01 Apr 2010.

To cite this article: H. D. Einhorn Dr.Ing. & E. Newbery (1940) ELECTRO-MAGNETIC INDUCTION INWATER, Transactions of the Royal Society of South Africa, 28:2, 143-160

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ELECTRO-MAGNETIC INDUCTION IN WATER.

By H. D. EINHORN, Dr.Ing.

(Communicated by E. NEWBERY.)

(With eight Text-figures.)

(Read August 16, 1939.)

CONTENTS. PARE

I. INTRODUCTION . . . 143

11. METHOD: (1) Arrangement. . . 144 (2) Reason for Method. . 146 (3) Magnetic Field . . 146 (4) Transformer E.M.Fs. . 146 (5) Valve Voltmeter and Har-

monics . . . 148 (6) Procedure and Computa-

tion . . . . 149

111. RESULTS : (1) Induced Voltage and

(a) Flow Speed . . . 150 ( 6 ) Magnetic Field . 150

(c) Electrode Distance. (d) Quantitative Rela-

tions . . . (2) Conductivity . . . (3) Position of Electrodes . (4) Different Electrodes . (5) Oscillography. . .

IV. PRACTICAL APPLICATIONS . V. CONOLUSION . . . .

APPENDIX 1. ON TRANSFORMER E.M.Fs. . . . .

APPENDIX 2. CALCULATION OF

CIRCULATING CURRENTS . APPENDIX 3. CORRECTION FOR

ERROR INTRODUCED BY DOUBLE-EARTHING . .

PAQE 150

151 152 152 152 153 154 165

155

156

159

I. INTRODUCTION. The electro-magnetic laws as applied to solid metal conductors are

the fundamentals of electrical engineering, well known and verified in experiment and practice every day since they were stated by Faraday.

How the same laws are to be applied to fluid %on-metallic conductors has hardly been investigated,* and it is the object of this paper to answer questions such as: What voltage is induced in an electrolyte flowing in a magnetic field 1 What are the quantitative relations 1

Can we measure this voltage '?

* Experiments on gaseous and solid non-conductors moving in a magnetic field have been carried out by R. Blondlot (C.R., 133, 1901, p. 778); H. A. Wilson (Phil. Trans., A, 204, 1905, p. 121); L. Slepian (Ann. d. Phys., 45, p. 861).

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144 Transactions of the Royal Society of South Africa.

It is by no means obvious what to expect. While the opinion * has been voiced that “the free movement of ions may produce voltages higher than in an equivalent width of metal conductor,” the opposite t is that “with short poles, the ions, due to their limited mobility, may hardly have had time to move before they are beyond the field, and the p.d. set up would then be very small.”

The latter view can be illustrated as follows:- Suppose water moves in a uniform magnetic field of intensity B (Gauss)

a t a speed v (cm./sec.), then the induced electric field E (V/cm.) will be B V ~ O - ~ and will cause a lateral velocity v’ of the ions = mBv10-8 cm./sec. where rn is the mobility of the ions, which in the case of the H ion may be taken as 3x10-3 cm./sec.p.V/cm.$ Hence, in a field of 15,000 gauss, d = 2 x lo-%, i.e. the angle of drift would be very small and a 5-cm. electrode would receive ions only from a mm. thick zone close by; but as the flow speed in this boundary layer is unknown (presumably very small) the ultimate effect is doubtful.

On the other hand, calculations carried out by the author (Appendix 2) lead to the result that the electric field set up is independent of the conductivity, hence independent of the mobility of the ions.

An experimental investigation seemed to be worth while for the sake of obtaining a definite answer to the problem.

The only experiments hitherto, as far as the author knows, have been conducted by Baradays who tried to measure voltages induced in the Thames due to the magnetic field of the earth. His results were in- conclusive and the experiment was soon abandoned. The effect of disturbances, such as motor-electrolytic actions, was in his case probably excessive, and the means to measure very small voltages inadequate a t his time.

The experiments reported in this paper have been carried out on a smaller scale under laboratory conditions.

11. METHOD. (1) Arrangement.-The experimental arrangement is shown in fig. 1.

The fluid runs from a drum containing an indicating float through a rubber pipe to a flattened glass tube, which is placed in the airgap of an electro- magnet. The potential difference between two electrodes inserted into

H. J. Northfield, Electrician, 1936, p. 40. t G. W. 0. Howe, Electrician, 1936, p. 14. In metals, the mobility of electrons is about 104 times greater (see Pohl, Physical

Diary, January, 12th to 14th, 1832. Principles of Electricity and Magnetism, p. 250, 1930).

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Electro-Magnetic Induction in Water. 145

the glass tube a t right angles to the directions of flow and of magnetic field is measured.

The flow speed was determined by meam of the level indicator and a

R I1 I <

I

- - - - . 4 a

r? \ I OUCf/a w I I

Fro. 1 (Container drawn to half the scale of magnet).

stop-watch, the cross-sectional areas of the drum (A) and of the flattened part of the glass tube (a) being known.

If the indicator travels the distance Ax cm. in At seconds, then the flow speed between the magnet jaws is

A Ax Ax v = - -=go- . 2a At At

(From measurements A = 1000 cm.2; a = 5.55 cm.z.)

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(2) Reasons for Measuring Method adopted-The main difficulties

(i) The voltages to be expected are small (a few millivolts), since the %ow speed and the magnetic field intensity are limited.

(ii) The internal resistance of this “low voltage generator” is high (a few thousand ohms). Therefore the amount of power available to feed an instrument is exceedingly small.

(iii) Disturbances, such as electrolytic and motor-electrolytic effects threaten to supersede the electro-magnetic effect to be investigated.

foreseen are as follows:-

These difficulties were overcome by employing an A.C. Valve-Voltmeter for measuring the E.M.Fs. and using an Alternating magnetic field. The Valve-Voltmeter combines the advantages of sensitivity and very small power consumption.

The use of an alternating field enables electrolytic effects to be eliminated * and facilitates the design of the Valve Voltmeter.

The drawback in using A.C. is that disturbing E.M.Fs. are induced in the Voltmeter leads (transformer effect), which have to be compensated.

(3) Magnetic Field.-The electro-magnet is fed from the 50 c / s muni- cipal supply over isolating transformers in such a way that the mean potential level of the windings could be adjusted with respect to the earth potential. The field has been investigated by means of search coils, connected to the Valve Voltmeter, resp. to an oscillograph.

The field distribution along the %ow axis is shown in fig. 7 (dotted line). The time variation of the field is almost sinusoidal. A small third

harmonic does not affect the flow induced voltage much; its disturbing effect by direct induction into the voltmeter circuit is discussed below.

(4) Compensating of Transformer E.M.Fs.-A difficulty due to the use of an A.C. Field has been mentioned above; the voltmeter leads act as the secondary of a transformer and voltages are induced which have to be eliminated. Two methods are possible.

(a) The voltmeter leads are arranged in such a way as not to enclose any flux. That means taking one connection up the airgap and making it mechanically adjustable. The occupation of valuable airgap space by connection leads is a disadvantage, of course, otherwise this solution has proved t o be simple and practicable.

(a) Two leads are connected to the one electrode, one being taken out on each side of the magnet; they are shorted over a resistance, and the

As an additional safeguard against motor-electrolytic effects, silver electrodes, coated with silver chloride, were used for the greater part of the experiments (see Newbery and Smith, Trans. Electrochemicd Society, vol. 73, 1938, p. 266).

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-

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FIQ

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voltmeter terminal is connected to the rheostate slider, which is adjusted to obtain compensation (see fig. 2 ~ ) . This method does not waste airgap space.

( 5 ) Valve Voltmeter and Harmonics.-A battery valve voltmeter con- sisting of two amplifier stages and one grid rectifier (fig. 2 A ) is employed to measure the induced voltages.

With a matching resistance of 5 megohms, the power absorbed with the instrument is negligible,

The sensitivity as seen from the calibration curves (fig. 2B) is well adapted to the expected voltage range.

A better approach to linearity may be achieved by employing three amplifier stages and a rectifier instrument or anode rectifier, but the

For explanation see Appendix 1.

-+ Reading.

no. B~.-Calibration curve.

arrangement used safeguards against over-load, as the incremental deflection of the indicating milliammeter decreases with increasing applied voltage.

The calibration was carried out with a sinusoidal 50 CIS voltage and the flow induced voltage is also practically sinusoidal. But i t was found that a third harmonic (ix. 150 c/s) voltage of about .08 mV appears across the valve voltmeter terminals when the magnet is switched on, which could not be eliminated by compensation. I ts value is independent of the flow speed, but as it is superimposed on the flow induced voltages it is apt to falsify the results unless a correction is made. To do this on grounds of theoretical consideration is hardly possible as the grid rectifier instrument reads neither RMS nor peak nor average values. The following experimental way was adopted:-

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Non-sinusoidal waves of known composition supplied from two independent 50 and 150 c/s sources were applied to the valve voltmeter and the influence of the third harmonic on the reading noted. The measured results were then corrected accordingly.

The phase angle of the harmonic had to be considered and was ascer- tained by means of the oscillograph.

(6) Procedure and Computations.-The valve voltmeter is switched on, the zero position adjusted (by means of resistance R, fig. 2 ~ ) . Then the magnet is switched on and a minimum deflection obtained by means of resistances R, and R, (in fig. 2 ~ ) , i .e . the transformer E.M.Fs. are compensated out, but for the mentioned third harmonic. After opening the water-tap a reading was taken, and after closing it compensation and zero adjustment checked up.

Although the valve voltmeter does not take any appreciable current the voltage measured is bound to be smaller than the E.M.F. generated, on account of circulating currents, which constitute a kind of internal loading. An estimate of the influence of internal circulating currents and of double-earthing on the terminal voltage is made in Appendixes 2 and 3.

111. RESULTS. (1) Blow Speed ; Magnetic Field; Electrode Distance.-The main con-

clusion can be stated in advance as follows: In contradiction to some doubts mentioned a t the outset, the E.M.F. induced in 0owing water is just as in metallic conductors:

E = B v d l x 10-8 Volts, s: where dl is an element of path between the electrodes in cm., B the magnetic 0ux density in gauss, v the flow speed in cm./sec.

Considering that in the arrangement investigated, B and v are practically constant in space between the electrodes, the induced E.M.F. is simply

E = B 1 v 10-8.

If B is alternating with time, E will be alternating too and of similar

A voltage V, equal to this E.M.F. diminished by internal voltage drops, wave form.

can be measured V = E k, k,,

where k, and k, are correction factors, slightly smaller than 1, to allow for the voltage drop due to internal circulating currents (k,) and due to double earthing (kJ. They can be estimated mathematically (see

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Appendixes 2 and 3) and are independent of B and v, but vary slightly with 1.

The measured voltage should therefore be proportional to the field and to the speed. The following experimental results corroborate these statements :-

(a) The induced voltage is directly proportional to the flow speed. This relation was established for different electrodes, different magnetic

'4 t

--3 Water Velocity (cm./sec.).

FIG. 3.-Induced Voltage and Flow Speed. (a ) 8 mm. electrodes on bare rods. ( b ) 8 mm. electrodes on insulated rods.

2.5 om. apart. B =800 Gau~s.

fields, different fluids, and different electrode distances for speeds from 10 to 80 cm./sec. The graph, fig. 3, shows two typical sets of readings (the difference between the two lines is explained below, 111, (4)).

( b ) The induced voltage is proportional to the mean field intensity (fig. 4 ~ ) . Each voltage-speed ratio value in figs. 4~ and 4~ is the mean value of a series of measurements a t different speeds (slope of line in fig. 3; the inaccuracy is due to the inaccuracy of the field measurements).

(c) The induced voltage is very nearly proportional to the electrode spacing (fig. 4 ~ ) . A perfect proportionality was not expected.

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( d ) Quantitative Relations.-In order to verify the above statement quantitatively, let us compare a calculated and measured voltage value

200 400 600 800 I --+ Flux Density (Gauss).

FIG. 4A.-Induced Voltage and Magnetic! Field. 8 mm. Electrodes on insulated rods, 1.95 cm. apart.

DO

I 2 3 4 -+ Electrode Distance (cm.).

FIG. 4~.-Induced Voltage and Electrode Distance. 8 mm. Electrodes on insulated rods. B =800 Gauss.

for the foIlowing data: B = 800 gauss; v = 50 cm.jsec. ; 1 = 2-5 cm. E = 1.00 mV.

Hence

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Circulating currents cause a voltage drop of 5 per cent. (see Appendix 2), hence k, = -95. To allow for double earthing in the case of small electrodes, 2.5 cm. apart, 8 correction factor k, =-90 is appropriate (see Appendix 3). Hence V = -86 mV.

The two values agree well within the accuracy achieved. The disagreement may, moreover, be explained by a slight excess of water speed between the electrodes over the average velocity on which the calculation is based.

This clears up the main questions asked a t the outset; but various other matters suggested themselves for investigation.

(2) Conductivity.-Comparative tests with tap water of 17,000 ohm cm., and a salt solution (NaCl) of only 200 ohm cm. specific resistance, show that the conductivity of the fluid does not influence the induced voltage, which is measured between the electrodes.

This verifies the theoretical result (Appendix 2) that the voltage drop due to internal circulating currents is independent of the conductivity. If an external load is applied, on the other hand, the terminal voltage is reduced and varies with the conductivity.

(3) Position of Electrodes i n Magnet Field.-The question was asked: How does the relative position of the electrodes and the magnet affect the measured voltage? It is to be expected that the voltage picked up a t the edge of the field is smaller than that a t the centre. But it has to be determined whether there is any difference between a displacement of the electrodes upstream or downstream with respect to the centre of the magnetic field.

For very high speeds and high conductivity fluids (say mercury) the magnetic field should be distorted * by the circulating currents so as to be virtually shifted downstream (an analogy to the “ armature reaction” in machines). But a quantitative estimate shows that this effect is exceedingly small in the present case where the speeds of flow are moderate and the resistivity high.

Another thought will be: Is there any lag in establishing an electric field, or will the charges be separated immediately the fluid enters the field? A trend of thought similar to that outlined in the introduction may lead to the fallacy that the sluggishness of the ions may cause some lag, but the experiment showed (within the limits of the accuracy achieved) that for a given displacement it makes no difference whether the electrodes are displaced upstream or downstream.

(4) Different Electrodes.-The following figures when studied in con-

The measured voltage (see fig. 3, curve b ) is V = 4 8 mV.

* This effect in a 2-mm. copper plate is extensively treated in H. Hertz’ thesis on But the conductivity is there loB “Induction in Rotating Spheres” (fig. 11, a and b) .

greater than in our case.

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junction with the electric field distribution (fig. 7) represent an additional, though rough, experimental check on the calculations in Appendix 2.

Measurements were made with the following electrodes :- (A) Round Silver-silver chloride-Disks (8 mm. diam.) a t the end of

l&-mm. silver rods, the rods insulated by a shellac coating. (B) Same as (A) but rods bare. (C) Flat Silver-silver chloride-Plates (55 mm. x 10 mm.) a t the end

(D) Brass Plates (55 mm. x 10 mm.), both ends slightly bent up, on of 1-mm. bare silver rods.

bare brass rods.

The following voltages have been measured with the electrodes 2.5 cm. apart, a t 800 gauss; a corrected value (see Appendix 3) is also given to allow for the influence of double earthing:-

Electrode. A. B. C. D.

Voltage per 1 m. per sec. accer. veloc. 1-77 2.02 1.67 1.77 mV. Corrected voltage per 1 m. per sec. . 1-97 2.25 14'7 1-88 mV.

The electrodes (A) do not distort the electric field, and the voltage between two points of the flow space is picked up correctly to be compared with calculated values.

With electrodes ( B ) a higher voltage is obtained as the electrode distance is virtually increased (see also fig. 3).

With electrodes (C) and (D) a smaller voltage is measured, partly because the influence of the rods is relatively smaller, partly because the increased circulating currents from the ends of the plates, which extend into regions of smaller magnetic field, cause a higher internal voltage drop.

The difference between (C) and (D) is mainly due to increased flow speed between the electrodes in the case of (D), but the fact that the bent- up electrodes lie more or less in an equipotential line of the undistorted field (see fig. 7, Appendix 2) may also influence the result.

It should be noted for practical applications that small electrodes pick up larger voltages than extended ones.

( 5 ) 0sciZlography.-By converting the grid rectifier part of the valve voltmeter into an additional amplifier stage and feeding over a 22 : 1 step- down transformer into a Duddell type oscillograph, oscillograms have been taken with a 100 mA loop (if a more sensitive loop had been available, a series resistance could have been used in place of the transformer).

The curves revealed the disturbances which could not be eliminated, as third harmonics, and by also showing their phase angle indicated the proper way of correcting the results.

The flow voltages themselves proved to be pretty nearly sinusoidal, the

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154

harmonics in the magnetic field being small. are discussed below.

Transactions of the Royal Society of South Africa.

Oscillograms of water surges

IV. PRACTICAL APPLICATIONS. Although the experimental arrangement constitutes the purest form of

a hydro-electric generator, any power applications are out of the question, mainly on account of the high resistivity of electrolytes and the resulting low efficiency; the difficulty of adding up the small voltages gained (problem of the unipolar machine) is a further drawback.

As the voltage is proportional to the speed and independent of conductivity fluctuations,

There is B greater scope in the field of measurement.

C.

. . . . . . . . . . . . . F I G . 5A.

‘P FIG. 5B.

the development of an electro-magnetic flow speed indicator is possible; but it would hardly be competitive in cases where simpler and more direct methods exist.

There remains one sphere where the electric magnetic measuring method may prove extremely useful: the investigation of varying water speeds. The well-developed technique of electric oscillography can be applied to hydraulic problems such as water-hammer, valve actions, instabilities of flow, etc.

To show this a number of oscillograms have been taken while releasing or stopping a water-flow. Fig. 5A shows the sudden release of the water column in the experimental flow circuit (fig. 1).

A mathematical analysis assuming the friction to vary with the square of the speed shows that the relation between speed ( w ) and time ( t ) should be v=vmax tanh Bt; the coefficient B has to be determined from the dimensions of the flow circuit. The calculated curve (c in fig. 5) agrees very well with the measured rate of speed increase.

Fig. 513 shows the effect of stopping water-flow suddenly. Surging back is clearly seen a t the instant indicated by the arrow. Reversal of flow is

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proved by the 180-degree phase shift of the induced voltage. The residual wave-line after stopping, partly due to poor compensation, partly due to the third harmonic, affects the appearance of the oscillogram more than its value for evaluation.

The main object of these oscillogranis, taken without great preparation, is to show the possibilities of the method for the investigation of hydraulic transients.

V. CONCLUSIONS. The induced E.M.F. is proportional to the speed, the magnetic induction,

and practically proportional to the length of conducting path. The measurable voltage is reduced by an internal voltage drop, but as it is still proportional to the speed and independent of the conductivity, an electro- magnetic method of measuring flow speeds is possible. The electrodes should be made small for best sensitivity. Oscillography of the induced voltages in a moving fluid may open a field of research on hydraulic transient phenomena.

Grateful acknowledgment is made to the Research Grant Board which Thanks are also due to Professor financed the experiments carried out.

B. L. Goodlet, Mr. N. H. Roberts, and Dr. R. Guelke, for helpful advice.

APPENDIX 1. On Transformer E.M.Fs .

The left-hand side of fig. 2~ can be redrawn as in fig. 6. If El, E, are the E.M.Fs. induced in each section of the circuit and R,

and R, the respective rheostat sections, R, the voltmeter resistance, including the resistance between the electrodes, the currents I,, I,, I as shown, then

El = IIRl i- IR,, E, = IZR, - IR,, - - -

1 =I, - I,. VOL. XXVIII, PART IT. 11

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Eliminating I, and I, and solving for I:

Now if R, >> R, and R, IR, = EIR, - E2Rl.

Rl + R, This is the voltage measured due to transformer E.M.Fs. It can be made zero by making R,/R, = E,/E,.

APPENDIX 2 . Calculation of Circulating Currents.

The electric field induced in the moving medium causes currents t o circulate.

A knowledge of these currents is desirable in order to estimate the internal voltage drop and to gain knowledge about the actual physical electric field.

The following assumptions are made:- (1) The magnetic field does not vary with time. ( 2 ) It does not vary in its own direction, the Z-axis. (3) Otherwise its space distribution is given by a double Fourier Series * (4) The fluid flows only in the direction of the positive x-axis. (5) The current flow region is terminated by four planes y = & b and

( 6 ) External currents, e.g. due to double-earthing or voltmeter load,

Assumptions ( 2 ) , (4), and (5) reduce the problem to a two-dimensional one.

for E, modified by the velocity distribution.

z = f d , which form insulating boundaries.

are not considered.

Notation.- E = Intensity of electric field.

E, = Intensity of externally induced field = BvlO-8. i = Current density. A, B, C, P, Q, coefficients of Fourier Series.

p = ~ 1 2 1 and q = 7 ~ / 2 b . k = Conductivity.

1 , b = Length and breadth of flow space.

Fundamental Equations.- From Assumption (1) follows :

aE, aE c u r l E = O = - - Y . . . (1) a, a x

From Assumption ( 2 ) follows: ai, ai

. ( 2 ) &vi=O=- + -'. . ax a,

* See H. Hertz, Induction in Rotating Spheres, Mi~c. Papers, 1896, p- 66.

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Further on:

. (3) E x = - i, and E , = - + E ilf k k O‘

Let the distribution of the externally induced field Eo be:

E, = E ’ C A, cos mpx x C B n cos nqy, m n

where

There are no even harmonics in the y direction in order to make the field between the lines y = b and y = 2b an image of the field between y = 0 and y = b.

The coefficients A and B are chosen to reproduce the true distribution of the magnetic field as near as possible.

Multiplying (5) out, one obtains:

m=O, 1, 2 , 3 , . . . n = l , 3, 5, 7 . . . . (4) There can be no sin terms as the field is symmetrical.

E, = E ’ C Q,, cos mpx cos nqy. . (5) mn

Solution.-Let the two components of the current density in the direction of the x and y axis be:

i, = k E ’ C Pmn sin rnpx sin nqy mn

and i, = k E ’ c Q,, cos mpx cos nqy. . - (6) mn

Differentiating (6) and substituting in ( 2 ) yields one set of equations for the Fourier Coefficients :

and substituting (6) in (3), differentiating suitably and substituting in (1) another set :

mpprnn = RqQmn, (7)

nqPmn + mpQmn= - mpcrnn. * (8) From (7) and (8) the coefficients P and Q are readily obtained:

- (9) - Cmn and Qmn= p =--

mn p m qn -+- $7” Pm

- Cmn

By substituting (9) in (6) the current density a t any point is obtained. A system of current lines can be calculated by means of a simple

integration as - I = k E ’ C sin mpx COB nqy, .

mn mp where the line I = 0 passes the origin x = 0, y = 0.

Equipotential lines are also obtained by integration as

v = - E’C + Qmn cos mpx sin nqy, . mn nq

where the potential along the line y = 0 is zero.

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l----- - - - - * - I

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E l e c t ~ o - M a g n ~ ~ Induc~ion in Water. 159

The voltage drop due to circulating currents is given by Q,, which is

It is independent of the conductivity.

Numerical Example.-Current and equipotential lines are calculated

negative (see equation 9).

for a simplified field distribution,

where p = v / l 2 cm. and q=7r/4 cm., and the result is pictured in fig. 7. By

taking a greater number of harmonics into account, a voltage drop of about 5 per cent. has been determined for a distribution very near to the one in the experiment.

APPENDIX 3. Correction for Error introduced by Double-earthing.

There are two types of circulating currents causing an internal voltage drop, with the effect that the measurable voltage is smaller than the E.M.F.

B = BmaX(-35 + .50 cos p~ + -15 cos 2 p ~ ) ( l . 1 5 cos qy - a15 cos 3qy),

The voltage drop a t the magnet centre would be about 9 per cent.

Fro. 8A.

I I

FIG. 8B.

First, there are internal circulating currents within the fluid (see Appendix 2); second, there are external circulating currents due to double- earthing.

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2014


To earth certain parts of the fluid circuit, particularly the outflow tubing, is inevitable, and earthing one lead of the V.V. was found advisable to prevent disturbing waves being picked up or deflections being caused by touching parts of the circuit.

An accurate calculation of the quasi-stationary current flow similar to that carried out for the internal circulating currents would offer considerable difficulties. The reduction in terminal voltage is therefore assessed by means of the equivalent circuit shown in fig. 8~ on the basis of resistance measurements.

The problem is depicted in fig. 8 A .

If the E.M.F., induced linearly between the electrodes, is E, a voltage

E r/R, + dFR cot h d- v =

will be measured, where R, is the voltmeter resistance, while the values for T and R have to be chosen to suit the measured resistance values according to the following equations:-

Resistance between electrodes Resistance from one electrode to earth = dr% cot h d s

= 2 d & tan h hdrp.

Some resistance values and correction factors V/E, for R, =5MQ 21 00,

Electrodes . . El. distance . . 66 mm. flat.

2.5 cm. ~

Resistance measured between electrodes . between one electrode and earth between bothelectrodesand earth

Calculated resistance T . 2 2 I , R . .

Correction factor (k,) . -94

~

4000 24000 23000

4100 22500

8 mm. round. 2.5 cm.

10000 32000 30000

10200 29000

_--__

.go

8 mm. round. 2 cm.

8200 32000 30000

8400 29000

-91, Dow

nloa

ded

by [

UQ

Lib

rary

] at

16:

13 2

1 N

ovem

ber

2014

Documents

ELECTRO-MAGNETIC INDUCTION IN WATER