A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 16 of 27

(S C specific resistance of a ) "T = : | substance }'

628.] If the units of length, mass, and time are the same in the two systems, the number of electrostatic units of electricity con tained in one electromagnetic unit is numerically equal to a certain velocity, the absolute value of which does not depend on the magnitude of the fundamental units employed. This velocity is an important physical quantity, which we shall denote by the symbol v.

Number of Electrostatic Units in one Electromagnetic Unit. For*, C, 11, 5), £, (£, v.

Form, ^ .0, 93, <£, 21, -•

v

For electrostatic capacity, dielectric inductive capacity, and con ductivity, v*.

For electromagnetic capacity, magnetic inductive capacity, and

resistance, —5- •

p2

Several methods of determining the velocity v will be given in Arts. 768-780.

In the electrostatic system the specific dielectric inductive capa city of air is assumed equal to unity. This quantity is therefore

represented by -^ in the electromagnetic system.

R 2,

244 DIMENSIONS OF UNITS. [629.

In the electromagnetic system the specific magnetic inductive capacity of air is assumed equal to unity . This quantity is there fore represented by —$• in the electrostatic system.

Practical System of Electric Units.

629.] Of the two systems of units, the electromagnetic is of the greater use to those practical electricians who are occupied with electromagnetic telegraphs. If, however, the units of length, time, and mass are those commonly used in other scientific work, such as the metre or the centimetre, the second, and the gramme, the units of resistance and of electromotive force will be so small that to express the quantities occurring in practice enormous numbers must be used, and the units of quantity and capacity will be so large that only exceedingly small fractions of them can ever occur in practice. Practical electricians have therefore adopted a set of electrical units deduced by the electromagnetic system from a large unit of length and a small unit of mass.

The unit of length used for this purpose is ten million of metres, or approximately the length of a quadrant of a meridian of the earth.

The unit of time is, as before, one second.

The unit of mass is 10~~n gramme, or one hundred millionth part of a milligramme.

The electrical units derived from these fundamental units have been named after eminent electrical discoverers. Thus the practical unit of resistance is called the Ohm, and is represented by the resistance-coil issued by the British Association, and described in Art. 340. It is expressed in the electromagnetic system by a velocity of 10,000,000 metres per second.

The practical unit of electromotive force is called the Volt, and is not very different from that of a DanielPs cell. Mr. Latimer Clark has recently invented a very constant cell, whose electro motive force is almost exactly 1.457 Volts.

The practical unit of capacity is called the Farad. The quantity of electricity which flows through one Ohm under the electromotive force of one Volt during one second, is equal to the charge produced in a condenser whose capacity is one Farad by an electromotive force of one Volt.

The use of these names is found to be more convenient in practice than the constant repetition of the words ' electromagnetic units,'

629.]

PEACTICAL UNITS.

245

with the additional statement of the particular fundamental units on which they are founded.

When very large quantities are to be measured, a large unit is formed by multiplying the original unit by one million, and placing before its name the prefix mega.

In like manner by prefixing micro a small unit is formed, one millionth of the original unit.

The following table gives the values of these practical units in the different systems which have been at various times adopted.

FUNDAMENTAL UNITS.

PRACTICAL SYSTEM.

B. A. REPORT, 1863.

THOMSON.

WEBER.

Length, Time,

Mass.

Earth's Quadrant, Second, 10-11 Gramme.

Metre, Second, Gramme.

Centimetre, Second, Gramme.

Millimetre, Second, Milligramme.

Resistance

Ohm

IO7

IO9

IO1

Electromotive force

Volt

IO5

IO8

10U

Capacity Quantity

Farad

Farad (charged to a Volt.)

io-7 io-2

io-9 io-1

io-10

10

CHAPTER XL

ON ENERGY AND STRESS IN THE ELECTROMAGNETIC FIELD.

Electrostatic Energy.

630.] THE energy of the system may be divided into the Potential Energy and the Kinetic Energy.

The potential energy due to electrification has been already con sidered in Art. 85. It may be written

r=is(**), (i)

where e is the charge of electricity at a place where the electric potential is ty, and the summation is to be extended to every place where there is electrification.

If fj ffj Ji are the components of the electric displacement, the quantity of electricity in the element of volume dx dy dz is

where the integration is to be extended throughout all space.

631.] Integrating this expression by parts, and remembering that when the distance, r, from a given point of a finite electrified system becomes infinite, the potential ty becomes an infinitely small quantity of the order r*1, and that/, g, h become infinitely small quantities of the order r~2, the expression is reduced to

where the integration is to be extended throughout all space.

If we now write P, Q, R for the components of the electromotive

dty d^ city

force, instead of -- — , -- — and -- =- , we find dx dy dz

(5)

633-] MAGNETIC ENERGY. 247

Hence, the electrostatic energy of the whole field will be the same if we suppose that it resides in every part of the field where elec trical force and electrical displacement occur, instead of being confined to the places where free electricity is found.

The energy in unit of volume is half the product of the electro motive force and the electric displacement, multiplied by the cosine of the angle which these vectors include.

In Quaternion language it is —4/9(5 3).

Magnetic Energy.

632.] We may treat the energy due to magnetization in a similar way. If A, J5, C are the components of magnetization and a, /3, y the components of magnetic force, the potential energy of the system of magnets is, by Art. 389,

Cy]dxdydzt (6)

the integration being extended over the space occupied by mag netized matter. This part of the energy, however, will be included in the kinetic energy in the form in which we shall presently obtain it.

633.] We may transform this expression when there are no elec tric currents by the following method.

We know that da db do

Hence, by Art. 97, if

cm d& cm

f. o .. ( R\

as is always the case in magnetic phenomena where there are no currents,

' =0, (9)

the integral being extended throughout all space, or

jjl{(a + lTtA)a + (P + lTtB)p + (y+±'nC)y}dxdydz = 0. (10) Hence, the energy due to a magnetic system

248 ENERGY AND STRESS. [634.

Electrokinetic Energy.

634.] We have already, in Art. 578, expressed the kinetic energy of a system of currents in the form

T=^(pi\ (12).

where p is the electromagnetic momentum of a circuit, and % is the strength of the current flowing round it, and the summation extends to all the circuits.

But we have proved, in Art. 590, that p may be expressed as a line-integral of the form

where F, G, H are the components of the electromagnetic mo- mentum, §C, at the point (xy z), and the integration is to be ex tended round the closed circuit s. We therefore find

2 *"' J \ £?<$ ds ds'

If ^, z;, w are the components of the density of the current at any point of the conducting circuit, and if S is the transverse section of the circuit, then we may write

. dx .dy . dz

i — = uS, i^ = vS, 2-v = ^£, (15)

ds ds ds

and we may also write the volume

Sds = dxdydz, and we now find _

T = i / // (Fu + Gv + Hw) dxdydz, (16)

where the integration is to be extended to every part of space where there are electric currents.

635.] Let us now substitute for u, v, w their values as given by the equations of electric currents (E), Art. 607, in terms of the components a, /3, y of the magnetic force. We then have

where the integration is extended over a portion of space including all the currents.

If we integrate this by parts, and remember that, at a great distance r from the system, a, /3, and y are of the order of mag nitude r~3, we find that when the integration is extended through out all space, the expression is reduced to

/^7 dH\ fflG dF] 7

637.] ELECTROKINETIC ENERGY. 249

By the equations (A), Art. 591, of magnetic induction, we may substitute for the quantities in small brackets the components of magnetic induction a, b, c, so that the kinetic energy may be written 1 /././.

T= — JJJ(aa + 6p + cy)da!dydz9 (19)

where the integration is to be extended throughout every part of space in which the magnetic force and magnetic induction have values differing from zero.

The quantity within brackets in this expression is the product of the magnetic induction into the resolved part of the magnetic force in its own direction.

In the language of quaternions this may be written more simply,

where 33 is the magnetic induction, whose components are «, b, c, and JQ is the magnetic force, whose components are a, (3, y.

636.] The electrokinetic energy of the system may therefore be expressed either as an integral to be taken where there are electric currents, or as an integral to be taken over every part of the field in which magnetic force exists. The first integral, however, is the natural expression of the theory which supposes the currents to act upon each other directly at a distance, while the second is appro priate to the theory which endeavours to explain the action between the currents by means of some intermediate action in the space between them. As in this treatise we have adopted the latter method of investigation, we naturally adopt the second expression as giving the most significant form to the kinetic energy.

According to our hypothesis, we assume the kinetic energy to exist wherever there is magnetic force, that is, in general, in every part of the field. The amount of this energy per unit of volume

is -- '— S S3 $3, and this energy exists in the form of some kind

o 77

of motion of the matter in every portion of space.

When we come to consider Faraday's discovery of the effect of magnetism on polarized light, we shall point out reasons for be lieving that wherever there are lines of magnetic force, there is a rotatory motion of matter round those lines. See Art. 821.

Magnetic and Electrokinetic Energy compared. 637.] We found in Art. 423 that the mutual potential energy

250 ENERGY AND STRESS. [638.

of two magnetic shells, of strengths $ and $', and bounded by the closed curves s and / respectively, is

cos e , — as as ,

where e is the angle between the directions of ds and ds', and r is the distance between them.

We also found in Art. 521 that the mutual energy of two circuits s and /, in which currents i and i' flow, is

-if

cos e 7 .. f ds ds .

If i, i' are equal to (/>, </>' respectively, the mechanical action between the magnetic shells is equal to that between the cor responding electric circuits, and in the same direction. In the case of the magnetic shells, the force tends to diminish their mutual potential energy, in the case of the circuits it tends to increase their mutual energy, because this energy is kinetic.

It is impossible, by any arrangement of magnetized matter, to produce a system corresponding in all respects to an electric circuit, for the potential of the magnetic system is single valued at every point of space, whereas that of the electric system is many- valued.

But it is always possible, by a proper arrangement of infinitely small electric circuits, to produce a system corresponding in all respects to any magnetic system, provided the line of integration which we follow in calculating the potential is prevented from passing through any of these small circuits. This will be more fully explained in Art. 833.

The action of magnets at a distance is perfectly identical with that of electric currents. We therefore endeavour to trace both to the same cause, and since we cannot explain electric currents by means of magnets, we must adopt the other alternative, and explain magnets by means of molecular electric currents.

638.J In our investigation of magnetic phenomena, in Part III of this treatise, we made no attempt to account for magnetic action at a distance, but treated this action as a fundamental fact of experience. We therefore assumed that the energy of a magnetic system is potential energy, and that this energy is diminished when the parts of the system yield to the magnetic forces which act on them.

If, however, we regard magnets as deriving their properties from electric currents circulating within their molecules, their energy

639-] AMPERE'S THEORY OF MAGNETS. 251

is kinetic, and the force between them is such that it tends to move them in a direction such that if the strengths of the currents were maintained constant the kinetic energy would increase.

This mode of explaining magnetism requires us also to abandon the method followed in Part III, in which we regarded the magnet as a continuous and homogeneous body, the minutest part of which has magnetic properties of the same kind as the whole.

We must now regard a magnet as containing a finite, though very great, number of electric circuits, so that it has essentially a molecular, as distinguished from a continuous structure.

If we suppose our mathematical machinery to be so coarse that our line of integration cannot thread a molecular circuit, and that an immense number of magnetic molecules are contained in our element of volume, we shall still arrive at results similar to those of Part III, but if we suppose our machinery of a finer order, and capable of investigating all that goes on in the interior of the molecules, we must give up the old theory of magnetism, and adopt that of Ampere, which admits of no magnets except those which consist of electric currents.

We must also regard both magnetic and electromagnetic energy as kinetic energy, and we must attribute to it the proper sign, as given in Art. 635.

In what follows, though we may occasionally, as in Art. 639, &c., attempt to carry out the old theory of magnetism, we shall find that we obtain a perfectly consistent system only when we abandon that theory and adopt Ampere^s theory of molecular currents, as in Art. 644.

The energy of the field therefore consists of two parts only, the electrostatic or potential energy

W = \jjj(Pf +

and the electromagnetic or kinetic energy T= ~

ON THE FORCES WHICH ACT ON AN ELEMENT OF A BODY PLACED IN THE ELECTROMAGNETIC FIELD.

Forces acting on a Magnetic Element.

639.] The potential energy of the element dx dy dz of a body magnetized with an intensity whose components are A, B, C, and

252 ENERGY AND STRESS. [640.

placed in a field of magnetic force whose components are a, /3, y, is

Hence, if the force urging the element to move without rotation in the direction of a? is X1dxdydz,

and if the moment of the couple tending to turn the element about the axis of x from y towards z is L dxdydz,

L = By-C($. (2)

The forces and the moments corresponding to the axes of y and

z may be written down by making the proper substitutions.

640. J If the magnetized body carries an electric current, of

which the components are u3 v, w, then, by equations C, Art. 60S,

there will be an additional electromagnetic force whose components

are X2, Y%, ZZ) of which X2 is

X2 = VG — wb. (3)

Hence, the total force, X, arising from the magnetism of the

molecule, as well as the current passing through it, is

+vc-«6. (4)

dx dx

The quantities a, 6, c are the components of magnetic induction, and are related to a, (3, y, the components of magnetic force, by the equations given in Art. 400,

a = a -f 4 TT A,

£=/3 + 477.£, (5)

C = 7+477(7.

The components of the current, u, v, w, can be expressed in terms of a, /3, y by the equations of Art. 607,

dy d(3 4 TT u — —- j-

dy dz

da dy

4;TTV = -= -~-

dz dx

dp da

TT 4/7rw = -f- — -T

Hence dx dy

(6)

_ ' dx } dx n dx

1 ( da -.da da 1 d 1

= — \a T +b—+c~---- (a*+(32 +y2)}- (7)

47T ( dx dy dz 2 dee. }

641.] THEORY OF STRESS. 253

Multiplying this equation, (8), by a, and dividing by 47i, we may add the result to (7), and we find

(9)

also, by (2), i = ((J-/3) y-(c-y)/3), (10)

= ~(iv-eft), (11)

where X is the force referred to unit of volume in the direction of #, and L is the moment of the forces about this axis.

On the Explanation of these Forces by the Hypothesis of a Medium in a State of Stress.

641 .] Let us denote a stress of any kind referred to unit of area by a symbol of the form Phk) where the first suffix, h, indicates that the normal to the surface on which the stress is supposed to act is parallel to the axis of h, and the second suffix, ft , indicates that the direction of the stress with which the part of the body on the positive side of the surface acts on the part on the negative side is parallel to the axis of k.

The directions of h and k may be the same, in which case the stress is a normal stress. They may be oblique to each other, in which case the stress is an oblique stress, or they may be perpen dicular to each other, in which case the stress is a tangential stress.

The condition that the stresses shall not produce any tendency to rotation in the elementary portions of the body is

P - P

^hk — rWi'

In the case of a magnetized body, however, there is such a tendency to rotation, and therefore this condition, which holds in the ordinary theory of stress, is not fulfilled.

Let us consider the effect of the stresses on the six sides of the elementary portion of the body dx dy dz, taking the origin of coordinates at its centre of gravity.

On the positive face dy dz, for which the value of % is \ dx, the forces are —

254

ENERGY AND STRESS.

[641.

Parallel to x,

dP.

Parallel to y, (Pxy + * -^f dx} dydz = Y+x, .

(12)

Parallel to

(P«+ 4

The forces acting on the opposite side, — X_X9 —Y_x) and — Z_x, may be found from these by changing the sign of dx. We may express in the same way the systems of three forces acting on each of the other faces of the element, the direction of the force being indicated by the capital letter, and the face on which it acts by the suffix.

If Xdxdydz is the whole force parallel to x acting on the element,

Xdxdydz = XH

,£P.

whence

d

dx dx

^P + ^

dy vx dz

(13)

If Ldxdydz is the moment of the forces about the axis of x tending to turn the element from y to 0, Ldxdydz =

whence L = Pyg — Pzy . (14)

Comparing the values of X and L given by equations (9) and (11) with those given by (13) and (14), we find that, if we make

= --_(aa-±(<S

1

TTJ 1

p —

--% — A ~

~k

= ~T-C^

+r

i

4 77 1

= ^va"/'

I

(15)

the force arising from a system of stress of which these are the components will be statically equivalent, in its effects on each

642.]

MAGNETIC STRESS.

255

element of the body, with the forces arising from the magnetization and electric currents.

642.] The nature of the stress of which these are the components may be easily found, by making the axis of x bisect the angle between the directions of the magnetic force and the magnetic induction, and taking the axis of y in the plane of these directions, and measured towards the side of the magnetic force.

If we put <£) for the numerical value of the magnetic force, 33 for that of the magnetic induction, and 2 € for the angle between their directions,

a = *y cos e, /3 = «£) sin e, y •=. 0, a — 33 cos e, b = — 33 sin e, c

1 - 2 i '2

4 jf

(17)

p _ p — p _ p _ o

yz ~~ zx zy — -*• xz

Pxv = — - 33 <£> cos e sin e,

Pyx = — — - 33 4p cos e sin e. Hence, the state of stress may be considered as compounded of —

(1) A pressure equal in all directions = -— «&2.

8 77

(2) A tension along the line bisecting the angle between the directions of the magnetic force and the magnetic induction

(3) A pressure along the line bisecting the exterior angle between these directions = — 33 § sin2 e.

(4) A couple tending to turn every element of the substance in the plane of the two directions from the direction of magnetic

induction to the direction of magnetic force — - - 33 <£) sin 2 e.

When the magnetic induction is in the same direction as the magnetic force, as it always is in fluids and non-magnetized solids, then e = 0, and making the axis of x coincide with the direction of the magnetic force,

256

ENERGY AND STRESS.

[643. (18)

and the tangential stresses disappear.

The stress in this case is therefore a hydrostatic pressure - - «£j2,

combined with a longitudinal tension — 33 <£) along the lines of

f 4 TT

force.

643.] When there is no magnetization, 33 = $3, and the stress is still further simplified, being a tension along the lines of force equal

to -— <£)2, combined with a pressure in all directions at right angles

. 1

to the lines of force, numerically equal also to - — 43 2- The com ponents of stress in this important case are

Pxx = —(a*-(3*-y P = — ( 2-a2-/3

** 8 77 ^

yz zy ^^

(19)

PX = Px = JLal3t

4 7T

The force arising from these stresses on an element of the medium referred to unit of volume is d d

f -J-PVZ+ -rP™>

ay " dz

Y_ d

=

1 C da d/3 dyl 1 ( d(3 dal ' 1 C dy da) ^da d(3 dy\ 1 /da dy\

__

dy

fa

dy

Now

da d(3 dy

-7- + ~r + -T dx dy dz

da. dy

-j- -y-

dz dx dft da

-j =- = 4 77 W-

ax dy where m is the density of austral magnetic matter referred to unit

645-] TENSION ALONG LINES OF FORCE. 257

of volume, and v and w are the components of electric currents referred to unit of area perpendicular to y and z respectively. Hence, X = am+ vy — wj3

Similarly Y = fim + wa— uy,

(Equations of Electromagnetic (20)

Force.) Zi = ym-i-vip — va.

644.] If we adopt the theories of Ampere and Weber as to the nature of magnetic and diamagnetic bodies, and assume that mag netic and diamagnetic polarity are due to molecular electric currents, we get rid of imaginary magnetic matter, and find that everywhere

• = 0,and *? + *0 + ?y=0, (21)

dx dy dz

so that the equations of electromagnetic force become, X = v y — w /3,

Y—wa-uy} (22)

Z = ujB—va.

These are the components of the mechanical force referred to unit of volume of the substance. The components of the magnetic force are a, /3, y, and those of the electric current are u, v, w. These equations are identical with those already established. (Equations (C), Art, 603.)

645.] In explaining the electromagnetic force by means of a state of stress in a medium, we are only following out the con ception of Faraday"*, that the lines of magnetic force tend to shorten themselves, and that they repel each other when placed side by side. All that we have done is to express the value of the tension along the lines, and the pressure at right angles to them, in mathematical language, and to prove that the state of stress thus assumed to exist in the medium will actually produce the observed forces on the conductors which carry electric currents.

We have asserted nothing as yet with respect to the mode in which this state of stress is originated and maintained in the medium. We have merely shewn that it is possible to conceive the mutual action of electric currents to depend on a particular kind of stress in the surrounding medium, instead of being a direct and immediate action at a distance.

Any further explanation of the state of stress, by means of the motion of the medium or otherwise, must be regarded as a separate and independent part of the theory, which may stand or fall without affecting our present position. See Art. 832.

• Esrp. Res., 3266, 3267, 3268. VOL. TT. S

258 ENERGY AND STRESS. [646.

In the first part of this treatise, Art. 108, we shewed that the observed electrostatic forces may be conceived as operating through the intervention of a state of stress in the surrounding medium. We have now done the same for the electromagnetic forces, and it remains to be seen whether the conception of a medium capable of supporting these states of stress is consistent with other known phenomena, or whether we shall have to put it aside as unfruitful.

In a field in which electrostatic as well as electromagnetic action is taking place, we must suppose the electrostatic stress described in Part I to be superposed on the electromagnetic stress which we have been considering.

646.] If we suppose the total terrestrial magnetic force to be 10 British units (grain, foot, second), as it is nearly in Britain, then the tension perpendicular to the lines of force is 0.128 grains weight per square foot. The greatest magnetic tension produced by Joule * by means of electromagnets was about 140 pounds weight on the square inch.

• Sturgeon's Annals of Electricity, vol. v. p. 187 (1840) ; or Philosophical Magazine, Dec., 1851.

CHAPTER XII.

CURRENT-SHEETS.

647.] A CURRENT-SHEET is an infinitely thin stratum of con ducting matter, bounded on both sides by insulating1 media, so that electric currents may flow in the sheet, but cannot escape from it except at certain points called Electrodes, where currents are made to enter or to leave the sheet.

In order to conduct a finite electric current, a real sheet must have a finite thickness, and ought therefore to be considered a conductor of three dimensions. In many cases, however, it is practically convenient to deduce the electric properties of a real conducting sheet, or of a thin layer of coiled wire, from those of a current-sheet as defined above.

We may therefore regard a surface of any form as a current-sheet. Having selected one side of this surface as the positive side, we shall always suppose any lines drawn on the surface to be looked at from the positive side of the surface. In the case of a closed surface we shall consider the outside as positive. See Art. 294, where, however, the direction of the current is defined as seen from the negative side of the sheet.

The Current -function.

648.] Let a fixed point A on the surface be chosen as origin, and let a line be drawn on the surface from A to another point P. Let the quantity of electricity which in unit of time crosses this line from left to right be $, then </> is called the Current-function at the point P.

The current-function depends only on the position of the point P, and is the same for any two forms of the line AP, provided this

s z

260 CURRENT-SHEETS. [649.

line can be transformed by continuous motion from one form to the other without passing through an electrode. For the two forms of the line will enclose an area within which there is no electrode, and therefore the same quantity of electricity which enters the area across one of the lines must issue across the other.

If s denote the length of the line AP, the current across ds from

left to right will be — ds.

If </> is constant for any curve, there is no current across it. Such a curve is called a Current-line or a Stream-line.

649.] Let }f be the electric potential at any point of the sheet, then the electromotive force along any element ds of a curve will be

d^ , f-d*,

ds

provided no electromotive force exists except that which arises from differences of potential.

If ^ is constant for any curve, the curve is called an Equi- potential Line.

650.] We may now suppose that the position of a point on the sheet is defined by the values of </> and [r at that point. Let dsl be the length of the element of the equipotential line ^ intercepted between the two current lines <£ and <j> + d<l>, and let ds2 be the length of the element of the current line $ intercepted between the two equipotential lines ty and \fr + d\lf. We may consider ds} and dsz as the sides of the element dty d^r of the sheet. The electromotive force — d\l/ in the direction of ds2 produces the current d<p across dslf

Let the resistance of a portion of the sheet whose length is ds2t and whose breadth is dsl} be ds2

(T —.- J

0*1

where <r is the specific resistance of the sheet referred to unit of area, then ds.2 7

'*-zf'

, ds-, ds.2

whence jj- = <r yf -

a<j) d\l/

651.] If the sheet is of a substance which conducts equally well in all directions, dsl is perpendicular to ds2. In the case of a sheet of uniform resistance or is constant, and if we make \jr' = a\f/, we shall have ds: __ d(j>

d9t~~ d+'*

and the stream-lines and equipotential lines will cut the surface into little squares.

652.] MAGNETIC POTENTIAL. 261

It follows from this that if fa and i/r/ are conjugate functions (Art. 183) of cj) and \f/t the curves fa may be stream-lines in the sheet for which the curves x/// are the corresponding equipotential lines. One case, of course, is that in which fa = \f/' and \j/i = — <£. In this case the equipotential lines become current-lines, and the current-lines equipotential lines *.

If we have obtained the solution of the distribution of electric currents in a uniform sheet of any form for any particular case, we may deduce the distribution in any other case by a proper trans formation of the conjugate functions, according to the method given in Art. 190.

652.] We have next to determine the magnetic action of a current-sheet in which the current is entirely confined to the sheet, there being no electrodes to convey the current to or from the sheet.

In this case the current-function 0 has a determinate value at every point, and the stream-lines are closed curves which do not intersect each other, though any one stream-line may intersect itself.

Consider the annular portion of the sheet between the stream lines $ and <j)-{-b<p. This part of the sheet is a conducting circuit in which a current of strength 8 $ circulates in the positive direction round that part of the sheet for which c/> is greater than the given value. The magnetic effect of this circuit is the same as that of a magnetic shell of strength 8 $ at any point not included in the substance of the shell. Let us suppose that the shell coincides with that part of the current-sheet for which 0 has a greater value than it has at the given stream-line.

By drawing all the successive stream-lines, beginning with that for which $ has the greatest value, and ending with that for which its value is least, we shall divide the current-sheet into a series of circuits. Substituting for each circuit its corresponding mag netic shell, we find that the magnetic effect of the current-sheet at any point not included in the thickness of the sheet is the same as that of a complex magnetic shell, whose strength at any point is C-{-(f), where C is a constant.

If the current-sheet is bounded, then we must make C 4- <£ = 0 at the bounding curve. If the sheet forms a closed or an infinite surface, there is nothing to determine the value of the constant C.

• See Thomson, Camb. and Dub. Math. Journ., vol. iii. p. 286.

262 CURRENT -SHEETS. [653.

653.] The magnetic potential at any point on either side of the current-sheet is given, as in Art. 415, by the expression

= ^-

where r is the distance of the given point from the element of surface dS, and Q is the angle between the direction of r, and that of the normal drawn from the positive side of dS.

This expression gives the magnetic potential for all points not included in the thickness of the current-sheet, and we know that for points within a conductor carrying a current there is no such thing as a magnetic potential.

The value of H is discontinuous at the current-sheet, for if &j_ is its value at a point just within the current-sheet, and Q,2 its value at a point close to the first but just outside the current-sheet,

&2 = Hj + 4 TT $, where </> is the current-function at that point of the sheet.

The value of the component of magnetic force normal to the sheet is continuous, being the same on both sides of the sheet. The component of the magnetic force parallel to the current-lines is also continuous, but the tangential component perpendicular to the current-lines is discontinuous at the sheet. If s is the length of a curve drawn on the sheet, the component of magnetic force

T

in the direction of ds is, for the negative side, —T^J and for the

2

positive side, — =-^ — —^ + 4 -n -f • ds ds ds

The component of the magnetic force on the positive side there

fore exceeds that on the negative side by 4 TT -~ - At a given point

ds

this quantity will be a maximum when ds is perpendicular to the current-lines.

On the Induction of Electric Currents in a Sheet of Infinite

Conductivity. 654.] It was shewn in Art. 579 that in any circuit

where E is the impressed electromotive force, p the electrokinetic momentum of the circuit, R the resistance of the circuit, and i the current round it. If there is no impressed electromotive force and

no resistance, then ~ = 0, or p is constant. tit

656.] PLANE SHEET. 263

Now 7;, the electrokinetic momentum of the circuit, was shewn in Art. 588 to be measured by the surface-integral of magnetic induction through the circuit. Hence, in the case of a current- sheet of no resistance, the surface-integral of magnetic induction through any closed curve drawn on the surface must be constant, and this implies that the normal component of magnetic induction remains constant at every point of the current-sheet.