A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 15 of 27

point, having the same instantaneous position, referred to the fixed and the moving axes respectively, then

dx __ §x duo , ^

~di = bi + ~di' with similar equations for the other components.

By the theory of the motion of a body of invariable form,

bx

— = « + wa*

}> (2)

Since F is a component of a directed quantity parallel to x, if — r— be the value of -=- referred to the moving axes,

dl" (ZFbv dFby clFbz dF

Substituting for -=- and -y- their values as deduced from the dy dz

equations (A) of magnetic induction, and remembering that, by (2), d bx d ly d bz

= °' = a>3' =~^

_b_x d^b^ d_by dffbz d bz dt ~ dx U dx bt + dx U fy bt + dx ~U + dx *i

b , bz dF

224 ELECTKOMAGNETIC FIELD. [6OI.

Ifweaowput

dF' dV *z dF

_^=j H

Of 01 Of

-.

The equation for P, the component of the electromotive force parallel to a?, is, by (B),

referred to the fixed axes. Substituting the values of the quanti ties as referred to the moving axes, we have

dy> dz> dF d(* + V) (9)

Cdt~^Tt"dt dx for the value of P referred to the moving axes.

601.] It appears from this that the electromotive force is ex pressed by a formula of the same type, whether the motions of the conductors be referred to fixed axes or to axes moving in space, the only difference between the formulae being that in the case of moving axes the electric potential # must be changed into vI/ + 4//.

In all cases in which a current is produced in a conducting cir cuit, the electromotive force is the line-integral

taken round the curve. The value of * disappears from this integral, so that the introduction of SP' has no influence on its value. In all phenomena, therefore, relating to closed circuits and the currents in them, it is indifferent whether the axes to which we refer the system be at rest or in motion. See Art. 668.

On the Electromagnetic Force acting on a Conductor which carries an Electric Current through a Magnetic Field.

602.] We have seen in the general investigation, Art. 583, that if a?x is one of the variables which determine the position and form of the secondary circuit, and if XL is the force acting on the secondary circuit tending to increase this variable, then

. ,-v

Since ^ is independent of xlf we may write

602.] ELECTROMAGNETIC FORCE. 225

(3)

and we have for the value of Xlf

ds

Now let us suppose that the displacement consists in moving every point of the circuit through a distance b% in the direction of #, b% being any continuous function of s, so that the different parts of the circuit move independently of each other, while the circuit remains continuous and closed.

Also let X be the total force in the direction of x acting on the part of the circuit from s = 0 to s = s, then the part corre-

7 ~V

spending to the element ds will be -=- ds. We shall then have the

following expression for the work done by the force during the displacement,

/dX ^ f d / ~.dx ~dy -rTdz\ „

rbatk s= LI -j— ( F-7~ + G -f + J2V) 6# ds, (4)

ds 2J dbsn^ ds ds ds'

where the integration is to be extended round the closed curve, remembering that 80? is an arbitrary function of s. We may there fore perform the differentiation with respect to b x in the same way that we differentiated with respect to t in Art. 598, remem

bering that dx dy dz

-= - = 1, -y£- = 0. and -= — = 0. (5)

dbx '

We thus find

The last term vanishes when the integration is extended round the closed curve, and since the equation must hold for all forms of the function bas, we must have

dX . / dy -, dz\ /P,N

— = »| ((?££._), (7)

ds 2V ds ds'

an equation which gives the force parallel to x on any element of the circuit. The forces parallel to y and z are

dT . f dz dx\ .

— = lAa— -- C-=-)* (8) d* 2V ds ds'

dZ . ^dx dy^ , .

•j- = 4f^-3 — «•/•!• (9)

ds 2\ ds dx'

The resultant force on the element is given in direction and mag nitude by the quaternion expression i2Vdp$$, where i2 is the numerical measure of the current, and dp and 53 are vectors

VOL. II. Q

226 ELECTROMAGNETIC FIELD. [603.

representing the element of the circuit and the magnetic in duction, and the multiplication is to be understood in the Hamil- tonian sense.

603.] If the conductor is to be treated not as a line but as a body, we must express the force on the element of length, and the current through the complete section, in terms of symbols denoting the force per unit of volume, and the current per unit of area.

Let X, Y, Z now represent the components of the force referred to unit of volume, and u, v, w those of the current referred to unit of area. Then, if S represents the section of the conductor, which we shall suppose small, the volume of the element ds will be Sds, and

n = -^ - - . Hence, equation (7) will become

S(vc-w6), (10)

(Equations of Electromagnetic (C)

or X = vc —wb.

Similarly Y= wa — uc,

, r/ 7 Force.

and Z — ub — va.

Here X, J", Z are the components of the electromagnetic force on an element of a conductor divided by the volume of that element ; n, v, w are the components of the electric current through the element referred to unit of area, and #, b, c are the components of the magnetic induction at the element, which are also referred to unit of area.

If the vector § represents in magnitude and direction the force acting on unit of volume of the conductor, and if (£ represents the electric current flowing through it,

en)

CHAPTER IX.

GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD.

604.] IN our theoretical discussion of electrodynamics we began by assuming- that a system of circuits carrying electric currents is a dynamical system, in which the currents may be regarded as velocities, and in which the coordinates corresponding to these velocities do not themselves appear in the equations. It follows from this that the kinetic energy of the system, so far as it depends on the currents, is a homogeneous quadratic function of the currents, in which the coefficients depend only on the form and relative position of the circuits. Assuming these coefficients to be known, by experiment or otherwise, we deduced, by purely dynamical rea soning, the laws of the induction of currents, and of electromagnetic attraction. In this investigation we introduced the conceptions of the electrokinetic energy of a system of currents, of the electro magnetic momentum of a circuit, and of the mutual potential of two circuits.

We then proceeded to explore the field by means of various con figurations of the secondary circuit, and were thus led to the conception of a vector 2[, having a determinate magnitude and direction at any given point of the field. We called this vector the electromagnetic momentum at that point. This quantity may be considered as the time-integral of the electromotive force which would be produced at that point by the sudden removal of all the currents from the field. It is identical with the quantity already investigated in Art. 405 as the vector-potential of magnetic in duction. Its components parallel to x, y, and z are F, G, and H. The electromagnetic momentum of a circuit is the line-integral of $1 round the circuit.

We then, by means of Theorem IV, Art. 24, transformed the

Q 2

228 GENERAL EQUATIONS. [605.

line-integral of £1 into the surface-integral of another vector, 53, whose components are a, d, c, and we found that the phenomena of induction due to motion of a conductor, and those of electro magnetic force can be expressed in terms of 53. We gave to 53 the name of the Magnetic induction, since its properties are iden tical with those of the lines of magnetic induction as investigated by Faraday.

We also established three sets of equations : the first set, (A), are those of magnetic induction, expressing it in terms of the elec tromagnetic momentum. The second set, (B), are those of electro motive force, expressing it in terms of the motion of the conductor across the lines of magnetic induction, and of the rate of variation of the electromagnetic momentum. The third set, (C), are the equations of electromagnetic force,, expressing it in terms of the current and the magnetic induction.

The current in all these cases is to be understood as the actual current, which includes not only the current of conduction, but the current due to variation of the electric displacement.

The magnetic induction 53 is the quantity which we have already considered in Art. 400. In an unmagnetized body it is identical with the force on a unit magnetic pole, but if the body is mag netized, either permanently or by induction, it is the force which would be exerted on a unit pole, if placed in a narrow crevasse in the body, the walls of which are perpendicular to the direction of magnetization. The components of 53 are #, #, c.

It follows from the equations (A), by which a, b, c are defined, that da M (i^^

dx dy dz

This was shewn at Art. 403 to be a property of the magnetic induction.

605.] We have defined the magnetic force within a magnet, as distinguished from the magnetic induction, to be the force on a unit pole placed in a narrow crevasse cut parallel to the direction of magnetization. This quantity is denoted by ȣ), and its components by a, /3, y. See Art. 398.

If 3 is the intensity of magnetization, and A, B, C its com ponents, then, by Art. 400,

a = a -f 4 TT A, c = y+4-n C.

(Equations of Magnetization.) (D)

6o6.] MAGNETIC EQUATIONS. 229

We may call these the equations of magnetization, and they indicate that in the electromagnetic system the magnetic induction 33, considered as a vector, is the sum, in the Hamiltonian sense, of two vectors, the magnetic force .£), and the magnetization 3 multi plied by 47T, or 33 = £ + 4?r3.

In certain substances, the magnetization depends on the magnetic force, and this is expressed by the system of equations of induced magnetism given at Arts. 426 and 435.

606.] Up to this point- of our investigation we have deduced everything from purely dynamical considerations, without any reference to quantitative experiments in electricity or magnetism. The only use we have made of experimental knowledge is to re cognise, in the abstract quantities deduced from the theory, the concrete quantities discovered by experiment, and to denote them by names which indicate their physical relations rather than their mathematical generation.

In this way we have pointed out the existence of the electro magnetic momentum §1 as a vector whose direction and magnitude vary from one part of space to another, and from this we have deduced, by a mathematical process, the magnetic induction, 33, as a derived vector. We have not, however, obtained any data for determining either 51 or 33 from the distribution of currents in the field. For this purpose we must find the mathematical connexion between these quantities and the currents.

We begin by admitting the existence of permanent magnets, the mutual action of which satisfies the principle of the conservation of energy. We make no assumption with respect to the laws of magnetic force except that which follows from this principle, namely, that the force acting on a magnetic pole must be capable of being derived from a potential.

We then observe the action between currents and magnets, and we find that a current acts on a magnet in a manner apparently the same as another magnet would act if its strength, form, and position were properly adjusted, and that the magnet acts on the current in the same way as another current. These observations need not be supposed to be accompanied with actual measurements of the forces. They are not therefore to be considered as furnishing numerical data, but are useful only in suggesting questions for our consideration.

The question these observations suggest is, whether the magnetic field produced by electric currents, as it is similar to that produced

230 GENERAL EQUATIONS. [607.

by permanent magnets in many respects, resembles it also in being- related to a potential ?

The evidence that an electric circuit produces, in the space sur rounding it, magnetic effects precisely the same as those produced by a magnetic shell bounded by the circuit, has been stated in Arts. 482-485.

We know that in the case of the magnetic shell there is a potential, which has a determinate value for all points outside the substance of the shell, but that the values of the potential at two neighbouring points, on opposite sides of the shell,, differ by a finite quantity.

If the magnetic field in the neighbourhood of an electric current resembles that in the neighbourhood of a magnetic shell, the magnetic potential, as found by a line-integration of the magnetic force, will be the same for any two lines of integration, provided one of these lines can be transformed into the other by continuous motion without cutting the electric current.

If, however, one line of integration cannot be transformed into the other without cutting the current, the line-integral of the magnetic force along the one line will differ from that along the other by a quantity depending on the strength of the current. The magnetic potential due to an electric current is therefore a function having an infinite series of values with a common difference, the particular value depending on the course of the line of integration. Within the substance of the conductor, there is no such thing as a magnetic potential.

607.] Assuming that the magnetic action of a current has a magnetic potential of this kind, we proceed to express this result mathematically.

In the first place, the line-integral of the magnetic force round any closed curve is zero, provided the closed curve does not surround the electric current.

In the next place, if the current passes once, and only once, through the closed curve in the positive direction, the line-integral has a determinate value, which may be used as a measure of the strength of the current. For if the closed curve alters its form in any continuous mariner without cutting the current, the line- integral will remain the same.

In electromagnetic measure, the line-integral of the magnetic force round a closed curve is numerically equal to the current through the closed curve multiplied by 4 TT.

607.] ELECTRIC CURRENTS. 231

If we take for the closed curve the parallelogram whose sides

are dy and dz, the line-integral of the magnetic force round the

parallelogram is ^y dp

^dy dz

and if u, vf w are the components of the flow of electricity, the current through the parallelogram is

u dy dz.

Multiplying this by 47r, and equating the result to the line- integral, we obtain the equation

dy dz with the similar equations

do, dy ( (Equations of /-™\

4 7T V = -= ~- ) Electric Currents.) W

dz dx

dp da

dx dy J

which determine the magnitude and direction of the electric currents when the magnetic force at every point is given.

When there is no current, these equations are equivalent to the condition that adx + fi dy + y dz = D£l,

or that the magnetic force is derivable from a magnetic potential in all points of the field where there are no currents.

By differentiating the equations (E) with respect to x, y, and z respectively, and adding the results, we obtain the equation du dv dw

. I I . Q

dx dy dz

which indicates that the current whose components are u, v, w is subject to the condition of motion of an incompressible fluid, and that it must necessarily flow in closed circuits.

This equation is true only if we take #, v, and w as the com ponents of that electric flow which is due to the variation of electric displacement as well as to true conduction.

We have very little experimental evidence relating to the direct electromagnetic action of currents due to the variation of electric displacement in dielectrics, but the extreme difficulty of reconciling the laws of electromagnet ism with the existence of electric currents which are not closed is one reason among many why we must admit the existence of transient currents due to the variation of displace ment. Their importance will be seen when we come to the electro magnetic theory of light.

232 GENERAL EQUATIONS. [6o8.

608.] We have now determined the relations of the principal quantities concerned in the phenomena discovered by Orsted, Am pere, and Faraday. To connect these with the phenomena described in the former parts of this treatise, some additional relations are necessary.

When electromotive force acts on a material body, it produces in it two electrical effects, called by Faraday Induction and Con duction, the first being most conspicuous in dielectrics, and the second in conductors.

In this treatise, static electric induction is measured by what we have called the electric displacement, a directed quantity or vector which we have denoted by £), and its components by/*, #, k.

In isotropic substances, the displacement is in the same direction as the electromotive force which produces it, and is proportional to it, at least for small values of this force. This may be expressed by the equation i

<T\ -IT- rr, (Equation of Electric /-pry

4 IT ' Displacement.)

where ^is the dielectric capacity of the substance. See Art. 69.

In substances which are not isotropic, the components /, #, h of the electric displacement 2) are linear functions of the components P, Q, -K of the electromotive force (£.

The form of the equations of electric displacement is similar to that of the equations of conduction as given in Art. 298.

These relations may be expressed by saying that K is, in isotropic bodies, a scalar quantity, but in other bodies it is a linear and vector function, operating on the vector (£.

609.] The other effect of electromotive force is conduction. The laws of conduction as the result of electromotive force were esta blished by Ohm, and are explained in the second part of this treatise, Art. 241. They may be summed up in the equation

ft = C (£, (Equation of Conductivity.) (G)

where (£ is the intensity of the electromotive force at the point, $ is the density of the current of conduction, the components of which are p, q, r, and C is the conductivity of the substance, which, in the case of isotropic substances, is a simple scalar quantity, but in other substances becomes a linear and vector function operating on the vector ($. The form of this function is given in Cartesian coordinates in Art. 298.

610.] One of the chief peculiarities of this treatise is the doctrine which it asserts, that the true electric current (£, that on which the

614.] CURRENTS OF DISPLACEMENT. 233

electromagnetic phenomena depend, is not the same thing as $, the current of conduction, but that the time- variation of 2), the electric displacement, must be taken into account in estimating the total movement of electricity, so that we must write,

(£ = £+2), (Equation of True Currents.) (H)

or, in terms of the components,

dt dg

j V

dk

(H*)

611.] Since both $ and 2) depend on the electromotive force ($, we may express the true current (£ in terms of the electromotive force, thus

or, in the case in which C and K are constants,

w = CR+ —- KC-j-'

47T dt

612.] The volume-density of the free electricity at any point is found from the components of electric displacement by the equation ^f dg dk

613.] The surface-density of electricity is

where /, m, n are the direction-cosines of the normal drawn from the surface into the medium in which f, g, li are the components of the displacement, and /', m' ', n' are those of the normal drawn from the surface into the medium in which they are f', /, //.

614.] When the magnetization of the medium is entirely induced by the magnetic force acting on it, we may write the equation of induced magnetization, $$ = /*«£), (L)

where p is the coefficient of magnetic permeability, which may be considered a scalar quantity, or a linear and vector function operating on «£j, according as the medium is isotropic or not.

234

GENEKAL EQUATIONS.

615.] These may be regarded as the principal relations among the quantities we have been considering. They may be combined so as to eliminate some of these quantities, but our object at present is not to obtain compactness in the mathematical formulae, but to express every relation of which we have any knowledge. To eliminate a quantity which expresses a useful idea would be rather a loss than a gain in this stage of our enquiry.

There is one result, however, which we may obtain by combining equations (A) and (E), and which is of very great importance.

If we suppose that no magnets exist in the field except in the form of electric circuits, the distinction which we have hitherto maintained between the magnetic force and the magnetic induction vanishes, because it is only in magnetized matter that these quan tities differ from each other.

According to Ampere's hypothesis, which will be explained in Art. 833, the properties of what we call magnetized matter are due to molecular electric circuits, so that it is only when we regard the substance in large masses that our theory of magnetization is applicable, and if our mathematical methods are supposed capable of taking account of what goes on within the individual molecules, they will discover nothing but electric circuits, and we shall find the magnetic force and the magnetic induction everywhere identical. In order, however, to be able to make use of the electrostatic or of the electromagnetic system of measurement at pleasure we shall retain the coefficient //, remembering that its value is unity in the electromagnetic system.

616.] The components of the magnetic induction are by equa tions (A), Art. 591, dH dG

n — —

a/ — — -y-

dy dz

dF dH o — — --- — dz dx

dF

dx dy The components of the electric current are by equations (E),

Art. 607,

dy aft

4 77 U — V- 7- >

0* &

da

dz d(B

~

dx

dy

=£

dx da

~~

dy

6l6.]

VECTOR-POTENTIAL OP CURRENTS.

According to our hypothesis a, b, c are identical with respectively. We therefore obtain

If we write

235 i, fift /uy

tffo? dy dy2 dz2

dF dG dH

J = -j- + -r + ~r >

ax dy dz

dzdx

we may write equation (1),

Similarly,

dJ

4 TT ja v = -- + V2 #»

If we write F'=- fff U- dx dy dz, ~|

-, j

where r is the distance of the given point from the element xy z, and the integrations are to be extended over all space, then

(7)

The quantity x. disappears from the equations (A), and it is not related to any physical phenomenon. If we suppose it to be zero everywhere, / will also be zero everywhere, and equations (5), omitting the accents, will give the true values of the components of 51.

• The negative sign is employed here in order to make our expressions consistent with those in which Quaternions are employed.

236 GENERAL EQUATIONS. [617.

617.] We may therefore adopt, as a definition of 2[, that it is the vector-potential of the electric current, standing1 in the same relation to the electric current that the scalar potential stands to the matter of which it is the potential, and obtained by a similar process of integration, which may be thus described. —

From a given point let a vector be drawn, representing1 in mag nitude and direction a given element of an electric current, divided by the numerical value of the distance of the element from the given point. Let this be done for every element of the electric current. The resultant of all the vectors thus found is the poten tial of the whole current. Since the current is a vector quantity, its potential is also a vector. See Art. 422.

When the distribution of electric currents is given, there is one, and only one, distribution of the values of 31, such that 31 is every where finite and continuous, and satisfies the equations V2§1= 47Tf*<£, fl.VSl = 0,

and vanishes at an infinite distance from the electric system. This value is that given by equations (5), which may be written

Quaternion Expressions for tJie Electromagnetic Equations.

618.] In this treatise we have endeavoured to avoid any process demanding from the reader a knowledge of the Calculus of Qua ternions. At the same time we have not scrupled to introduce the idea of a vector when it was necessary to do so. When we have had occasion to denote a vector by a symbol, we have used a German letter, the number of different vectors being so great that Hamilton's favourite symbols would have been exhausted at once. Whenever therefore, a German letter is used it denotes a Hamil- tonian vector, and indicates not only its magnitude but its direction. The constituents of a vector are denoted by Roman or Greek letters.

The principal vectors which we have to consider are : —

Constituents.

The radius vector of a point .................. p x y z

The electromagnetic momentum at a point 2[ F G H

The magnetic induction ..................... 53 a I c

The (total) electric current .................. (£ u v w

The electric displacement ..................... 2) f g h

6 1 9.] QUATEKNION EXPRESSIONS. 237

Constituents.

The electromotive force ..................... (£ P Q R

The mechanical force ........................ g XYZ

The velocity of a point ........................ © or p so y z

The magnetic force ........................... «£) a /3 y

The intensity of magnetization ............ 3 ABC

The current of conduction .................. ft p q r

We have also the following scalar functions : — ,The electric potential ^. The magnetic potential (where it exists) 12. The electric density e. The density of magnetic ' matter ' m.

Besides these we have the following quantities, indicating physical properties of the medium at each point : —

(7, the conductivity for electric currents. K, the dielectric inductive capacity. fji, the magnetic inductive capacity.

These quantities are, in isotropic media, mere scalar functions of p, but in general they are linear and vector operators on the vector functions to which they are applied. K and JJL are certainly always self- conjugate, and C is probably so also.

619.] The equations (A) of magnetic induction, of which the

first is> dH dG

a = -= --- r-» dy dz

may now be written sg _ yyty

where V is the operator

. d . d -, d %-j- +7-7- + £-7-1

dx * dy dz

and Vindicates that the vector part of the result of this operation is to be taken.

Since 21 is subject to the condition $ V 2[ = 0, V§[ is a pure vector, and the symbol V is unnecessary.

The equations (B) of electromotive force, of which the first is

, . dF d* P = cy—oz -- - --- r- ,

dt dx

become @= F®33 — $ — V*.

The equations (C) of mechanical force, of which the first is

v , d^> dil

JL = cv — mv — e — -- m -7— j dx dx

become = 7 $ 33 —

238 GENERAL EQUATIONS. [619.

The equations (D) of magnetization, of which the first is

a — a 4- 4 TT A, become 33 — <$ 4- 4 TT 3.

The equations (E) of electric currents, of which the first is

dy d(3

4 TT u — -/ -- fi dy dz

become 4 -n & =

The equation of the current of conduction is, by Ohm's Law,

£ = <7<g. That of electric displacement is

3. = -?-K®.

4 7T

The equation of the total current, arising from the variation of the electric displacement as well as from conduction, is

<£ - S + 2X When the magnetization arises from magnetic induction,

SB = M£.

We have also, to determine the electric volume-density,

e = £V$). To determine the magnetic volume-density,

•m = S V 3.

When the magnetic force can be derived from a potential

= - V 12.

CHAPTER X.

DIMENSIONS OF ELECTRIC UNITS.

620.] EVERY electromagnetic quantity may be defined with reference to the fundamental units of Length, Mass, and Time. If we begin with the definition of the unit of electricity, as given in Art. 65, we may obtain definitions of the units of every other electromagnetic quantity, in virtue of the equations into which they enter along with quantities of electricity. The system of units thus obtained is called the Electrostatic System.

If, on the other hand, we begin with the definition of the unit magnetic pole, as given in Art. 374, we obtain a different system of units of the same set of quantities. This system of units is not consistent with the former system, and is called the Electro magnetic System.

We shall begin by stating those relations between the different units which are common to both systems, and we shall then form a table of the dimensions of the units according to each system.

621.] We shall arrange the primary quantities which we have to consider in pairs. In the first three pairs, the product of the two quantities in each pair is a quantity of energy or work. In the second three pairs, the product of each pair is a quantity of energy referred to unit of volume.

FIRST THREE PAIRS.

Electrostatic Pair.

Symbol.

( 1 ) Quantity of electricity . . . . e

(2) Line-integral of electromotive force, or electric po

tential E

240 DIMENSIONS OF UNITS. [622.

Magnetic Pair.

Symbol.

(3) Quantity of free magnetism, or strength of a pole . m

(4) Magnetic potential ...... H

ElectroJcinetic Pair.

(5) Electroldnetic momentum of a circuit . . p

(6) Electric current ....... C

SECOND THREE PAIRS.

Electrostatic Pair.

(7) Electric displacement (measured by surface-density) . 3)

(8) Electromotive force at a point . . . (£'

Magnetic Pair.

(9) Magnetic induction * ..... 33

(10) Magnetic force .;•« » ..... $

Electrokinetic Pair.

(11) Intensity of electric current at a point . . . (£

(12) Vector potential of electric currents . . .51

622.] The following relations exist between these quantities. In the first place, since the dimensions of energy are , and

those of energy referred to unit of volume , we have the

following equations of dimensions :

(1)

(2) Secondly, since e, p and 51 are the time-integrals of C, fi, and (£

Thirdly, since E, 12, and p are the line-integrals of @, .£>, and 91 respectively,

Finally, since et C, and m are the surface-integrals of $), 6, and respectively,

625.] THE TWO SYSTEMS OF UNITS. 241

623.] These fifteen equations are not independent, and in order to deduce the dimensions of the twelve units involved, we require one additional equation. If, however, we take either e or m as an independent unit, we can deduce the dimensions of the rest in terms of either of these.

(3) and (5) [j,] = M=

(4) and (6)

(10)

624.] The relations of the first ten of these quantities may be exhibited by means of the following arrangement : —

e, 2), «£), C and 12. E (£, 33, m and p.

The quantities in the first line are derived from e by the same operations as the corresponding quantities in the second line are derived from m. It will be seen that the order of the quantities in the first line is exactly the reverse of the order in the second line. The first four of each line have the first symbol in the numerator. The second four in each line have it in the deno minator.

All the relations given above are true whatever system of units we adopt.

625.] The only systems of any scientific value are the electro static and the electromagnetic system. The electrostatic system is

VOL. II. ft

242 DIMENSIONS OF UNITS. [626.

founded on the definition of the unit of electricity, Arts. 41, 42, and may be deduced from the equation

which expresses that the resultant force (£ at any point, due to the action of a quantity of electricity e at a distance L, is found by dividing e by 7/2. Substituting the equations of dimension (1) and (8), we find

whence \e\ = \L* If* T^} , m = in the electrostatic system.

The electromagnetic system is founded on a precisely similar definition of the unit of strength of a magnetic pole, Art. 374, leading to the equation ^ m

• : = L* '

J/ whence

e-] ri ^J - -^ J

and [e] =

in the electromagnetic system. From these results we find the

dimensions of the other quantities.

626.] Table of Dimensions.

Dimensions in

c, , , Electrostatic Electromagnetic Symbol Sygtem System

Quantity of electricity .... e [Z* M * T~l] \L* M*.

Line-integral of electro- | ^ ^ M- T~^ \ti H* T~*.

motive force 3

Quantity of magnetism -\

Electrokinetic momentum t . $m I [tf M*\ \L* M* T~1].

of a circuit ) *

Electric current C [L* M* T

Magnetic potential ) ' {Q,

Electric displacement | _ [T-^M^Tl[ IT'

Surface-density

Electromotive force at a point @ [^"M/^7-1] [ZJtf I7"2].

Magnetic induction 53 [IT^] [i;-^^-1].

Magnetic force § [L* M* T~] [L~ M* I'1].

Strength of current at a point (£ [Z~* If * T"2] [^^ If* Tl] .

Vector potential 31 [Z-!f]

628.] TABLE OF DIMENSIONS. 243

627.] We have already considered the products of the pairs of these quantities in the order in which they stand. Their ratios are in certain cases of scientific importance. Thus

Electrostatic Electromagnetic Symbol. System. System.

e lT2\

-=- = capacity of an accumulator . . q [Z] T~ I

/•coefficient of self-induction *\ -^- = j of a circuit, or electro- > L ~T~\ \f*

(. magnetic capacity J

2. _ ( specific inductive capacity | ^ r _ ¥=: ( of dielectric \

33 r^72!

-£- = magnetic inductive capacity . . ju y2 M-

4P L^ J

x? r- yr —i p T — 1

-— = resistance of a conductor .... R -=- "TT