Electron Theory of Matter

Hn flDemoriam

Xibraq? of

3onas

ffiernarfc

professor of

matbanson

p basics

Carnegie flnstitute of

o

CAMBRIDGE PHYSICAL SERIES

THE

ELECTRON THEORY OF

MATTER

CAMBRIDGE UNIVERSITY PRESS C.

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CHAPTER

1

THE ORIGIN OF THE ELECTRON THEORY The Electron Theory of atomic theory.

of Matter

It differs

may be looked upon as a form from the form of the atomic theory

with which chemistry is familiar, especially in that it makes the ultimate atoms minute geometrical configurations of electric

A

charge, instead of particles of uncharged matter. large number of different lines of inquiry, often closely interwoven, have led to

the adoption of such a view of the structure of matter. Of these lines of inquiry, however, three may be considered

different

pre-eminently conspicuous.

In the

first place,

although the electron theory has made most it is a logical development

rapid progress in the last two decades, of the views held a century ago by especially of the views to which .

Davy* and

Berzeliusf and

Faraday J was led by

chemical discoveries made somewhat

his electro-

Davy concluded, from phenomena known in his

later.

a general review of the electrochemical day, that the forces between the chemical atoms were of electrical

Shortly afterwards a complete system of chemical structure depending on the same idea was developed by Berzelius and, origin.

original form Berzelius's electrochemical theory was have much in common with insufficiently elastic, its main features The laws of electrolysis on the views modern the most subject.

although in

its

discovered by Faraday led to an important advance by pointing for electricity; for they distinctly to an atomic constitution

showed that each chemical atom invariably transported either a quantity of electricity or an integral multiple of that *

Phil.

Tram.

p. 1 (1807).

| Mtim. Acad. Stockholm (1812)

;

Nicholson'** Journal, voln. xxxiv.

(1818).

$ Kxp.

lies.

R. K. T.

5377, 5*23,

001, 713, 821 arid especially 852,

8(>).

THE ORIGIN OF

4

ai^j./^

inn,

The amount is as follows Stated briefly the method cloud the in may be the whole of the drops rf water condensed on he of supersaturate produced by calculated from the degree the for formula of Stokes s The :

totalcharge.

|

application

known expansion. rate of

fluid gives the average size of of a sphere in a viscous number n of the these data determine the total

fall

Thus

each drop.

that practically all the assumed, for sufficient .reasons, total charge ne on all The ion. one and only one drops contain out of the chamber them the ions could be determined by sweeping the strength an electrometer before the expansion took place, It

drops

is

into

same as in the condensation Thus the charge on a single ion was obtained by in this way Thomson showed that the

the of the source of ionisation being experiments.

Proceeding

division.

negative

ions liberated in air

from radium each carried the

by Roentgen rays and by the ft rays same charge as the hydrogen ion in

electrolysis.

ions investigated in these experiments are rather cated structures and are not identical with the electron.

The

compliIn the

produced by ultra-violet light falling on a was shown by Thomson that the particles when first, These emitted have the same value ofe/m as the cathode rays. case of the ionisation

metal

it

would not be likely to aggregate together in the presence of gas Wilson* showed that the negative ions molecules, and C. T.

R

from ultra-violet light behaved exactly like those from the other The inference ionising agents in his condensation experiments.

from these experiments therefore is that the particles which form the cathode rays and which are emitted during photoelectric action carry a charge equal to that of the

hydrogen atom in Experiments by Townsendf, on the rate of fall of the clouds produced when the gases evolved from chemical actions in the wet are allowed to bubble through water, occurring way had previously led him to conclude that the ions present in such gases carry the same charge as a hydrogen ion in electrolysis.

electrolysis.

This conclusion has been strengthened by other methods of determining the charge on an electron. One of these depends on the theory of the radiation of electromagnetic energy from hot, bodies.

The theory *

of this

Phil Trans. A.

method vol.

will

be considered in the

cxcn. p. 403 (1899).

t Phil Maq. Feb. 1898.

THE ORIGIN OF THE ELECTRON THEORY

5

One of the recent methods, which is due. to Rutherford f and Geiger, depends upon the properties of radioactive substances. These are found to emit positively charged bodies, called a particles, which carry twice the charge e of an electron and are able to sequel*.

produce a large number of new ions when they pass through a gas. By magnifying this secondary ionisation by means of an auxiliary electric field and also using a very sensitive electrometer, Rutherford

was able to detect the ionisation produced by a single a particle. When a very weak radioactive preparation was used the a particles were emitted at times separated by rather wide and irregular intervals, and as the effect produced by each one separately could be detected, the number emitted by a given amount of the radioactive substance in a given time could be measured. The only other datum which is required to measure e is the quantity of positive electricity which is carried away from the same quantity

of the

preparation by the a rays. obtained by other experiments.

The

This had previously been

drop method has recently been improved by H. A. Wilson]: and R. A. Millikan. The former showed that the charge on the drops could be deduced from the rate of fall under falling

gravitation combined with different electric fields, without makinguse of the degree of supersaturation whilst the latter showed how ;

the drops of water could be replaced by drops of a non- volatile oil. The drops of oil have the great advantage that they do not and by allowing a sufficient number of electrons to evaporate :

combine with them and applying a supporting electric field which just balances the gravitational force, they can be kept under observation for an indefinite length of time. In this way Millikan has shown that the method is capable of yielding results of very groat precision. All the throe methods bust mentioned are quite accurate and exhibit an excellent agreement. It is claimed that the charge e on an electron is known to within 1 per cent. Millikan's|| latest value is = 4r<Sl x 10~ 10 E.s. unit or 1/60 x 10~"JO E.M. unit. *

See chap. xv. t Roy. Soc. Proc. A. Phil.

Mag. VI.

vol. LXXXI. pp. 141,

vol. v. p.

429 (1903).

163 (1908).

THE ORIGIN O*

Application

to the

Atomic Theory.

other important consequences. These experiments have led to of an element is one gram atom Since the charge carried by and the charge from electrolytic experiments, accurately known shown to be been has ion a monovalent electrolytic carried

by

the number of atoms that of a gaseous ion, it follows that equal to to the same degree known atom of any substance is in one

gram

of accuracy as

e.

Since the charge which

by one grain 9*649 x 10 E.M.

carried

is

:{

atom of a monovalent element in electrolysis is one gram atom of any units it follows that the number of atoms in element

is

23 6'02 x 10

to liberate Also, since the charge required C. and 760 mms. is 04327 101. at

.

half a cubic centimetre of unit, it follows that

the

H

2

number

of molecules in one cubic centi-

metre of any gas under standard conditions of temperature and pressure

is

1'60

---.

These values are in agreement with the comparatively inaccurate estimates which had previously been given by methods bused on the kinetic theory of gases and other considerations.

Millikan was also able to observe the changes produced by the combination of single ions with the drops. These experiments, as well as those of Rutherford with the a rays, furnish a very direct. and convincing proof of the atomic theory of matter arid electricity. The consequences of the atomic theory of matter have recently been strikingly verified by experiments in other directions.

Perrin* has shown that the irregular motions of minute suspended in fluids are in accordance with the requirements of the particles kinetic theory of gases.

determination of the

A

study of these motions also leads i a number of molecules in one grain 'molecule of

any element and thus to a determination of

by Perrin

The value

e.


btai ne
mplicated distribution of charge may obviously be obtained by le integration of the amount arising from each volume element, calculation it is necessary to integrate for each electric intensity separately and combine the the of >mponent isults according to the rule for the composition of forces. This i

making the

solution and subsequent composition of vectors is often trouble>me as well as clumsy, and it is not necessary for the calculation

the electric intensity.

'

may be dispensed with by the known as the Potential.

It

jction of another function

intro-

The Potential.

The Potential

is

defined as the work divided

hen an infinitesimal

electric

charge

is

by the charge brought from some stanThe standard position is

ard position to the point in question. sually taken to be a point at an infinite distance

away from

larged bodies. The value of the potential calculated in this way Lust be independent of the path of approach to the point under msideration, otherwise an indefinite amount of work could be Dtained by

making the charge move round a

closed

contour

assing through the point under consideration and the standard This would be contrary to the law of conservation :>sition. ? The potential is a function of each point in space and energy. Such a function is known assesses magnitude but not direction. 3

a scalar point function.

The

electrostatic potential is single-

allied.

Let

P

ich that '

1

at P. to

}

Q be two points at an

PQ

is

Let

dV

Q and

let

infinitesimal distance d$ apart

and

in the direction of the resultant electric intensity be the increase in the potential in passing from

the direction

P

>

Q be

considered positive.

Then

ELECTRIC INTENSITY

lfi

T

:-

,

- AW*.

*. that

A'- -

J

AND POTENTIAL

The value

of the

component of

whose inclination v wfc'iMtv at P in any other direction be less than

.fa

It

,

.

i,,,K *,i

!*

*C >i

/'

;

to

#

always J,-coB0. This laid off in the direction infinitesimal length an A*.***

7

, then from Q length Q In this from on so and at point to point. Q .'*.)iuiit iiiMwity the that such in line tangent to space ** skill draw a curved

H4i

run

*[, *

at

any point

The curve

that point.

!*v*n-i

electric intensity is also finite at internal points. n1 c?T-\r> CiTi r\f TO rl a ^ onH rliTnrJiri T' iri'K~v

-fl-/-i

01 >-k-v

1

m

Consider TTT-I-V

T\QT4-ei

AXD POTENTIAL

ELECTRIC INTENSITY

00

7)V I',

iml

V

t

As we have

as before.

can be

2

small

c

By

enough p/r *

with

0,

just shown,

made

so that the latter

is finite.

-^

as

large as

We

we

may be disregarded

Thus

in tin*

_ 3Jj } CT

vanishes

when

intensity

at

f rfr fsin 0d0 f Jo Jo

47rJo e

We

= 0.

internal

conclude

that

the

'obtained

by

therefore

may be

points

the potential

Gauss's Theorem.

The consideration of the distribution of the to

in

surfaces

in

space

any surface whatever Let

X

and

leads let

to

electric intensity

interesting

results.

dS be an element

of

it.

be the component of the electric intensity at dS, along the to the element, being reckoned positive if it is in the

N

drawn outward from the surface. We now prove that the integral JJNdS taken over any closed kv Is equal to e, where e is the algebraic sum of the charges

direction of the normal

within the surface.

ELECTRIC INTENSITY AND POTENTIAL Consider

P

point

where ~~

PG

is

(9

~ 2

first

the intensity

*S

the -angle between t ^le

S

^

angie

E

i

at

l

We

within the surface.

Q due

have

PQ

S =e l

If there are a large

number of

normal component of

to a charge e l at a l

^PQP- and

and the normal

dS

subtended by

far as the single charge e l is concerned,

21

at P.

Thus, so

we have NidS =

point charges e ly e,

etc.,

Now

to dS.

p -~-

da)^

the resultant

electric intensity

= e d^ -f e^do)* -f e^do> -f = ^TT^! ff do) = 4-7r&> etc. ff e = e +^ + ^ + ... = e ......... ...... (3). //JV dS

4*7rN dS

so that

Now

l

. . .

s

l

c?ft>!

,

e.2

.

,

r

So that

1

This result can obviously be extended from a series of point charges to a continuous distribution in the same manner as that

employed

in dealing with the potential.

It remains to prove that charges outside the closed surface It is evident that contribute nothing to the surface integral. element of solid doo conical angle arising from an external every

charge will cut the closed surface an even number of times. The value of NdS for the intersections of the cone by the surface will

be alternately positive and negative since the direction of the electric intensity is constant in space but alternates in sign with reference to

the successive normals.

The numerical value

NdS is the same for successive intersections, being So that the

surface integral

is

equal and opposite elements. conclude that the value

~

of

6

equal to

do>.

divided up into a series of pairs of Its value is therefore zero, and we

offfNdS over any

closed surface

is

equal

to the charge inside.

This result

is

known

as Gauss's

Theorem.

means of calculating the value of the electric intensity arising from various symmetrical Thus in the case of a uniformly distributions of electric charge. at any point external to the shell the intensity charged spherical

The theorem

is

of great value as a

It also follows with the shell. the point and concentric that the intensity must be inu the symmetry of the problem of radius r is 4an*> the electric Since the area of a sphere radial centre of the charged shell is the from r at a distance

E

= by 4ir^#

riven K

;

so that 2

#=e/47rr

........................... (4).

force vanishes inside

that the way we may prove

In a similar

case of a sphere These results may be extended to the volume so that layers equidistant from the throughout its Thus we may show, foi to are charged equal density. uniform solid a inside force sphere of electricity sample, that the

the shell

varies

the distance from the centre.

its

The

application

Theorem to the tubes of force As we have seen, a tube bounded by a surface which is the

of Gauss's

on p. 16, is instructive. a tubular region of the lines of force.

of

is

to

Let us apply Gauss's Theorem a portion of such a tube, terminated at each end by equiThe lines of force run along the tubulai surfaces. so that at each point

I

the component of the intensitj Over the ends the resultanl

to these surfaces vanishes.

!

*'leetric

at

E

Let it be and S2 being the

to the surfaces. intensity will be normal

the end where the cross section

is

E

S13

z

The value of ffNdfc corresponding quantities at the other end. I: nver the whole surface considered is clearly 1 S1 2 S2

E

E

.

a region where there are no electric = E^S2 thus the electric inthat so this vanishes, rharges at is as the area of cross section o: any point inversely Minify the tube of force

is

in

E&

the tubes

!

i

.f

;

force at that point.

Fwk-T the conditions contemplated in electrostatics the surface nt hi t r if electricity must be an equipotential surface other would be currents of electricity flowing from one par Lr w\rt nv tt another. The tubes of force must therefore star .

;

:\

.

i

;r..iii

ji *h

*;1* u*t '*

-

Now apply Gauss's Theorem t< a tube of force and its continuation int< the conductor and terminated

such a surface.

landed by if

by equipotentia ne in>ide and the other outside the conductor. Th< j:rh>ity vanishes over all the surface inside the conducto: ijnr.nal

component vanishes over the tubular surface

TIP \alue

of

jjNdS

is

thus

eual

to the val

&

A

ELECTRIC INTENSITY AND POTENTIAL end

this quantity over the

23

This is equal to the charge the surface the area of its intersection by the tube of inside

which

;

We

w here 7

is

section. a-

is

E& =

have

we make

the

end

section approach indefinitely near to the charged surface, $! so that

= &>,

force.

therefore

If

2

F

,

a-F]

^d ^ = +

ft
.

(ll

apply this result to a minute cavity of any shape

o.harcrftd

bodv.

We

have seen

23) that

(p.

ll^dS^

jlone

this equality is

follows that

it

cavity

the shape and size of the independent of

327

327

327

.................. ( 13 >

+ ^-i^-/9 -^7+yr oz da? 2

2

dy

It is the This equation is known as Poisson's equation. the satisfied differential form of the potential by equation

At pints where there are no that we get Laplace's equation

Ex ,Ey,E

z,

This

0, so charges we have p as a particular case of it.

the components of E, are given

often

is

=

electric

V 2 7=0

written

the equation

is

We may

E=(EX9 Ey9 18g) = -gradV.

V 2 7=-/>

or

dEx dEy + Hx Hy This equation

by

in the form

dEz __ ~dz~ P

'

often abbreviated to div

E = p,

the operator

acting upon any vector denoting the sum of the results of the of the operator on each of the com-

' closed snrface

il-ustnit^ '*

theorem that the surface is

e qual to the

volume

volume density inside (Gauss's Theorem, p 23) tht method we shall first apply it to the simpler case

tn^iiur ewrdinates.

ELECTRIC INTENSITY AND POTENTIAL Consider the element coordinates,

volume

of

whose centre

is

27

in rectangular coordinates of the

dxdydz,

The

x, y, z.

angular points of the element are x \

,

,

dx dif

,

dz.

;

so that the total

W^"

_

Theorem By

thus derive

Equation

.

inside or to the charge equal equ

M. is HUB a

^

V7

P~5?+

in rectangular

V W

coordinate,

Coordinates.

the mdius

makes with

Fig. 5.

Let-

PQflSrWTIT be

the element of volume.

surfaces: the intersection of the following

}.v

and r + idr;

radii

plane passbg through about the axis

Oz and

c^axally 4. ^

th..-

tie respectively.

elements of

MIL

forrm

The mlius

(2)

of semi-angles

The coordinates

volume are

fl-irfi?

ai

of the angular points

P = r - %dr, 9 - 4 d9,

$

- ^ d^>, and of

by the intersection and the sphere of radius r is clearly r sin

of the circle formed

C'*n* of serai -angle

Thv*

is

two spher

two planes passn r-^dr with a fi 0^ making angles <J>-$(Z$ and ^+id

0'

\

and over the inner

dr) sin 6

or r (r

dO

d

to

the

fluxes are therefore

+

and

The

J

%dr)sindd(j>x

(r

intensities

dr}.

2 dr

dr \

19F V -\--~dr

total flux over these

~

*

x

,

and

RSVW

The element

^rrdd) respectively. 2 o

arc

PQTU

of the plane surfaces

the

of

mean normal

intensities are

8 f, r

1

The areas

1

,

7

of the

The

dr x rdd.

137 A

9

f jr

I

dV

.

each other and to

surfaces are equal to

fluxes are therefore equal to ,

sin 6

\d(f>

^

+ drdOfiV

,

alld

their

^

2

13 8 F

(^-2

7

A

to**)'

sum being

L_ r2 sin2 5

We

thus find for the total flux over

all

.

2

3c/>

the six surfaces of the

element of volume

By Gauss's Theorem

this is

equal to the charge inside,

which

is

pdr = p Si>

that

r cr

This i

TIH-

i

s

therefore Poissou's

nemtor

^-

-

r3r

_,

+

r3tf

'

1

U hich

Equation in spherical coordinates,

^ + ,-=

3fl

F

'

Laplace

s

operator Vi takes in this

ELECTRIC INTENSITY AND POTENTIAL

31

Cylindrical Coordinates.

other system of orthogonal coordinates similarly. For instance in the case

may be

Any

treated

dr

of cylindrical coordinates r, 6, z the element of volume is bounded

by (1) two coaxal r and r

cylinders of radii

dr with their axes coincident with the axis of z, (2) two 4-

Fig

6.

planes inclined at an angle dd to one another and passing through the axis of z, (3) two parallel planes perpendicular to the axis of z and at a distance dz apart.

The volume

of the element is clearly dr x rdO x dz. The of the element being V, the flux over the at the centre potential outer cylindrical surface will be

and over the opposite fr

V

face

d ^^} 2 / edzx^(V~l^dr} dr V 2 or J

}

the total flux over the two faces being *

*>

**>

The area

of the plane inclined faces

intensity over

_1-

them +

d

^ d6

^ V

r do

(

c)u

\

i

r or)

>*-

dr dz, the mean normal

is

}

and the

total flux over

J

them

The area

of the faces perpendicular to

total flux over

them =

r dr

dO dz

-=

Qz

is

dr x rdd and the

.

oz-

So that the

flux over the

L_ (

dr

4.

_13F r dr

Thus the form which coordinates

whole six faces

is

+

= rp

rj

__ ^^ 36/ 2

(

dz 2

x rdrdddz.

}

Poisson's Equation takes in cylindrical

is _

2

+ '

-

"

__~

.(16).

ELECTKIC INTENSITY

32

AND POTENTIAL

The Uniqueness of the Solutions.

and rectangular branches of

in spherical, cylindrical Laplace's Equation is of the greatest importance in many

mathematical physics. that the

first

It is clear

from the preceding discussion

derivatives of the solutions of this

equation represent

a vector which flows out from a series of points uniformly in all Its applicahility to the theory of radiation, of condirections. duction of heat and electricity, to attraction as well as to electrical

hydrodynamics and gravitational and magnetic attractions is at

obvious.

The

of

equations arises from the them we only need to be given the value

differential utility of the foregoing

feet that if

V over

we can

solve

certain surfaces in order to obtain the

complete

distri-

This result depends upon the satisfies the equation theorem, which we shall now prove, that if V 3 V = - p throughout any region of space and has certain assigned "button of electric force in

the

field.

V

over surfaces bounding the region, then it is the function which satisfies these conditions. For if not let

only

V also

satisfy

the same conditions and let us write

the expression for Green's

and since

RT

Bnt

Theorem

V^F = V4r = - p

the surfaces,

we

this

is

V V in

Then

throughout the space and

F-

V=

find

a sum of squares so it can only vanish if each We thus have separately.

integral

term vanishes

U= V=

in equation (9).

;

ELECTRIC INTENSITY AND POTENTIAL

(V -

Since fl

F')

~ (F-

F')

dS

33

when

vanishes

also

is

given over the surface S, it follows that F is unique except for an additive constant when the value of p is assigned throughout the

and that of space r also,

3F ^

over the boundaries

the electric intensity

It

so that, in this case

:

on is

would lead us too

determined uniquely. consider the functions

to

afield

far

(Fourier's Series, Spherical and Zonal Harmonics and Bessel's Functions) which are the solutions of Laplace's Equation approFor the development of this priate to particular problems. interesting subject the reader may be referred to Byerly's Fourier s

Series

and Spherical Harmonics. Total Energy of a System of Charges.

We may

find the total energy of a system of charged bodies and potentials as follows. Since the

in terms of their charges

potential at any point of the field

taken over

all

equal to

is

1

1

-

1

the charged bodies in the

dr +

J *j

field, it will

dS

I

'

jjj

j

f

be re'duced

to l/n of its value if all the charges are reduced in the ratio n:l. Let n be any very large number, and suppose that initially all the

charges are at an infinite distance from, one another. Bring up l/n If F is the final of each element of charge to its final position. the at will this operation any potential point change during potential '

F

The work done

from

to

will lie

between

.

and

Vp

dr.

will lie II n of all the charges &

1 up the element p dr

in bringing

The work done

between

and

/

bring up a second potential at

??th

part of

If this process

is

1

1

n-JJJ all

between

F/??, 1

;

(

1

Vp r

dr.

ZSTow

and the work done in

Vpdr and ^

1 J

j

Vp

continued the work done in the 5th stage will

between 1 ffr

Vpdr and

up

This will raise the

the charges.

any point from V/n to 2

this second stage will lie

1

in bringing

( If -, >>- iii

Vpdr. '

dr. lie

ELECTRIC INTENSITY

34

The

AND POTENTIAL

work done in bringing up the whole of the charges from

total

will a state of infinite dissemination

between

lie

n

Vpdr

-tJJ

and This

is

of the system, equal to the total potential energy lies

which

between ,

,

Tr Vpdr and

When Iff/

n

is

increased indefinitely each of these values coincides

This

Vpdr.

the part due to the

is

are surface charges

we

complete expression for the total Is

volume charges.

If

have to add ^fJcrVdS. The energy of any system of charges

shall

therefore 1

r r r

T

i

r

an The Energy in the Field. In the preceding paragraph we have deduced an expression On energy in terms of the charges and their potentials. the view that electrical actions are transmitted a

for the

through medium, we should expect that the energy would reside in the medium. It is easy to obtain from the equation (17) an expression which admits of this interpretation.

Since p

=-V F 2

and

a-

But by Green's Theorem, allowing from the fact that the normal considered, this

is

that

-m-nt

the ,,t

we

for

the reversal of sign arising

is

now drawn

into

the space

equal to

'lljl&fo S--.

= dn *'

k

(19).

energy of the system the field contained an

is the same as if each amount of energy $E> per unit

ELECTRIC INTENSITY AND POTENTIAL

35

Stresses in the Meld.

Maxwell showed that the forces acting on any system of charged bodies could be attributed to a system of stresses in the medium in which they are embedded. The necessary and be the case is evidently that the resolved part, in any direction, of the resultant of all the forces acting on the parts of the system, arising from systems external to it, should be expressible in the form of an integral over any

sufficient condition for this to

and isolates the system. The alternative that would imply possibility part of the force was not transmitted across the boundary, through the action of the parts of the medium on one another, but arose fro m so-called action at a distance. surface which surrounds

Consider any surface S surrounding and isolating the system Let be the x component of the resultant of static charges e^ l e from all external electrical on force acting 1 arising systems. and the volume density at any point Then if is the potential p

X

V

where the integrations are extended throughout the volume The volume integrals will be capable of transenclosed by S. formation into integrals over the boundary surface S* if we can write "*"

dx [da?

dP

.

., lorm in the f

,

-=r-

4-

ox

w We have ,

dVyV = 1

-~- *-; ox ox-

x

'dy-

dQdE TTh

d

/3FV

^ ^" "o~ 2 ox \ox ]

dy-

'by

^

"" d_

\

.

oz

cy >

dx dy )

/3F 3F\

dy \div dy)

'by

1

dxdy d_

fdVY

2 dx \ dy)

__ dz 2

dz \dK dz )

'

'

2

das

v\

\dz

)

01 P.han

TT

ELECTRIC INTENSITY

8fi

will

Thus the integrand

be

in the

AMU

form desired

if

we put

of the resultant force parallel considering the components at similar surface integrals axes we should arrive

By

y and

to the

involving

^

''

-Ijffl'-f V-f Yi \3/J \3*/ ^w~2(l3y/

( 23 )>

/3Fy_,9FY) T~ O..

(24) \^^J>

lj/3Fy -r = -*U

tl

f

3F3F

.(25),

pxy and ^2.

In the new notation in terms of the to students of elasticity,

jp's,

which

will

be familiar

we may write

(26).

The

last integral is

taken over the enclosing surface, and I, m, n drawn away from the enclosed

are the direction cosines of dS,

volume.

Similarly

dg

+Z =

anl

1

l

1

If

+ mp yz + np zz ) dS

(28).

we adopt the standpoint that the action of the electric

charges on one another tliiii

(lpxz

(27)

|i xjr ,

p

ini

,

is

transmitted by the intervening medium, are the six components of the stress

pa pyxt pZXi pzy ,

which transmits the action. ice these quantities,

From the point of view of action at on the other hand, have no physical

ELECTRIC INTENSITY" AND POTENTIAL

37

In order to obtain a more definite picture of the physical nature of the supposed stresses let us consider the case in which

dS

equipotential surface, so that its normal is tangential to a line of force. Let the resultant electric intensity at dS be E, then

part of an

is

37 - -_

dV --

7ET IE

da

_

=r

mE

,

and

%z

pxx = ^E- (I- - m" - n-), = m - n - '^ Pyy - -m

and

_ - dV = nE,

dy *

-

~

2

?i

The components

2

I*

the

of

pyz = E-mn, = -n, Pzx x = E*lm.

),

per unit area across

force

dS

are

respectively

+ np2x = ^lE-f 71^^ = i wi

\ 2

-

............... ( 29 ).

lp x

Thus the resultant

traction

Next suppose that dS surface.

Its

Vf

l

The

A?

f

since

(ujtv paEflivZ) and

an-

f4j

equal

a

at

see (p. 57) these relations

shall

the furces which would be exerted on a unit charge shapes made in the dielectric.

in cavities of certain

jhtint

Taking the

if

east*

E

first,

E

we observe that

the force which

is

wnuH

K/ vxt-nel s^ch a cavity, owing to the existence of polarization, "itiiMl : the two ends, and the contribution from these r*

*

-.uit ehar-.je

'j

in

?

.1

*ii

r

1

^

-

ins-i ir

^

S.

Since similar expressions are obtained from the other two terms of it follows that

the integral

Polarized Shells.

A

shell is

polarized

a superficial distribution of polarization.

It may be regarded as a region bounded by two surfaces at an infinitesimal distance apart and carrying opposite charges on the

In general the direction of the polarization at any orientated in any manner with reference to the point may be normal to the surfaces, but the only case which is of any impor-

two

sides.

that in which the resultant polarization is always directed normal to the shell at every point. Such shells are the along said to be normally polarized, and we shall confine our attention tance

to

is

them.

are of great importance in the theory of electro-

They

magnetism.

Let AD,

EC

mally polarized positively

and

charged.

Let

point,

be a section of the surfaces bounding the nor-

AD being

shell,

EG

P

OP being

negatively

be a distant

equal to

r.

any point in the substance of the shell and is the nor-

is

ON

mal.

are

AD, QR, AB, DC, etc. infinitesimal. The angle

PON=0. Fig. 15.

Let

t

be the thickness and

P the polarization of

AD=QR = BC by equivalent to a shells it is

the shell at 0.

Denote the element of area

Then the element of the shell ADGB is doublet whose moment is PtdS. In dealing with dS.

convenient to introduce a

new quantity

called the

strength of the shell.

The strength of a shell at any point is equal to the product of the intensity of the polarization of the shell by the thickness of the shell at that point.

We

shall

moment

$

is

denote

equal to the

Now

by

moment

consider the

^

We

it

.

Then-^Ptf.

of a portion of the shell

have seen

(p.

whose area

Since is

PtdS

is

the

dS, the strength

of the shell per unit area.

P

potential at arising from the shell. 58) that the potential at a point distant r

DIELECTRIC MEDIA

due

to a doublet of

moment u

is

U3

-(-],

where

4?r OS \ rj

differentiation along the axis of the doublet.

element of the

shell

u,~6d>S and ^

= 3s

~

denotes

cs

In the ease of the ,

~

where

denotes

en

8/2

differentiation along the direction of the outward normal to the Thus the potential due to the element positive face of the shell.

dS

is

and that due

to the

whole

shell is

)dS where the surface integral

is

(8\

extended over the whole of the

positive surface of the shell.

The most important case which arises is that in which the has the same value at every point of the shell The strength

then said to be uniform or of uniform strength. a case may be taken outside the integral, and shell is

In such




is

................................. l91

the solid angle subtended by the entire shell at the

point P.

We

shall next calculate the potential energy of the shell in the First consider the potential energy of a doublet OQ which Let e at 0. carries a charge 4- e at Q and Q be the field.

V

and Q

potentials at

the doublet

Then the

respectively.

,

V

potential energy of

is

eVQ -eV = e(VQ - 7 = e.OQ*?=p*Co C6 )

7

ar -~-

ex

where

p. is

the

moment

cosines of its axis

5.

+w ar + n ar\ cz

)

cy

of the doublet

Now

apply

1A

............ (10),

;

and

Z,

m, n the direction

this result to the case of the

THE ELECTRON THEORY OF

04

dS

area Considering the element of

shell.

polarized

of the shell

x ~- so that the potential energy of the whole shell ,

tis

-

Since

is

the force outward along the normal from the

tf/i

of the

side

positive

number of lines

which thread the surface from the positive is the It is thus equal to Ny where

N

to the negative side.

number

the surface integral represents the

shell,

of force

of tubes of force

which leave the

shell

by the positive

side.

Polarization on the Electron Theory.

The

electron theory furnishes a very natural explanation of The chemical atoms out of which matter is built polarization. are regarded as consisting of a large number of electrified particles. The behaviour of these particles is considered to be different

according to whether the substance

quite a conductor or an insulator

In conductors, part, at any rate, of the electrons are very smallest electric field is sufficient to cause

,-, i,_ ,,

>i* i-tinn

'

electrons as distributed

in positions of stable equilibrium.

1

'li.

,,f

/

DIELECTRIC MKD1A they are pulled back by forces of the same nature as those which held them in equilibrium before the external field was applied. In the position of equilibrium which finally results, the force exerted on the electron by the external force tending to restore

it

field will just

balance the

to its original position of equilibrium.

It is clear from what has been said that the displacement of the ultimate electrified particles, which occurs in a dielectric when it is exposed to the action of an electric field, is equivalent to the

creation of so

many

doublets, one for each particle.

We

have

seen that the polarization which occurs in the dielectric under the same circumstances can be represented as due to the development of doublets in each element of volume. We shall now consider the whole matter from a more quantitative standpoint ; as a result of our investigation we shall see that the results of the polarization

theory can be obtained just as well from the properties of the doublets which develop from the displacement of the electrons.

Actual and Mean Values.

We

have already pointed out (p. 9) that in the electron theory we have to deal with different elements of electric charge A somewhat similar in different classes of problems which arise. distinction arises in connection with

many other physical

quantities

which determine the nature of the electric field. For instance in the discussion of this and the preceding chapter we have regarded the induction, the polarization and the electric intensity as vectors whose magnitudes changed only very gradually as we moved from one part of the

field to another.

We

have always thought of them

as though any alterations in their magnitudes which might occur, in a distance comparable with the distance between two molecules,

could safely be considered as negligible, provided the two points compared were both in the same medium. This method of treatment obviously becomes inadequate when our view of the phenomena is so highly magnified as to take into account the effects of individual electrons or even atoms.

between two

So

far,

parallel planes filled

we have considered the space with dielectric, when the planes

region in which the electric intensity has the same magnitude and direction at every point. It is clear, however, that the actual electric intensity, are

maintained at different

potentials,

as

a

THE ELECTRON

gg

an infinitesimal a unit charge occupying the force exerted on in both magnitude and direction volume will constantly change At places which are to another. from one part of this space actual force will be enormous close to an electron the sufficiently

with what

cornpirecf

may have any

we have

and

called the electric intensity,

it

direction whatever.

what can be the use of a conception of tempted to ask so much at variance with what we the electric intensity which is The answer is, of course, that most of believe to be the reality. with the our methods of experimenting are so coarse, compared these enormous differences atomic scale, that they do not detect of atomic magnitudes. order the of distances within which occur for the most part measure only the Our experimental arrangements a large number of atoms. values over

One

is

spaces containing

average

is not because possess validity as such far so values but because, experimental they are the true enable us to detect, happens as if the

The reason why our average values

everything

arrangements

were the true values. average values It remains to specify the average values we have been dealing Let < reprewith more accurately than we have done hitherto. sent one of the scalar functions or a component of one of the

which determine the state of the electric

vectors,

field.

For

fxiimpk $ may be the electrostatic potential at a point. Let r lie any small volume so chosen that its linear dimensions are large distances but small compared with the ei.anpared with atomic 1

,

distances within which changes in

are perceptible by the usual Then methods. the experimental average value of may be Ivfin ti! as the value of
lK.w* that

V-

F=-p

;

and since

=-grad

V,

etc.

of

F=-p,

it

#=-grad

V.

Thus

DIELECTBIC MEDIA

67

the average forces and potentials are the same as those which would obtain if the actual charges were replaced by a distribution of density equal to the average density at every point. It is clear that the induction, polarization and electric intensity in a dielectric are average values in the sense indicated, and that the results that we have deduced are valid if this is understood.

Potential due

to

the Displaced Electrons.

We

have seen that in the presence of an electric field the electrons are displaced, the positive in the direction of the field and the negative in the opposite direction. shall see that the displacement thus

We

produced of

is

moment

equivalent, for each electron, to the creation of a doublet = es, where e is the charge and s the displacement, p,

This doublet will contribute to the potential at

of that electron.

a distant ^ point

P

an amount ~~ ^~ ;

4-7T

)

,

and

if

there are v such

ds \rj

doublets per unit volume the total potential to which they will will be give rise at the point

P

i

err

d

n\,

^ os \rj \\\vp,[-\dT. 47rjJJ In general the different electrons in the atom will be variously situated so that they will not all undergo the same displacement a given electric field. We may divide them up into classes, the electrons in a class being characterised by a given value of for a given field. Suppose there are n such classes and let vp

s in all

s

,

p

jji

and

class.

sp

denote the values of

Then

it is

v,

p, s for the electrons of the

pth

clear that

(14).

We

shall

now

consider the relation between the

moment /^

of

The the doublets and the electric field which produces them. exact form of the relation between the restoring force and the displacement will depend on the arrangement of the electrons in At present our knowledge of this arrangement is very the atom. limited but, in any event, the restoring force must be a function of the displacement, which vanishes when the displacement is zero

THE ELECTRON THEORY OF

g$

from Taylor's Theorem that for small displacements the force must be proportional to some odd power of the

It follows

restoring

displacement

and since the frequency of the natural vibrations of by their optical properties, is independent of

;

bodies, as exemplified

natural to suppose that this power is the first. We shall assume, therefore, that when an electron of the pih class is displaced a small distance sp the restoring force is equal to

the amplitude,

-f

X p Sp,

it is

where \p

a positive quantity which

is

librium

constant and

When

the state of equiattained, the pull of the external electric field on the The oc component of balanced by the restoring force.

characteristic of this class of electrons.

electron

is

is is

3F the force on the electron

is

ep

where

,

^

V is

the actual part of

the potential at the electron whose charge is ep which arises from If scp is the # component of sp the presence of the external field. then, provided the reaction to the displacement is independent of Its

direction in space,

measured

the x component of the restoring force

in the positive direction of

%

is

--

os

p

and the equi-

Xj>

libriiini

value of this displacement

is

^--X^ Xow

..................... (15).

a moment's consideration shows that

when a charge

placed a distance xp the electrical effect is exactly the that which is produced by the creation of a doublet whose

For the displaced system e^,. which is obtained when a doublet

is

distance a

way

c.

.incidt-s

IN

^

that

apart

xp

is

ep is dis-

same as

moment

absolutely identical with that ep at a consisting of charges

is

superposed on the original system, in such displacement^ and the charge -ep

coincides with the

with the original Thus the displacement (15) charge +eP Mjuivalent to the creation of a doublet whose moment is .

dV 111

!h " "!>*-"*'

an

electric field the

medium

is

unpolarized charges forming the *\> .line SJMVIU.S is zero. Thus the potential V due to the p polarized ni^miii is that which arises from the totality of the doublets which

"ii'J

-

th- jMtt-niial due to the distribution of

DIELECTRIC MEDIA

From

equation (5) for the potential due to a single doublet we SFp, the part of Fp which arises from the doublet the x component of whose moment is given by (16), is

see that

T^TT

{(j3Cp

OXp

Now in addition to being made up of electrons the matter with which we are dealing possesses a coarser type of structure which we may term molecular. Each unit of molecular structure, which we may fying

it

by the

refer to as a molecule without necessarily thereby identi-

with the more definite chemical molecule,

is

characterized

fact that, referred to its axes of

electrons satisfy

symmetry, similarly situated identical structural conditions. In considering a

structure of this kind

clearly absurd to

it is

endow the

electrons,

we have

done, with the property of suffering a hypothetical restoring force which, for a given displacement, is independent of the direction of that displacement in space. As we are confining as

ourselves to the case of non-crystalline substances this difficulty may be overcome by taking sp pj yp ,zp to be the average

^x

displacement of all the electrons which belong to the class p in a given small region and recollecting that all directions for the axes

symmetry of the molecules are equally probable. Suppose that each molecule contains n electrons, so that p has all the values from 1 to n, then P the average contribution to P which arises from a molecule at the point a, b, c is of

V

&V

*-*

A

4"7r

^"D p "M p

p=1

i *-\

f\

l

/

*

(dap da p \rj

^"7

If the dimensions of the element of

pared with r

=

coordinates of

{(x

-

P and

molecules, the part

volume dr

is clearl

a)-

+ (y

if it,

dVp

+ (z

2

&)

^T"

~

'

I

cbp cbp \rj

'

^ ccp ccp \r

volume dr are small com2 where x, y, z are the }-,

c)

nevertheless, contains a large of

VP

which

\

arises

number

of

from the element of

where v

is

number

the

of molecules in unit volume.

'(37 3

/T\

.

373

Thus

fl^

r'--** 3 (17).

with By comparison

components

equations (7) are

of the polarization

=-!/

2

and (16) we find that the

P

o y

^

.(18).

P =-v 2

2 3_F

z

=

*

P= thus equal to the sum of the moments of all The dielectric coefficient doublets in unit volume.

T/ie polarization is

the equivalent

K

given by the relation

is

.(19).

Thus da

For crystalline media, \p will take different values for the different and c because the axes of the molecules are definitely

directions a, 6

orientated in such substances.

Xow if we

average over a large

number

?>V

of molecules, -

under

dap the sign of summation in (19) does not become equal to the value of in the

cif

same

region.

these depends

I'K-etrons in

for this.

The

first

upon the defimteness of the arrangement of the The electron whose type we have

the molecule. the suffix

indicated

There are two reasons

by p always subject, owing to the definite structure uf the molecule, to certain geometrical relationships with the other electrons in the same molecule This fact is not taken iiei-i

is

unn of in the definition of

F and

of -5-

oa

in*k'j*ndenT ni'.k'cule

and

of is

the

.

The second reason

is

arrangement of the electrons within the caused by the molecular rather than by the

DIELECTRIC MEDIA

71

electronic structure of matter. The nature of this second factor can be most readily brought out by considering the dielectric properties of an ideal kind of matter whose imaginary molecules

are so simply constituted that the

first

shall therefore consider the case of

We

factor does not occur.

a substance whose molecules

are monatomic and whose atoms give rise only to a single doublet each, under the influence of the electric field.

Case of the Ideal Simple Substance.

We

suppose each molecule of the substance to consist of a and that, under the influence of the external field, each atom, single atom develops a single doublet placed at its centre. The forces acting on one of the electrons whose displacement gives rise to this doublet consist of (1)

the restoring force called into play by

its

displacement,

the force arising from the charges in the field, including (2) the doublets of the polarized medium not situated in its immediate

neighbourhood, the force arising from neighbourhood of the atom. (3)

When

It is clear

is

equilibrium

the doublets in the immediate

established

from our discussion of the

electric intensity in dielectric

media that

(2) is equal to the electric intensity

remains to discover the nature of consideration

as

centre -^

cs

comparable with

enough

it.

=

dV .

It

Co

About the doublet under

describe a sphere

,3F

small that the value of

(3).

-

whose radius

is

so

in a distance does not vary appreciably rr

At the same time the sphere must be big number of molecules. The force (3) will

to contain a large

be equal to the force exerted by the doublets in this sphere on the electron under consideration. This will only be true provided the

dimensions of the sphere are within the assigned limits otherwise this force will not be independent of the radius of the sphere. :

To

calculate the

magnitude of

(3)

we suppose the

spherical

THE ELECTKUJN

72

On account of this, and the centre of a small spherical cavity. fact that the doublets behave on the average like the to owing the equivalent polarization P, the ^doublet at the centre will be a force, additional to E, whose amount is determined acted on by

on the walls of the cavity. by the equivalent polarization charge at any point, on the surface area cos This is equal to per unit an angle with the makes of the sphere, the radius to which

P

direction of P.

The

resultant force

over the spherical surface

is

/Y

thus

1

1

due to the whole distribution

&P -

j

2

2

cos 0dS,

where the integral

This extends over the surface of the sphere whose radius is r. of radius the of the is and to sphere. independent equal |Pe

is

The remaining part of (3) consists of the force which would be caused by the doublets which we have removed, if they had not been removed. This will depend very much on the geometrical arrangement of the atoms among one another. In certain parA doublet situated at a point ticular cases this force vanishes.

whose coordinates are

#, y, z,

with respect to the centre as origin,

and whose moment has components equal to to a force at the centre, whose x component

JJL

X , /^,

^,

will give rise

is

*

,

4TT

where If the

r2

=# + 2

2 ?/

4- z*.

atoms are arranged fortuitously so that any one position

in the sphere is as likely as

mean

another the

values

Hey

= ~xz =

and

- r2

2

__ -

:3t/

-r

2

'

_ 3^ - r 2

-V

3 L =-

(ofi -f

It

u~ 4- z-}

J

2

3r2

=-

follows that the force

arising from the doublets which we out of the cavity vanishes on the average, if the atoms are arranged fortuitously. The same is true if they are in regular cubical order* It follows, in either case, if thv atoiti* haw the .simple constitution we have that the

liaw

ivitiitYed

imagined,

*

H.

A. Lorentz, Theory of Electrons, p. 306.

DIELECTRIC MEDIA force (3)

When

=^Pe.

73

the molecules have a less symmetrical

distribution, the additional force arising from the molecules which we took out of the spherical cavity will still be proportional to the

polarization P, so that we can represent the more complex cases if we replace the factor -J by an unknown factor a depending on the

configuration of the molecules.

Returning to the more symmetrical distribution, we see that the total force acting on the electron at the centre of the atom under consideration is, on the average,

..................... (20),

so that

comparing with formulae

-P,

K

,

whence

If

-

we apply

(19), since

n=I,

/0 _

P=-

,

and

and

(16), (18)

*

x

..................... ^ } (21),

1

this formula to the case of a gas,

we

see that the

only one of the quantities on the right hand side which varies with the density of the gas is v, the number of molecules per cubic centimetre.

K K

This

is

proportional to the density, so that for a gas

1

-

+2

should be r to the densitv. proportional r

The

results of ex-

periments are in agreement with this formula within the limits of experimental error, although the experimental measurements of the dielectric constants of gases are not very exact.

When we come dispersion of light, which K is replaced

to consider the

we

phenomena

of refraction

and

shall see that a very similar formula, in

2 by n connecting the ,

refractive index n with

It seems the density, can be developed along similar lines. advisable to postpone the detailed discussion of the experimental evidence for and against these formulae until the optical phenomena

are considered, as the evidence will then be vA/r^

ol->oll

ri/-v\iT

fii-t-ii >v-k

-f/-\

/~/-\Y-

01

H

ti-v*

-i-T^ti

much more -h^or

/-\-T

f

n *^

complete. 4-Tt-/-v

voo O/-M-I o

THE ELECTRON THEORY OF

74

value why the average

of

is

^

not equal to

^

.

This one depends

itself and not on the on the complexity of the atomic structure constitution. atomic an The matter that fact mere possesses

by considering a very contain a very large to atom an Suppose exaggerated case. If such an atom is held. all number of electrons very loosely nature of this factor can best be realized

field it will behave like a conductor of the placed in an electric same size and shape so that there will be no field acting on the ;

The

electrons in the interior.

atom

will

move

electrons towards the outside of the

so as to shield those inside

from the action of the

The same effect will also occur to a smaller extent even when the number of electrons is comparatively small external electric force.

and their displacements are inconsiderable. It is clear that the a small volume of the average value of the force throughout material

is

different

from the average value taken over a particular

type of electron.

The

force acting on an electron inside a molecule will arise from the charges outside the molecule and partly from the pirtly doublets inside the molecule itself. We can regard each molecule

as equivalent to a simple atom possessing the same average electric moment, so that the force acting on an electron inside a molecule arising from external causes will

be

e

(E -f aP) where the constant :

depend on the geometrical configuration of the molecules, when the distribution is a fortuitous one as taking the value in a fluid. The way in which the second part of the force on the a will

internal electron depends upon the external field may be realized by considering the conditions which are necessary in order to

change the displacements of

all

the electrons in a given ratio. so that

The displacements are proportional to the forces acting, this means that the force acting on an electron in the field changed in the same ratio at every point.

will

be

Now

the force arising from a given doublet is proportional to the moment of that doublet, so that the part of the force acting on the given electron which arises from other doublets in the same atom will be altered in the

same

ratio as the total force at

that the difference

part of the force

between

which

is

any point in the field. It follows and the total force, which is the

this

of external origin,

must be changed in

DIELECTRIC MEDI^

75

the external field may change, the force acting on any assigned electron will always be changed in the same proportion. This result may be represented by putting M -'pv"p

o

CSp

where

Lp

is

v-*-*

"r

*--

................(

i

Lo

)

5

a constant characteristic of the pth class of electrons.

Comparing with

p.

73 we see that

tf(E + aP)

=P=

(

K -l)E,

so that

and

~~

q

1

1

- a2z/ \pLp ep

z

_

= av 2

\p Lp e/

('25).

P=I

\p Lp and

ep

,

but they

will

vary for different electrons in the same molecule have the same value for corresponding electrons in

may

different molecules of the

The expression on the

same substance.

may therefore be represented by a summation over each molecule multiplied by the number of molecules of the sub-

right hand

side

We

stance in unit volume.

therefore find that

(26),

where k

We

is

a constant and p

shall

significance

find

that

is

the

when we come to The investigation

the density of the substance. coefficients

\p have an important phenomenon of optical

consider the

leading up to formula (25) will not apply to optical problems without modification, as the displacement of the" electrons in such cases is not necessarily always in phase dispersion.

with the corresponding

When we

"

force."

are dealing merely with the electrostatic behaviour f*xr\ a-flvYrrl fn -npo'lpr flip r-mnnlir^f irm*-

:

MAGNETISM

H (l 1

2

of magnetic intensity can be are of lines of force whose equations

The

1

iii*aiis

field

da___

(8

(4

The magnetic

1

The equation

1

dy

potential

satisfied

_

dz

fi at

by

ds

___

,^\

any point

O in

mapped out by

is

free space is

vn=o 1

5

If rf$

)

is

an element of

(4).

any closed surface not intersecting

a magnetic medium, that- is to say, one which lies entirely in a medium of permeability unity, and if I, m, n are the direction cosines of the normal to this element, then

taken over the whole of the closed surface

is

equal to zero.

This

the magnetic analogue of Gauss's Theorem for the free aether, and is true since each magnetic substance contains equal and is

magnetic charges. It follows that in free space the is a solenoidal vector. intensity magnetic c*

opposite

*

H

of a system of magnets may be obtained, as system of electric charges, by bringing up equal fractions of the final system, one at a time, from a state of infinite (0)

The energy

in the case of the

dissemination. se pirate

As we do not wish

magnetic charges

it is

to

contemplate the existence of

desirable to regard the disseminated

elements as magnets and not charges. This introduces the intensity of magnetization (IX) Iy Iz ) instead of the density of In this way the energy of charges into the final expressions. a system of magnets is found to be ,

the integral being taken throughout the magnetized matter.

It

follows from this expression, by a calculation similar to that carried out in the similar electrostatic case, that the energy per unit

volume of the

the free aether

field in

is

.............................. (6).

The '

points involved here are discussed at length by J Jeans --.--. -

MAGNETISM

The magnetic

(7)

be represented as arising from

forces can

the following system of Maxwell stresses in the aether

_

if/any-

(2Q\*

:

cQ\~]

Coy)

}

an an

= 80 8O d,x

"dz

fc


o)/4 was the solid angle subtended by the shell at the point where the An electric current is thus equivalent in potential was measured. its

magnetic action

4>7rA

to a

magnetic shell whose strength

is

equal to

times the intensity of the current.

There

is

an important difference between the field due to an and that arising from the pinmivalpnt, rna.o-nAtin

electric current

!l
the

iin/i'tion t|n;fi

to

Hence

since integration"

from

P

to

Q

P

along the same path adds nothing to The difference between the integral along PSQ and

ami kick again from Q to ill*-

is clear,

P

and ending at Q in each case, is all round the circuit in the

same integral taken

PSQ:

I from Q through R to

since the value of

/ taken from

tor the entire closed

cY

P

to

P.

contour we have

cl

oZ

Q

along

PRQ

is

ELECTROMAGNETISM

9o

where the surface integral extends over any surface bounded by the contour, and where the cosines are the direction cosines of the normal to the surface

at the .point of integration.

It

will

be

observed that the positive direction of the normal is towards that side of the surface from which a right-handed screw would move if it

were turned in the same sense as that of the integration

round the contour.

The First

Law

of Electrodynamics.

Let us apply Stokes's Theorem to the case in which whose components are x yt magnetic intensity

R

is

H H H

H

,

the

We

z.

then have

|

(Hx dx -f Hydy

H dz)

-4-

2

dHy 2 = n\(cH M -7=r~ --

^

nx

J](\dy dy

cH x --^-~ + fdff

cos nu J

\cz

oz )

cH

- -^x + fcH -^ to / 1t

\ I

\

But we have seen

tj

is

cy

7 ~,

J

that

\Hx dx + H where j

)

cos nerdS.

dy

+ H2 dz = 4TrAj,

H

we mean the vector whose components are the where by rot taken in succesquantities on the left hand side of equations (4), This equation is a vector equation, that is to say, of rot independently for each of the components sion.

it

H

holds

and j

respectively.

The Electric Current.

The electron theory regards the electric current which flows It supposes that in a conalong a wire as a convection current. duct ir there are a number of charged particles which do not 1

execute small displacements about a position of equilibrium, ias in the ease of those electrons which we considered when we were discussing the behaviour of dielectric media, but

which are able

move

In a freely from one part of the conductor to another. metallic conductor these particles are believed to be electrons and to

them from the bound which only undergo small displacements from their position of equilibrium when an electric field is made to act on thrin. The free electrons in a metal are believed to be in much tin.- sune condition as the molecules of a When we come, gas.

aiv culled "free electrons" to distinguish electrons,

later MIL in consider it

is YI

the evidence for this belief,

TV strong and that the resemblance

we

shall see that

a very close one. In th-cji>eot liquid electrolytes the charged particles are of atomic "i- nmircular dimensions, and in many cases of electric conduction

thi'Mi^li

In

gases this

all

is

is

the case also.

these cases the charged particles, of whatever nature,

aiv K-ln.'Yvd to be

moving about

irregularly in all directions,

they are not subjected to the action of an electric ^hyii fill- liMl-'ll does Tint fiUKi* !im' fi-i-now, ^f .-.r .,1,-,,,4- , :.L .-,+,*.-,

even field. _.

EL fcXTROM AGX ETISM

on the be moving in any average, the particles are just as likely one direction as in external electric an of other. effect The any field is to on definite drift, so a motion the superpose irregular that on the inuve in the the average t.
^s.

3y J

\

Since these relations are true for any surface bounded by the contour, the surface integrals must be identically equal so that ;

cE1 _cEI = _ cy

l

ct

a^_a^ = dz

x 1 cB_

cz __

4

ex

ct

^_^ = _j j j

an. I

(,2l

tii-M

in

the energy of the magnetic latHdH, which represents the bar. In a small the aether of the spice occupied by

jntrrvul of

time dt the increment in the magnetic energy in the

clone against the back electroThis must be equal to the work is work This motive force of induction.

Sothat

(I

+

t

B= H

Thus It

is

4-

well

/

in

to

we

H^A-.

accordance with Chap. VI and

A =-= l

4-TrA

c

understand clearly the difference between the are using and the two systems, the electrostatic

system of units system and the electromagnetic system respectively, which are most frequently used in books dealing with the theory of electricity,

The

unit of electric charge in the

electrostatic

system

is

V4?r

times our unit of electric charge; but it is not this difference so much as the difference in the units in which the magnetic quantities are

moment. obtained

On

measured which

from the unit of

electric

quantities are then obtained unity in the equation I

This

fixes

desirable to emphasize at the system the unit of current is

it is

the electrostatic

charge,

by giving

A

and the magnetic the arbitrary value

Hds = 4i7rAi.

the unit of magnetic force and so determines the unit

magnetic charge. In our units the measure of i is V4?r times greater and A is 4-Trc times less than in the electrostatic system, so f

that our unit of

magnetic force is c\/4?r times greater than the Since has the same value on all systems which have the same unit of mechanical force our unit of pole

electrostatic unit.

.strength is cV4-7r

mH

times smaller than the electrostatic unit.

The electromagnetic system of units sets nut

also

makes

A

1,

but

it

by denning the unit magnetic pole as that which repels an equal pule at unit distance with a force of one clvne. On this

113

ELECTROMAGXETLSM

system the unit of pole strength is therefore \ 4-jr limes, our unit and 1 c times the electrostatic unit. The electrostatic unit of pole thus greater than the electromagnetic unit by the on cms. per sec. It follows that the measure of

strength

is

factor 3

x 10

H

1?l

the electromagnetic system

is

1

"e

times

its

measure on the

electro-

Whence, by considering the equation \Hds = Tri true on both these systems, it follows that the measure ot

static system.

which e

is

on the electrostatic system

is c

times greater than on the electrounit, of electric charge

Thus the electromagnetic

magnetic system. is c times the electrostatic

unit,

and

c \ 4ir

times our unit of

electric charge.

when the unit of electric charge is defined as which that repels an equal and similar charge at unit distance with a force equal to 1/47T dynes, and the quantities which can be derived from it without making use of the two laws of electroIt follows that

magnetism are measured

In terms of units

this unit of electric charge

:

which are based on and when in addition the unit of

magnetic pole strength is defined as the strength of that pole which repels an equal and similar pole at unit distance with a force of 1/47T dynes,

and when the quantities which can be derived

from this without making use of the two laws of electromagnet ism are measured in units which are based on this unit of magnetic pole strength: then the two laws of electromagnetic induction

become

:

(1)

rot .0

(2)

=-

~ c

In the sequel

and

we

........................... ilSi.

ct

shall always use (17)

and i!8l rather than

(

(8).

Magnetic Force due

We

have seen (Chap,

shell of strength



to

an Element of Electric Current.

v, p.

85) that the force due to a magnetic can be represented as

at a distant point

P

shell. Since a arising from each element of the boundary of the current of strength i placed in a medium whose permeability is JJL

causes the same distribution of magnetic intensity as a shell of

1 1

= pi i\ it fallows that each element ids of an can be Actnr eiinvnt regarded as giving rise at every point in magnetic intensity of amount

*,} i

ELECTHt M AGXETISM

4

!^'!ii

*

*l

i

*

^=

JarA/jLi

;


= 2~ at

JJ dt

I

J

EP = nj+

_

When

ct

&r and, from

the value of

I

-^- S^c

to

Thus the

= 2?r- cE

if

,

{/,/-.

ct

!,

t

H

ffoE do) = 1 icE cir = 5-y. z 2a ct

-^ 47rJ J c?

j

I

-TT

J

PQ

11

cuts both planes \ve

ip

J ct

and opposite

)

(

10




Hurmuzescu 3*0010 x 10

10

Perot and Fabry 2'9973 x 10 10

The mean

of these quantities c

is

= 3-0001

x 10 10 cms./sec.

For the velocity of propagation of electromagnetic waves in Blondlot and Gutton J by

air the following values are collected

Blondlot 3-022 x 10

10 ,

:

2'964 x 10

10 ?

2'980

x 10 x 10 10 10

Trowbridge and Duane Mac-Lean

2-9911 x 10 10

Saunders

2-997

The mean

{"

3'003

2*982 x 10 10

these quantities

,

x 10 10

is

2-991 x 10 W cms./sec. * r

Ann. der Phys.

vol. xsxiv. p.

551 (1888).

Imports du Congrea de Physique,

Paris, 1900, vol. n. p. 267.

For the velocity of

light in free aether

Gornu* gives

as the

*0027 x 10 10 cins./sec. Dividing by most probable value 3'0013 1*000294 the refractive index of air referred to a vacuum, this gives for the velocity of light in air

-0027 x 10 10 cms./sec.

3-0004

The

velocity of electric

waves

known with much

is

less ac-

curacy than the other two quantities but they are undoubtedly all three identical in value within the limits of experimental error involved in each case.

Since the velocity of propagation light

in

magnetic

and

dielectric

is

c/V/4/c,

the velocity of

media should be inversely

proportional to the square root of the product of the magnetic permeability and the dielectric constant. Since it follows from the light that the refractive index n of a medium is inversely as the velocity of propagation of the light through it, it follows that for different media of the same magnetic per-

wave theory of

meability n-

oc

K.

This law has not been found to be even approximately verified In fact, a moment's confor the waves which constitute light. sideration shows that it

must be wrong,

since it

would make n

of dispersion shows that n is a function of fact that n* is not proportional to K is not to be regarded as an objection to the electromagnetic The theory on which it has been deduced is theory of light. constant, whereas the

phenomenon the wave-length. The

when

applied to the free aether but its scope is not wide enough properly to account for the optical behaviour of material media. The reason for this is that material media contain

exact

charged particles which are set into motion by the and magnetic forces of the light waves, and it is necessary consider the dynamics of these particles to account satisfactorily

electrically electric

to

for the optical

If

behaviour of such media.

we turn from

light

waves to the

electrical vibrations of

much

lower frequency emitted by the Hertzian oscillator the state of affairs is very different. The period of these vibrations is, as a rule,

great compared with the natural periods of the electrons that the motion of the

in the molecules of the substance, so *

Loc.

cit. p.

246.

ELECTROMAGNETIC WAVES

J-J4

rlrctn.ii>

ihi.-

i

!

hi-

the

same

magnitude as

.sum-

ihr

,.f

much

i>

litfht

wave.

as

it

the

would be under a steady instantaneous

field

value of that

Under these circumstances the material

nvated as u continuous medium of definite dielectric character the velocity of propac..rrtk-i'iit if, and fr waves of this be should media different inversely as the square roots gation in ran

In*

This conclusion

i.t'thr dielectric coefficients.

is

substantiated

by

Thus A. D. Cole* found the refractive

ivsuhs ofYxjieriinents. ind'X t* water t be 811 whereas tin-

its

dielectric coefficient

tc

= 80.

With other substances the agreement appears to be satisfactory within file nit her considerable limits of error of the determinations if ihe dielectric coefficients f. defer to the next chapter the consideration of the

\\\* >hall

of bodies towards light different than that predicted by the simple form of the electromagnetic

make the behaviour

causes which

There are, however, a theory which we have been discussing. of phenomena exhibited by electromagnetic waves in

number

matter which are partly true for light waves for true very long waves. The rest of this chapter strictly will be occupied with an account of some of these. their relation to

and

Properties of a Plane-Polarised Electromagnetic Wave.

A

solution of the equation

provided is

J-'-f

-M

2

-Hi 2

=L

-^ = a~-u

is.

"

The expression on the right-hand

side

a complex quantity, being equal to

The

real part

of a therefore represents a disturbance of

X and amplitude

which

wave-

propagated along the straight line .r/ = ?/'w = with constant .?/?? amplitude it Q and constant It is thus the velocity a. appropriate specification of a monochromatic train of plane waves of If we take wave-length X. the direction of propagation to be along the axis of z we shall length

is

t

*

*

IHVrf.

Ann.

vol. LVII. p.

290 (1896).

Fleming, Principles of Electric

Wave Telegraphy,

p.

320.

ELECTROMAGNETIC WAVES have

n

=

1

and

u

be equal to u Q e

will

125

T^~~>

an( j its real

part to u cos

Let the

will

for

(at

Ex

,

the x component of the

the wave front, and consider the train of

Ey = E

which

..................... (12).

2)

real part of u represent

electric intensity in

waves

^A-

=

z

The

0.

electric intensity in this train

be completely specified by the equations

>* ?

Ex = real part of

Ex = E

or

Q

It is clear that

ables such

cos

~(at-z) A.

~z

.................. (13).

any equation between functions of complex

as, for

vari-

example,

F

l

(a, iv)

= F.

2 (a!,

Real part of

Imaginary

part-

of

iy)

two equations

involves the separate truth of the

and

(at

^ = Real part of F

2

F = Imaginary l

part of F.2j

=V

1 would be equal to a real quantity, which is This principle effects considerable simplification in the working out of problems arising in connection with the propagation of waves, as it enables us to work with the complex solution

o therwise

i

absurd.

and then pick out the

The advantage

real parts at the end of our calculations. of this lies in the fact that the complex equations

are usually simpler than their real equivalents.

Suppose that we are dealing with the train of plane waves propagated along the axis of z. Each of the vectors x E>n Ez x z which serve to specify the state of the medium at any

E

,

,

H H H ,

, tJ

-~

~

(af r) ~* The values point at any instant must be of the form u Q e however are not independent but have to satisfy the six equations .

on

p.

116,

viz.

cH

z

cH,

= cEx ic

t

cy

cz

c

ct

dHx

cHz

K

cE

,

'

ELECTROMAGNETIC WAVES

12(5

>

As

eacli

9y

8^

c

?

?

?#

c

dt

C&y

C&x

?#

%

p d*2z

r ^ = ct

we have

.

i

9

2-7T

---

X

a,

x

e d

=

^r

9^

c

of the dependent variables const,

dy

is

of the form

^

*

= n0,

9

-~

=

oz

.

^

2-7T __

(14)

\

/C(7

, .

O IT

rr

L*

that e yrw

The

eqiicitinns

added

to

above

.(15).

may

all

them, but this would

have a constant of integration mean merely the superposition

) ,

Eg =

;

ELECTROMAGNETIC WAVES '

x

=0.

=

H,;

- cos

.....

JLtfl

6X.,e*M-y* "*--< ~*.\

a., p.,

and the reflects

wave by

!

Ex " = X^''-'-?^---^ /// =

/?" = --

0,

E, = !

V'

0.

=

Ct;

c^0,X,e^^'y^^' s ^^.

......

-/*!

#,"= - .JL

s i ri

ft,

A>*3'.v-^*;-:r^

.

sA*i

Since the boundary conditions have to be satisfied at of the time, for all of

tile that-

factor in these expressions

Thu>

them.

A

=

Jic/i

determined by the medium,

= A^

'./..

that

.so

Alsiu

,.

;

=

t/j

ci, ;

.

boundary conditions must be values of

y,

the velocity

Hence

A=

so that the exponential factor in u must be Thus fii sin ft = fi* sin ft =/3- sin ft= ;

=

sin ft, so that the angle of reflexion in equal magnitude to the angle of incidence. Also ft

sin ft

/?2

ci

is

and

common 3 sin ft.

4

2

must be

(ij

.(17).

sn Thus the

^-

satisfied also for all

to each of the vectors.

Therefore sin

values

all

must be the same

retractive index or the ratio of the sine of the angle

of incidence to the sine of the angle of reflexion is equal to the ratio of the velocities of propagation of the light in the t\vo

media. of sin

$i

Also with the convention as to the signs of : and G S and and sin ds which is here adopted, in conformity with

general usage, cos

The boundary

#,

=

cos

X

.

conditions will

now be

satisfied if

X^X, = X, cos al ul

a

ft

(X,

AV) "

=

cos 0,

X

19),

1

a.2 jjL2

sin ft (X.-r '

1

-

'

(IS),

AV)= a. '

sin

ft, ~

J,

r20).

"

"

2

9

-i

ELECTROMAGNETIC WAVES

&=

Sim-.- sin ft sin

i

and (20) are identical. equations (18)

>

we find S,himr ^nations .

li.HH.

A"

1

to K in calculating a from (28) does not exert any important influence on the result. The formula also does not the

suggest observed difference in colour between the incident and transmitted

These discrepancies arise from the fact that the current by discrete electrons, with the consequence that both and K are functions of p. The reason for this will be made

light. is

a

carried

clearer in the sequel

;

we

now

shall

phenomena which

consider the

attend the reflexion of light at a conducting surface.

Metallic Reflexion.

The problem of metallic reflexion is very similar to that furnished by the case of reflexion at the boundary between two The same conditions as to continuity of the insulating media. tangential electric and magnetic forces and of the normal electric and magnetic inductions have to be satisfied in both cases. The difference arises from the conducting power of the metallic medium, and we have seen that the type of theory proper to an insulating medium accounts for the propagation of waves in a conducting

medium

if

quantity #/ #2' its

we

replace the

icr^/p,

where

dielectric coefficient

cr.2

constant by the complex

dielectric is

and p

the conductivity of the medium, the frequency of the waves. It is

the boundary conditions cannot be fitted by the method previously adopted, the only change made being that the real quantity K2 in the former problem is replaced by natural therefore to see

if

f

'

the complex quantity K.2 where tcz is the real dielectric icr^jp, coefficient. We shall consider here only the case of waves polarized in the plane of incidence.

A

more complete discussion may be

found in Drude's Lehrbuch der Optik,

p.

334.

The incident wave being

Ex = X,e^ (, then

l9



^

are constant coefficients.

e's

of motion are given for each q

The equations

by the extended

Lagrange's equation*

dt\dqs j

They

if

dqs

are therefore /O

and

dqs

dqs

dq s

X varies as e

ipt

""

/O Q\

*

S\

i

i

the forced vibrations are given by B.

The natural

vibrations

be obtained when the external

will

X

Those corresponding to the electric intensity is equal to zero* be determined will therefore by the equation displacement qs

&? + VsQs + e They

be proportional to

will

Evidently

TTS is

e

complex and

i7rst

s

=

qs

(40).

where

if ive

put

TTH is

irs

a root of

= p + iks s p s

,

will

the frequency of the corresponding principal period and k8 decay factor. Since

z

we have

>

/3 S

k^)

(p s

p/ =

Hence and

-^

6 g /^

-I-

7^

= 0,

j~^

(41),

'

ft

=

ry

*

r

Lord Bayleigh, Theory of Sound,

s

vol.

i.

chap.

v.

be its

in wie ^^^ of volume. in any sufficiently large element electrons n the over all the volume unit in polarization of such elements If v is the number

The summations

will

be

=vS

e,

S

n

= fCo

where

Thus

a

for

medium

of unit

2*

the complex magnetic permeability

m is given by

refractive index

m = 1+ 2

-

where n ^_

~- 2

It

is

m r (a/

V

-f

evident that the refractive index

must be independent of

The

n and v except in the combination nv. particular values of n and v are arbitrary 'except that n has to be a sufficiently large number. The product nv is equal to the number of electrons in unit volume of the substance and is therefore a characteristic

The

constant.

requisite independence

is

secured by the fact that

on large the constants which enter into (44) keep summation s. Thus values of the for different themselves repeating in (44) is really a summation over the different principal modes of

when

n

is

mode is multiplied by the number of times The total number of terms, coincident equal to three times the number of electrons

vibration in which each it

or

occurs in unit volume. is

otherwise,

present in unit volume.

Formula (44) is of the Sellmeier type except for the inclusion of the dissipation term. With energy functions of the type now under consideration, the relation between the refractive index and the density of the substance is not an obvious one, since the constants A i/r g 2 and 8 will involve the density in virtue of the relations on pp. 173 175. A formula of the Lorentz type would, however, arise if we assume that the only part of the force on an electron which depends on the density of the medium is = aP, where a is a constant and P is the polarization. Formula (44) then becomes t9 ,

<j>

~

2

1

-a

'

w -1 a

(

)

,

-1

V/

2

where the constants are now somewhat portional to the density of the substance independent of it.

A s/

different.

and

\fr 8

and

is proare nearly

(j> 8

W

and F are not In the general case in which the functions T, of the values of x8 etc. sums reducible to squares simultaneously Sn simultaneous

are the solutions of

written

down

linear equations

in the form of determinants.

and can be

Consequently these

determinants enter into the expression for the refractive index and

make it difficult to handle except by approximate methods. In general the symmetrical coefficients which lie along the axes of the determinants are large compared with the remaining imsyminetrical coefficients

as a series of

magnitude.

sums

;

so that the determinants can be

expanded which decrease progressively in can be shown that the Lorentz and

of products

In this way

it

Sellmeier types of formulae result in virtue of approximations which are equivalent to the physical assumptions which have already been

We

shall

made

in

deducing them.

now return

to the behaviour in the

neighbourhood of

an absorption band and the residual rays, using the simpler formula (44) instead of (15). .,

m

19

near the Critical Frequencies. Absorption and Reflexion natural periods, In the neighborhood of one of the close a as approximation, sufficiently we shall have,

= Putting 7?i n on as find, p. 165,

(1

-

IK)

where n and

are real

/c

-n

and

so,yp=p89

positive

we

N

(48))

A=q

where

t

,

5=A

g

g

(50)

When

the absorption

wave-length we

is

small in a distance

get, as before, to a first

compared with one

approximation

Substituting the values of A, B, y and 8

we

find

^-

vanishes

if

we may put ty s2 =p, as a sufficient approximavalue of p for which the absorption is a maximum

In the fraction tion. is

Thus the

given by

Thus the corresponding true natural frequency is the frequency which the absorption is a maximum. It is somewhat less than the constant tyf which enters into the dispersion formula. for

We

shall

now turn

radiation reflected from

to the

problem of the intensity of the a surface of the substance under con-

,

ASSUMPTION AND SELECTIVE REFLEXION

sideration at normal incidence.

2 ?i

Substituting the values of

/c

2

?i

it is

_n

this

by given by

Denoting

already brought forward show that

~

(1

and

+

-?i/c

A:

2

)

-

/r,

IV

hJ

considerations

2?i 4- 1

found previously, we

get

2

\^-iY

_

l

1

f

/^7& + 2ABy

.

By

,

where

We menon

have seen that

for

substances which exhibit the pheno-

body colour U-KT is small compared with unity, and for it is necessary this to be the case for the particular value 7 = that 8 should be large compared with B. There is no guarantee of

that this will be the case with substances which give rise to the residual rays, since Nichols has shown that in the case of quartz the amount of the residual rays which are transmitted through

a slab of the substance only two to three wave-lengths thick is The value of the extinction incapable of experimental detection. coefficient, ntc, for

such substances,

unity or greater,

and

may

therefore be of the order

this corresponds to a value

comparable with that of

It does not

of

B

at least

seem

likely that there is any approximation of general application in the case of the residual rays which leads to any very marked simplification of

the formulae.

It is therefore necessary to evaluate the formulae

troublesome process. The involved are obtainable from the correspond-

in each particular case

constants

A

S.

and

B

and

this is a

ing constants in the usual Sellmeier dispersion formulae and so 2 2 also is (=7-f_p ). Determinations of 8 from the experimental

^

results

do not seem to have been carried out as

yet,

but the value

the most important factor in determining the proportion of the incident energy reflected. of 8

is

maximum

\

180

AND

DISPERSION, ABSORPTION

The of 7 or

of the curves precise nature

X depends upon

which express p 2 as a function

the values of the various

constants.

Nevertheless they always possess certain common features the example in the accompanying figure. are exhibited

by

which

The

ordinates represent the percentage of the incident energy which 2 and the abscissae are the values of 100 p is reflected, i.e.

they

6-5

;

7

8

7-5

95

8-5

10-5

4 Wave-lengths in 10~ cm.

Fig. 26.

wave-lengths (X = 2?rc/p) of the incident radiation. Alt the constants except s have the same value in each of the graphs = 2'05, A,. = 2-563 x 10~H 1, 2 and 3. The common constants are:

are the

^

and f/=4'53xl0 28 and in graph 3,

.

^=

indicates

the value

In graph 1, 1-42 x 10 13

.

X. of

,=0: in

The

X which

graph

2,

^=l'42xlO

vertical line at

corresnonds tn

la :

X=8'855

t,l^

^Mfi^i

AND SELECTIVE REFLEXION

DISPERSION, ABSORPTION

frequency tys

The points marked thus

.

mental measurements of 100 p 2

:

x, are Nichols's experi-

we

If

for quartz.

call

\ s the

=

^

and to the value 7 0, \ s will wave-length corresponding to be equal to the constant wave-length whose square enters into the denominator in the usual Sellmeier dispersion formula. It is usually assumed that \ s is identical with, or, at any rate, very close to the wave-length for which the energy is a maximum in the residual rays.

assumption

may

be

It is evident far

from being

from the figure that this justified.

Starting with values of X which are less than \ s (7 negative) a small value which gradually decreases to a very small has p* minimum. From this it rises very sharply to a maximum beyond it again diminishes, but more slowly than it rose. Thus the curves are far from being symmetrical about the position for which p 2 is a maximum. The maximum value corresponds to a value of \ which is distinctly less than \ s The positions of the

which

.

maximum and minimum may

of course be obtained

2 tiating the expression for p with respect to which results is of too high an order to be of

7,

by

differen-

but the equation

much practical use. minimum is very

It happens, however, that the position of the easily obtained

with sufficient approximation, since

we multiply the maining

it is

practically

minimum

If value of the numerator in p\ 2 2 top and bottom by 7 -f S the numerator re-

coincident with the

is

ID

The minimum value

of this

is

at 7

=

-

spending

minimum

2 provided S

is

value

of

p-

is

-.

jA.

The

corre-

JL

given very approximately,

rather small, by

Thus the position of this minimum and the corresponding value of p 2 should give an important check on the constants in the dispersion formulae.

CHAPTER IX THE FUNDAMENTAL EQUATIONS the electron theory may be equations of from the results of abstraction or regarded as a generalization assumes that matter is The electron theory Chaps, in, v and vi. but a distribution of electrified elements of volume in

THE fundamental

nothing space.

There are thus no magnetic charges in the sense in which

there are ultimate electric charges or electrons. fields which occur in nature arise entirely from the electrons.

The magnetic

motion of the The simplest assumption which we can make as to the

nature of the universal equations of the field is that they are identical with those which we have derived for the free aether containing electric charges.

an assumption, as it What we can be sure of

is

is

is

It is important to realize that this sometimes regarded as self-evident. that the fundamental equations must

degenerate into those for the free aether at points not in the immediate neighbourhood of material particles; but this is a very different thing from being sure that they are valid in the an atom or an electron. The assumption of their

interior of

universality

is

a hypothesis which will only be justified

conclusions to which

it

leads are in

if

the

agreement with deductions

from experiments.

We

therefore assume for the universal equations

p 0....,

"**--;

:

(1), (2),

W

THE FUNDAMENTAL EQUATIONS where the mechanical is

V

relative to the

force

on an

charge whose velocity

electric

measuring system

per unit charge,

is,

(5).

c

show that these equations, which are assoof Lorentz, are not inconsistent with any of our previous results. Looked at superficially they do appear to be inconsistent since, by simply writing average values in equations (1) to (5) we do not arrive at equations which are obviously identical with those which we found to comprise the behaviour of dielectric and magnetic media in Chaps, in, v and vi. It is necessary to

name

ciated with the

;

to be

remembered, however, that the vectors defined as the and magnetic intensities and inductions respectively, in those chapters, were all average values of the true electric and magnetic intensities but formed in different ways. When this difference is taken into account the discrepancy will be seen to It

is

electric

We

disappear.

shall

now

consider the equations in order from this

point of view.

Equation (1) is supposed to apply to any element of volume however small. The corresponding equation div D = p is an equation between average values, and only applies to an element In of volume which contains a very large number of electrons. order to compare large volume.

them

We

let

us integrate (1) over any sufficiently

have

1

7

pdr r where

=

dE,, dE,} -^ + JL+=l.

[f[\dE., \

.

7

7

dxdydz J

E

=

E

at any point of n represents the normal component of the surface. But we have seen that the induction n is the

D

D

n average value of the force in the flat cavity perpendicular to so that the normal induction is nothing else than the average value of the intensity taken over a surface perpendicular to it. ,

E

only strictly true provided the surface is so the that of polarization charges of a given sign excess large In other words, of the doublets whose inside of it is negligible.

This identification

is

axes are cut in two by the surface the difference between the number which leave their positive and those which leave their

negative ends inside must be negligible.

On

the other hand,

if

THE to

is not big enough satisfy indefinite. becomes induction of the

the surface

this condition, the

meaning

E=p

In any event we saw at the end of Chap, ill, that div that p represents the total average density of the always, provided arises from conduction or polarization electrons it charge whether = p is only The apparently inconsistent equation div or both. of the volume electrification the true provided the part of density which arises from the polarization electrons is left out of account.

D

= 0,

viz. div 5 Equation (15) of Chap, v, with div average values, and is consistent

is

H=

an equation over

for precisely the the same reasons as those which establish consistency of (1) and In this case we do not need to consider the possibility div D = p.

of an excess of magnetic poles of a given sign being situated For as the elementary magnets consist of inside the surface.

charges in motion,

electric

such a

way

it is

impossible to cut

them in two in The detailed

as to separate the equivalent charges.

formulation of the magnetic properties of bodies from this point of view will be left to a later chapter.

The equations obtained in Chapter VI also refer to average The equation which is equivalues of the dependent variables. valent to (3) is

In order to show that these equations are consistent to consider their geometrical interpretation.

analytical expression of the

fact that the

Each

it is

of

necessary

them

is

an

of the component, parallel to the contour, of the vector on the left, round any contour, is equal to the integral of the normal component of the vector on the right over any surface bounded the same contour.

Thus the

E

line integral

by

of (3 a)

is

the average value of the tan-

component of the

electric intensity taken round the to the of evidently equal It_is average value of the when derived from (3) because as in Chapter iv is equal to the average value of in a filamentous have

gential contour.

E

E

V

E

We

cavity. already seen that the average value of Sn over any surface is equal to the n over the same surface. average value of The equations (3)

H

and

(3 a) are therefore consistent with one another.

A<J>

THE FUNDAMENTAL EQUATIONS

of Quite similar considerations apply to (4) and equation (17) write we which may Chap. VI, .................. (4a) -

H

is

round a contour The average of the tangential component of the over corresponding evidently H, and the average of En

surface

There

is is

clearly

Dn

,

so that these equations are also consistent. In is worth remarking in this case.

one point which

surface bounded interpreting (4 a) it is desirable so to choose the cross it during the of electrons the none that contour the by

under consideration. Otherwise there is a contribution pV term owing to the motion of the electron. (4 a) then

interval to the

becomes inconsistent with contour.

This difficulty

the surface integral surfaces terminated by the

itself since

not have the same value over

may

all

does

same

be overcome either by choosing the

surface so that the polarization electrons do not cross it, or by taking the element of time large enough to include the average

value of effects arising from such translation.

This

is

the motion of an electron across the bounded surface to the creation of a separate

zero because is

equivalent charge in the unlike charge in the old

doublet with

new

its like

position of the electron, and its The creation of this doublet introduces a local term position.

which just wipes out the effect of the motion of the charge across the boundary. These remarks are pertinent to equation (4 a) only. Equation (4) is always consistent with itself, and is consistent with (4) when the latter is self-consistent. in the force

In comparing equation (5) with the corresponding equation

the agreement of the first term on the right is clear enough, hut the second requires fuller consideration. Here we have, to deal with the average value of taken a line to which // is

H

normal.

It

is

difficult to see

along

how

may be done directly, but, an indirect method may be employed. Considering the ease where (at least so far as average values are concerned) lei, this

E=Q

us apply the universal equation

F=-[V.H]

to find

i,he

force

material

the resultant force on the circuit

for

round the circuit p.

If the strength of the current is i

medium.

83

;

it

s.

\\Hn dS

is,

it

follows that (5)

and

that the average value of direction of

But, as

H

.

H]

taken

B.

is jjuH

En = ~\ En ds = K,E = D, s J

same

contour.

normal component of the

we have

seen,

(5 a) are consistent and, incidentally, taken along a line normal to the

H

From

this the

and magnetic vectors would

electric

[ids

similar to that in Chap, v, equal to i/c times the rate of

of course, the

universal magnetic intensity.

Thus

I

(5) gives

./

over any surface having the

Hn in this integral

The

o

t

By an argument

follows that the force is

change of

the value -

lead

analogy between the us to expect that

the suffix n denoting that the vector

is

perpendicular to the direction of integration.

By

dividing the space up by means of tubes of induction, it that the average values of the universal expressions %E* 2 for the electric and magnetic energy densities respec-

is clear

and

^H

2 2 This is only true provided \icE and |yu,J? neglect constant terms which may be regarded as representing the intrinsic energy of the electrons and of the molecular magnets.

tively are equal to

.

we

The Differential Equations

satisfied by the Vectors when Charges are present.

In Chapter

vn we

were concerned with the solution of equaand the extensions of them, which have just been considered, in the cases in which the density p of the charges was

tions (1) to (4),

zero. The results thus obtained naturally applied to the propagation of electromagnetic effects in insulators, including the free aether as a shall now consider the particular case.

everywhere

We

nature of the solutions in the more general case, when electric charges are present and contribute to the resulting phenomena by their motions and the forces they exert.

THE FUNDAMENTAL EQUATIONS The new satisfied

equations,

E

by

analogous to

is

true

A

are

A

if

if

we

first

K

1

-^

,

rot

dA z

dA y

dy

dz

dA x

dA z

dz

dec

easily

than

A,

The components

any vector point function.

is

which are

prove the general theorem

V*A = rot

A

grad div

of rot

^2

V 2 E=

and H, may be obtained rather more

would otherwise be the case

which

0.01

'

'

dAy

JdA a

dx

So that the x component, rotation of

A dy

example, of the rotation of the

for

is

_

dy \dx

dy

dz\ dz

da \dy

dx

dA y --r ~^

dA z \

fdA x

d = ^~~

"^

ox \tix

-r -^

dz J

oy

dz

X (&A ~-~_\

-}

ox-

Since a similar result follows for the other two components

grad div

A - V 2A = rot

rot

A

we have

............... ((>).

In order to obtain the differential equation satisfied by t and obtain

E we

differentiate equation (4) with respect to 2

#

^+

8'

Substituting the value of

a

T~ (pF) ,

,a//-. = crot-

H from (3) we get =~

whence, from (6) and

g

'

C

J

rot

(1),

In a similar manner, starting with (3)

we

find

(H).

Each of these equations is a vector equation and is equivalent each of the three components. to three separate equations between V are x ,Ey ,EZ) x y ,Hz and of E, Thus if the components x the Vz respectively, components are given by the fa, V

H H

E

H

,

,

,

Cartesian equations

,_

VH

,

'

and

I

_

--^~dF~~~c\W

......

~**

I

There are four other similar equations for the other components.

The nature

of the solutions of equations (9)

and (10) may be

discovered by considering the equation

**-where

o> is

which we

a function of

x, y, z

and

t.

.....................

In the

electrical



problems

a given function of these variables. The solutions of (11) have a certain degree of resemblance to the potential in the theory of attractions. The potential

V

satisfies

shall

have to consider

the equation

o>

is

V 3 F=p, where

p

is

attracting matter, measured in suitable units. the integral of this equation is

the density of the

As

is

well

known

Thus the potential at any point P is obtained if we take the element, pdr, of mass at any point, divide by ^irr where r is the distance from P, and integrate throughout space. We shall see that a precisely similar result holds for the functions ty which are the solutions of (11). The only difference lies in the fact that in we replace p in (12), not by the calculating the values of

^

instantaneous value of

the function on the right-hand side of (11), but by the value which this function had at the point of integration at an instant r/c previously, where r is the distance co,

from the point at which If in (11)

we introduce

equation becomes

E a

or

H

is

to

be calculated.

new independent

variable u

ict

the

THE FUNDAMENTAL EQUATIONS

The

left-hand side would be the value of

V

2

in rectangular i/r coordinates in a four-dimensional space so that the problem of finding the solutions of (11) can be looked upon as the problem of ;

finding the potential in a four-space.

(11) is

an example of a

number of electromagnetic equations whose symmetry is improved when the time t is replaced by the imaginary variable u = ict.

Kirclihoff 's Solution.

in

A very complete discussion of the solution of (11) was given 1883 by Kirchhoff* in connection with the theory of the pro-

pagation of

light.

As a preliminary

an auxiliary function % which

to solving (11) let us introduce

satisfies

the equation

This is the equation to which (11) reduces when the righthand side is put equal to zero and \jr is a function only of t and the distance r from a fixed point. If we put ^> = r^ (13) becomes 5

The most general

where

F

is

solution of this equation is (see p. 117)

any function whatever.

The two terms correspond

physically to disturbances velocity

c.

x=-F(t + -)

giving

where

We

propagated in opposite directions with shall only consider one of them and take

F is

Next

(14),

a perfectly arbitrary function.

consider the integral '

y^"^)

C&T

(15)

taken throughout a closed volume limited by an internal surface <j and an external surface 8. By Green's Theorem

Fig. 27.

the normals being directed into the enclosed volume.

have from

(15), (11)

We

also

and (13) 1

From

(16)

and (17)

m*s '

This

is

true for

all

to dt between limits r^a

r r /

values of

^ and

3^ on

3^\ _

j

dy _/jk 3t

it

with respect

We

tz .

y ^ onj

r

Let us integrate then get

t.

dn

^g.

A/ /C

3^\

f>

-

:

.(18).

Let us now return

may be any

This

suppose that

it

is

consideration of the function F.

to the

We

shall function of the argument t H- r/c. such a function that it takes the value zero

values of the argument except those in the immediate shall then have neighbourhood of the particular value zero. for all

We

F(&)=Q unless x lies between, let us say, shall also suppose that small quantity.

where

e

e is

a very

We

\ ................. ...... (19). .

Since

F

(so)

zero unless x

is

between

is

e

we

evidently also

have 6

^

J -oo

If the value of r

F(x)dx =

r F( -e

./

is

fixed rt+r/c

provided

to

+ r/c > e

finitesimal but

we

still

shall have, if

and ^

-f

suppose

co is

rfc


'

the value of

is

except when

t

infinitesimal interval

Now

let

t*

(o

at the instant

(oo)

dx

=1

t,

t

=

r/c.

......... (20),

This follows

between + e, F 0, and throughout this may be considered constant.

lies co

have a definite positive value and ^ a very large

^ being

definite negative value,

enclosed volume ^

4-


where

we make

r+e

%

vanish.

write the term containing

co

also

% which occur

do the values of

between +

in (18) in the

6.

form

fi

This in (20).

is

to equal i

1

1

jjj r

In a similar way

dr where

a)'

has the same meaning* & as

where

f

denotes the value of -~- at the point of integration un

)

\dn

I

(distant r from

P)

at the instant

to da integral with respect

We

may

=

t

The corresponding

r/c.

be similarly treated.

also have

dn

dn

(re

Thus

= where

1

ff (I) f J]dn\rJ^

1 ^ + Jjrcdn dS f

f

^

the value of

is

\|^'

'

f

J

V r

t}

f

\

*

+ -) * c)

at the point of integration at the

rut

instant

t

=-

c

.

The right-hand

integral

may be

integrated by

parts, giving

since

F(t + -} t~\

value of

vanishes at the limits.

Here

(-^-)

denotes the

i

-

Vv

at -the point of integration at the instant

t

=

,

in

C

accordance with our former notation.

The

left-hand side of (18)

may be

treated similarly, giving

rise to

>Y

Now

let

the surface

$

cr

,

become coincident with a sphere of ^

infinitesimal radius p about the point P.

left-hand side becomes

cp

=

Then on

^ ~~~

dp

and the

LVO

THE FUNDAMENTAL EQUATIONS

When

p

infinitesimal

made

is

to

become very small the terms in - become

compared with

-^r

So that the left-hand side

.

p

becomes identical with point P at the

instant

t

4^^ =

where

"SP

is

the value of ty at the

Hence, we have

0.

__3/i\ fa,(rjv

+ !&;/&

crdn(dt (21).

Now

let

the

surface

S

recede to an infinite distance and

suppose that at infinitely distant points the functions

^, ot

all

^ ^-,

and

have the value zero until a definite time T, then when

r becomes infinite the time

t=

,

to

c

which tlr', Y

{-},

and

\dnj

-(

\dt

]

J

T

so that every in the surface integral refer, is always less than } element of the integral is zero. This supposition is legitimate

physically since we always presuppose that physical phenomena are independent of past or present occurrences at an infinite at the point at the thus see that the value distance.

W

We

time

t is

P

equal to

r-

[({d T

(22), ^ '

47TJJJ r

where the integral is taken throughout space, and the value of each element of volume is that which it possessed at co for T

the instant

t

.

c

The Propagated Potentials.

we have just obtained physical interpretation of the result of the electric and the values means that It very simple. at any instant are at intensities any particular point The

is

P

magnetic

not, in general,

determined by

the state of the rest of the field at

The effects that particular instant, but by its previous history. at P, in so far as they are due to a particular element of volume distant r from P,

depend upon the

state of that

element of volume

than the instant considered. This time r/c is the time which would be required for light to travel from equal to The nature of the field distant element to the point P. at a time r/c earlier

the is

therefore such as

would

arise if

each portion of

which were propagated emitting disturbances tions

it

were constantly

from

it

in all direc-

with the velocity of light

When we come

to the actual calculation of

H

the values of

E

found that equations (7) and (8) in particular and of to, given by the right-hand values the to are unsuitable owing cases

it is

them, being somewhat complicated. The calculations may be simplified by the introduction of two new functions, the scalar sides of

and the vector potential U, from which

potential afterwards be derived

by

appropriate operations.

prove that JET =*

and if

JS;

and



U satisfy

rot

U

shall

..................... (24)

........................

v

prove first of all that a function This function is in fact

H = rot U.

r

.

For

if

(27)

is

true

dr

-

U always

exists

such

.................. (27). v '

we have r

Now

may now

the equations

i>

We

shall

........................ (23),

= ----grad(

-

that

E and H

We

4}7rJJ]

r

H

the values of

in the integral refer to the different is measured. points of integration and not to the point at which Let x, y, z be the coordinates of the at which is

and

a, 6, c

point the coordinates of the element of

^= a (-a) + (y-6) a

2

+(2f-.c)

a

U U

volume

required

Then

dr.

and the equation above

may be

written more clearly as

Ux = JL fff fij** 4-rr JJ J

\db

_ 9#_A dc ) V(a

_

dadbdc

- a ja~+ y~_ (~

&)-

+ (Z - CY

'

THE FUNDAMENTAL EQUATIONS

195

,

, (rot t

and

_ _L II ~ 4nr 1

=

f [f

/^ _ are not completely determined by the considerations

This that

is satisfied if

is

which have been brought forward. The only condition we have imposed on U except (24) is that it should satisfy the equation = rot U. Also $ may be any scalar quantity. If U and

H




THE ACTIVITY OF THE FORCES

202 Eeplacing rot

If

we

If

Cartesian equivalents this becomes

terms containing =-

,

~

and

re-

and integrate each term by parts this becomes

\ldydz (Ey H,

of the

its

collect together the

spectively c

H by

- Ez Hy ) + dzdx (EZ HX - EX HS )

m, n are the direction cosines of an element of surface dS = IdS, dzdx = mdS, boundary of the volume T we have dydz

I,

H

H -EX H Ex H -E Hx

dxdy = nd8, and Ev H,-Et v EZ X the at, y and .0 components respectively the integrated part

is

,

Z

,

of the vector

v

y

are

[EH]. Thus

equal to

where [EH] n denotes the resolved part of the vector [EH] along the normal to the element dS. The volume integral, after rearrangement, becomes

-I

and, since rot 2?

o rr

= --^-

and

H*= Hx*+ Hy + H*, *

this

may be

written

Thus

^ = c (([EH^dS 1

f f fa

^^ + 1^) dr

...... (1).

THE ACTIVITY

OBY

THE FORCES

the magnetic energy per unit volume of the It is aether and \E* is the electrostatic energy per unit volume. of this arid to be borne in mind that the investigation are

Now %H

2

is

H

E

the universal values of the forces introduced in the last chapter and are not the same inside material media as either of the defined as the intensities average values which we have previously and inductions in such media. The mean value, over a region of

the appropriate dimensions, of the present ^H* is identical with 2 2 and the mean value of the or or the former

^H

present

\E*

When E

is

and

%HB

B /^

identical with the former \tcE~ or

H are the universal functions

\ED

or

represents the rate at which energy is lost by the aether, or space, within the limited region r. Thus the whole of the work done by

the forces of the field on the electric charges is not covered by the energy lost by the aether in the immediate neighbourhood.

In general we have also to consider the quantity represented by the surface integral. Since the left-hand side of the activity equation is the rate at which work is done on the electric charges, and the volume integral represents the rate of loss of energy by

the electromagnetic field in the

enclosed volume, the

surface

integral which is equal to their difference must represent the rate at which energy flows into the region r from outside. Any possible alternative to this conclusion

would involve a denial of

the principle of the Conservation of Energy.

Poy ntiny 's Th eorem. The occurrence of the surface integral cff[HH] H dS in the equation of activity of the forces was first remarked by Poyntirig*, who gave to it a very definite physical interpretation. He pointed out that the behaviour of the field could be explained by the supposition that at every point there was a stream of energy

e<jual

per unit area to c[!.H], the direction of the stream being coincident with that of this vector, and therefore normal to the plane and H. The Poynting Flow of containing Energy thus vanishes when and // are coincident in direction, arid has a

E

E

THE ACTIVITY OF THE FORCES

204

value, other things

maximum

being equal,

when they

are at right of course, consistent

This interpretation is, another. angles to one with the equation of activity, and is, in fact, the most obvious It is, however, not the only interpretation. of it.

interpretation

For we may evidently add to

c

[E H] any vector .

R which satisfies

the condition that the surface integral of its normal component closed surface vanishes and the equation of activity will over :

any be satisfied.

still

We

know from Gauss's Theorem that this con-

we have div R = everywhere so that there number of vectors in addition to Poynting's which

dition will hold if

are an infinite

;

of activity. satisfy the equation It is interesting to consider to

what picture of the flow

of

energy we are led in typical instances on the supposition that it In the case of a straight wire coincides with Poynting's vector. carrying a current, for example, the electric force is parallel to the length of the wire, and the magnetic force is in circles about its

Thus the electric and magnetic forces are at right angles to one another, and the flow of energy is at right angles to both. That is to say, it flows perpendicularly into the wire from the

axis.

insulating

medium which surrounds

it.

Probably the most convincing case of the flow of energy in accordance with Poynting's vector is that furnished by the propaConsider a parallel beam of gation of electromagnetic waves.

We have seen that in such a beam the and magnetic vectors may be represented by

plane polarized light. electric

E = H = A cos (pt - x). It is

important to notice that the two vectors are always in

phase, and that they are equal in magnitude when expressed in the units used in this book. They are also at right angles to one .another. Thus the resultant flow of energy is perpendicular to

both

E and H

;

that

to say, it is along the direction of propaIt is equal at any instant to

is

gation of the light.

per unit area.

Its average value over a single period, or over

very large interval of time, o

l

T

TT

J

o

is

any

THE ACTIVITY OF THE FORCKS

Now \A*

mean energy present in unit volume of the this energy as flowing Poynting's Theorem represents is

the

wave, so that Since this is in accordance with the* c. along with the velocity of Poynting's Theorem to results of observation the application solid to radiation evidently rests on very questions relating grounds.

Forces exerted on the Charges.

In the last section but one we have considered the rate of working or "activity" of the forces acting on an enclosed electrical system. We shall now consider the value of the resultant force acting on On the electron theory of matter the results a similar system.

be applicable to any material system since, on this theory, the force acting on a material system is the aggregate; effect of the electric and magnetic forces which act on the electrons whieh will

constitute

it.

The

force exerted

on a unit charge-

is

K+

t

j

V II

j,

where V is the velocity of the charge relative to the system of instruments used to measure the forces. The reason for this

V

will be clearer later (see Chaps, xni particular specification of It will be observed that it is not inconsistent with

and xiv).

the deduction from the magnetic properties of electric currents

which led us to include the term c

force

on a charged body

has been considered well have taken

V

up

(p. 114).

[VH] So

to the present

in the- expression for the

far as is

any evidence which

concerned,

we

might, as

be the velocity of the charge' relative to the which we aether, might suppose to be absolutely fixed in space. to

When we come to consider the electrical and optical properties of bodies in very rapid motion, we shall see that the assumption that refers to the velocity relative to the measuring system

V

effects

very important simplifications.

The charge present

in the

element of volume dr he ing pdr,

the force exerted on this element of volume will

dF=p(jK+

l

\Vni\dr,

he

THE ACTIVITY OF THE FOKCES

206

and the equation

'

= divE.

since

whole -volume will be given by the vector

force on the

But

hence

V= c rot H

p

E.

and

^-//Jjdiv

-^-

,

ot

E + * [crot#. fl] -

.

If

BH-l =~ [^. rot #]

= - [rot .#.#],

and hence

F= -

~ [EH]

[[[c ot JJJ

.

dr +

4-

1

1

This

is

/

f ffdiv l

JJJ

.

F

.

L

J>

+ [rot H. H]} dr

/{divjfif. fl"

the total force on the volume

ponent of

E E + [rot E.E}\dr

Consider the x com-

r.

due to the third term of

Call

(2).

-H,"

1

...(2).

z

it

XH

.

Then

*--*