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.
F.
H0tttron:
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lontnm: H. K. LEWIS, 136
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28
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()(K)
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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
preeminently 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 ultraviolet 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 ultraviolet 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*ni
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 >kv
1
m
Consider TTTIV
T\QT4ei
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) = 47r&> 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.
FwkT 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
,
aF]
^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
ilustnit^ '*
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 semiangles
The coordinates
volume are
flirfi?
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
nJJJ 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
mnt
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 socalled 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 Vf 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 = Emn, = n, Pzx x = E*lm.
),
per unit area across
force
dS
are
respectively
+ np2x = ^lEf 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/ vxtnel 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
^

insi 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* itinn
'
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 ~~ ^~ ;
47T
)
,
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.
.incidts
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 jMttniial 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 noncrystalline 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'Ketrons 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 iieii
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
fiitii >vk
f/\
/~/\Y
01
H
tiv*
iT^ti
much more h^or
/\T
f
n *^
complete. 4Tt/v
voo O/MI 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\ aflvYrrl fn npo'lpr flip rmnnlir^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.onAtin
electric current
!l
the
iin/i'tion tn;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 righthanded 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 thcji>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 Kln.'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' fiinow, ^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
tiiM
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
4TrA
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 4Trc 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 cV47r
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 \ 4jr 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 = 5y. 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
= 30001
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 3022 x 10
10 ,
:
2'964 x 10
10 ?
2'980
x 10 x 10 10 10
Trowbridge and Duane MacLean
29911 x 10 10
Saunders
2997
The mean
{"
3'003
2*982 x 10 10
these quantities
,
x 10 10
is
2991 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.
30004
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 wavelength. 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
JJ4
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..rrtki'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 PlanePolarised Electromagnetic Wave.
A
solution of the equation
provided is
J'f
M
2
Hi 2
=L
^ = a~u
is.
"
The expression on the righthand
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 wavelength 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
~(atz) 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
27T

X
a,
x
e d
=
^r
9^
c
of the dependent variables const,
dy
is
of the form
^
*
= n0,
9
~
=
oz
.
^
27T __
(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*My* "*< ~*.\
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
wavelength 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 UKT 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 wavelengths 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 (=7f_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
65
;
7
8
75
95
85
105
4 Wavelengths in 10~ cm.
Fig. 26.
wavelengths (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,. = 2563 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, 142 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 wavelength corresponding to be equal to the constant wavelength 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 wavelength 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 selfevident. 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 selfconsistent. 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 righthand 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
lefthand side would be the value of
V
2
in rectangular i/r coordinates in a fourdimensional space so that the problem of finding the solutions of (11) can be looked upon as the problem of ;
finding the potential in a fourspace.
(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 righthand
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
lefthand 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.
lefthand 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 lefthand 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 righthand 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) + (y6) 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** 4rr 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 lefthand 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
BHl =~ [^. 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
**