REMARKS CONCERNING T H E CLAUSIUS-MOSSOTTI LAW * BY PIERRE VAN RYSSELBERGHE
I t is well known that the dielectric constant of a dielectric which is perject, homogeneous and isotropic obeys the Clausius-Mossotti law’,*13: €-I .-M c+2
d
Constant = P
=
in which e is the dielectric constant, M the molecular weight, d the density. The constant has the dimensions of a volume and is called the molecular polarization. It is defined by the relation 477c-Y P=-N 3
in which N is Avogadro’s number and CY is the molecular polarizability. The Clausius-Mossotti law can also be written as follows: € - I -€ + 2
na 3
in which n is the number of molecules per unit volume. When the Clausius-Mossotti law is written as above, the use of the Gauss C. G. S. system of units (dielectric constant and magnetic permeability of vacuum both equal to unity), or of the electrostatic C. G. S. system of units (dielectric constant of vacuum and velocity of light both equal to unity) is required. The law cannot be used as such in the electromagnetic system of units (magnetic permeability of vacuum and velocity of light both equal to unity) or the practical system of units (unit of length = 1 9 ~cm., unit of mass = 1 0 - l ~gm., unit of time = I second, magnetic permeability of vacuum and velocity of light both equal to unity). In these two syst,ems of units the numerical value of the dielectric constant of vacuum is 1/900. The purpose of this note is to give the Clausius-Mossotti law a form independent of the system of units and to derive it in a general way applicable to all dielectrics which are perfect, homogeneous and isotropic, including those which are permanently polarized. The derivations of the Clausius-Mossotti law usually given in text-books implicitly suppose that the Gauss C. G. S. or the electrostatic C. G. S. system of units are used and fortuitously give the correct result for permanently polarized dielectrics. * Contribution from the Chemical Laboratory of Stanford University. IO.F. Mossotti: Mem. di mathem. e fisica in Modena, 24 11, 49 (1850). R. Clausius: “Die mechanische U’grmetheorie,” 2, 62 (18791, P. Dehye: “Polare Molekeln,” 7 (1929).
“53
THE CLAUSIUS-MOSSOTTI LAW
A perfect, homogeneous and isotropic dielectric is a dielectric for which the polarization vector is given by
T=T,+kB
(4)
-
I, being the polarization corresponding to a total electrostatic field zero, H being the total electrostatic field, k being the electric susceptibility and of the coordinates.‘ The electroand for which k is indeuendent of static induction B is given by:
-
B
=
+ 4nT
cog
in which eo is the dielectric constant of vacuum. The vector (6)
is the total force acting on a punctual charge of unity when the volume of a spherical alveolus dug around the charge tends towards zero. It is called the
“electric resultant.”j Debye’s “inner field”
F6is related to R by: -
F
= e.R
(7)
We see that i? has the dimensions of an electrostatic field (e,-XL-?GMMT-’) while F has the dimensions of an electrostatic induction (e,+’4L-MM!.iT-1). Applying equation (4) we deduce from ( 5 ) and (6):
B
with e
=
eR + 4SL
e = e,
(8)
+ 4nk
is the dielectric constant of the dielectric. Wre see that
B - _-
e
+ 4“H10
when I, and H are parallel. Let us write: T=ncue,
(R+ZI)+(~
-?)I,
which is equivalent to: 1 = n a ~ + ( 1 -?)I, in which n is the number of molecules per unit volume and a the molecular polarizability. 5
Th. De Donder: “ThBorie mathkmatique de 1‘6lectricit6,” 90 Th. De Donder: loc. cit., p. 79. P. Debye: loc. cit., p. 6.
(1925).
PIERRE VAN RYSSELBERGHE
1154
From
(12)
we deduce: -
nae, 4nna H
I =
+ To
I--
3
Comparing this relation with ( 4 ) we see that: nae, 4nna
k=-
I - -
3
Introducing the value of I as given by ( 1 4 ) into ( 5 ) we find: -
B =
eon+
4anae, I
-
B
(I
Fi + 4*To
3
while equation (8) gives From (16) and
--4nna
=
ER4- 4 7 ~ 7 ,
7) we deduce : e - e, - 4nna PEo 3
e
and
+
which is the general form of the Clausius-Mossotti law. If instead of ( 1 2 ) we write:
T
=
nae,
(E + Z I )
(16) becomes:
and the correct Clausius-Mossotti formula could be deduced from this equation. (20) and ( P I ) are however both wrong because B is not equal to when To is different from zero. A correct reasoning would be to start by differentiating (s), (8) and ( 1 2 ) . One has then:
en
THE CLAUSIUS-MOSSOTTI LAW
I I j j
Replacing dT in ( 2 2 ) by its value deduced from (24) and equating the two expressions of d B one obtains again the Clausius-Mossotti formula. The dielectric constant is defined by equation (23), the constant of integration being 4 ~ 1 , ,as shown by ( I j ) . If 7, = o the dielectric is said to be soft. We have in this case: dB dH
e=-=-
B
H
summary
A general expression of the Clausius-Mossotti law, independent of the system of units and valid for all dielectrics which are perfect, homogeneous and isotropic has been esi ablished. Stanford University, Cahfornia.
