eS 
DIELECTRIC CONSTANT, ABSORPTION AND SCATTERING 299 
mental relation 
neem] 
where the cutoff wavelength A, is determined by the 
geometry of the guide and the type of wave. In the 
air-filled section of the guide 
n=[(BY- ene]. 
Consequently, from equations (93) and (94), the 
complex dielectric constant of the material under 
study becomes 
be (1/X.)? — (y2 / 2)? 
ic 6 95 
"(17 r0)? + (1/7)? oe 
where 
n=Jj = (96) 
for free space. Finally 
2=6 (2 i (97) 
The solution of equation (93) can be found from 
charts. It is claimed that with this method materials 
with very low dielectric losses can be investigated 
satisfactorily. ; 
Tue Resonator Q Mrruop*® 
Here the procedure consists in measuring the 
change of resonant frequency of a closed cylindrical 
resonator upon the insertion, along the axis, of a rod 
of the dielectric material in question. By observing 
the change of Q value resulting from the insertion of 
similarly dimensioned specimens of different mate- 
rials, it is possible to obtain comparative loss tangent 
values. The relevant theoretical relations are sum- 
marized below. 
By definition the Q value of a resonator system is 
given in convenient form by the relation 
energy stored 
energy loss per half cycle 
Both the energy stored and the energy loss can be 
computed from the field distributions within the 
resonator, and these are given, for a Tf wave, as 
He = AJ: (yp), (98) 
ji, es) (99) 
o + jue, 
where H, is the tangential magnetic field strength 
in amperes per meter. H, is the axial electric field 
strength in volts per meter, p is the distance of the 
point in question from the cylinder axis, y is the propa- 
gation constant 
? = new" — juon, 
w is the angular frequency, » the permeability in 
henrys per meter, and A is a constant determined by 
the strength of the exciting source. In the formulas 
(98) and (99) it was assumed that the walls of the 
resonator are of infinite conductivity so that no elec- 
tric intensity exists in them. This requires that 
E,(p = a) = Jo (ya) = 0, (100) 
where a denotes the radius of the resonator. This 
equation has an infinite number of real roots, the low- 
est being ya = 2.4048, and this determines the fun- 
damental resonant frequency and wavelength d,. If o, 
the conductivity of the dielectric, is neglected in com- 
parison with ¢w, the propagation constant becomes 
——— «2 
y¥=w Vue = —. 
bi We 
(101) 
e, is the dielectric constant of the material filling the 
resonator taken relative to air. Since A can be meas- 
ured, this dielectric constant may be derived from the 
relation 
2rde _ 9 4048 
2 2 
« = 0.146 () a () 
a do 
No appreciable error will be committed in using the 
preceding results for the practical case of dielectrics 
with low but finite conductivity. 
The Q of the filled cylindrical resonator is shown 
to be 
or 
(102) 
a 
a(1 +3,)tatans 
Here d is the wave-guide skin depth, 2z, is the axial 
length of the resonator and tan § = «/e, is the loss 
factor of the dielectric. Consequently 
1 1 
=———) 
tan 6 a 
where Q, is the Q of the air-filled resonator. 
It should be remembered in this connection that 
the theoretical Q, values, in general, are found to be 
considerably different from the measured ones. This 
tends to limit the reliability of the method. 
After having thus sketched the different methods 
used in the determination of the complex dielectric 
constant of substances of importance in wave propaga- 
tion, we turn now to the presentation of the data. 
Q = 
(103) 
(104) 
Liquip WATER 
Table 17 gives the results obtained recently on 
liquid water.??-585%.44 
It has been found by Saxton and Lane that the 
temperature variation of the dielectric constant in the 
range 0 to 40 C at 1.24 and 1.58 cm can be ac- 
