# Chapter 4:Behaviour of Dielectric Materials in ac and dc Fields¶

## Example 4.1,Page No:4.8¶

In :
import math

alpha      = 1.8*10**-40;           #polarisability of argon in Fm**2
e0         = 8.85*10**-12;          #dielectric constant F/m
N1         = 6.02*10**23;           #avagadro number in mol**-1
x          = 22.4*10**3;            #volume in m**3

#formula
#er-1=N*p/e0*E=(N/e0)*alpha
#calculation
N          = N1/float(x);                             #number of argon atoms in per unit volume in cm**3
N2         = N*10**6;                                 #number of argon atoms in per unit volume in m**3
er         = 1+((N2/float(e0)))*(alpha);              #dielectric constant F/m

#result
print'dielectric constant of argon = %3.7f'%er;

dielectric constant of argon = 1.0005466


## Example 4.2,Page No:4.9¶

In :
import math

#variable declaration
alpha       = 1.8*10**-40;       #polarisability of argon in F*m^2
E           = 2*10**5;           # in V/m
z           = 18;
e           = 1.6*10**-19;

#formula
#p=18*e*x
#calculation
p       = alpha*E;
x       = p/float(18*e);           #displacement in m

#result
print'displacement = %3.2e'%x,'m';

displacement = 1.25e-17 m


## Example 4.3,Page No:4.9¶

In :
import math

#variable declaration
E0    = 300*10**2;                  #local field in V/m
P1  = 3.398*10**-7;                 #dipole moment Coulomb/m
P2  = 2.124*10**-5;                 #dipole moment Coulomb/m
e0  = 8.85*10**-12;                 #permittivity in F/m

#formula
#E10Ci=E0-(2*Pi/3*e0)
#calculation
E10C1  = E0-((2*P1)/float(3*e0));       #local field of benzene in V/m
E10C2  = E0-((2*P2)/float(3*e0));      #local field of water in V/m

#result
print'local field of benzene=%3.2e'%E10C1,'V/m';
print'local field of water=%3.3e'%E10C2,'V/m';

local field of benzene=4.40e+03 V/m
local field of water=-1.570e+06 V/m


## Example 4.4,Page No:4.9¶

In :
import math

#variable declaration
p1 = 5.12*10**-34;          #p of benzene kg/m**3
p2 = 6.34*10**-34;          #p of water kg/m**3
e10C1  = 4.4*10**3;         #local field of benzene in V/m
e10C2  = 1570*10**3;        #local field of water in V/m

#formula
#p=alphai*e10Ci
#calculation
alpha1 = p1/float(e10C1);       #polarisability of benzene in F*m**2
alpha2 = p2/float(e10C2);       #polarisability of water in F*m**2

#result
print'polarisability of benzene = %3.2e'%alpha1,'F*m**2';
print'polarisability of water  = %3.2e'%alpha2,'F*m**2';
print'Note: mistake in textbok,alpha1 value is printed as 1.16*10**-38 instead of 1.16*10**-37';

polarisability of benzene = 1.16e-37 F*m**2
polarisability of water  = 4.04e-40 F*m**2
Note: mistake in textbok,alpha1 value is printed as 1.16*10**-38 instead of 1.16*10**-37


## Example 4.5,Page No:4.10¶

In :
import math

#variable declaration
e0     = 8.85*10**-12;          #abslute permitivity in (m**-3)*(kg**-1)*(s**4)*(A**2)
E      = 600*10**2;             #strength in V/cm
er1    = 2.28;                  #dielectric constant of benzene in coulomb/m
er2    = 81;                    #dielectric constant of water in coulomb/m

#fomula
#p=e0*E*(er-1)
#calculation
pB     = e0*E*(er1-1);        #polarisation of benzene in c/m**2
pW     = e0*E*(er2-1);        #polarisation of water in c/m**2

#result
print'polarisation of benzene = %3.2e'%pB,'c/m**2';
print'polarisation of water = %3.2e.'%pW,'c/m**2';

polarisation of benzene = 6.80e-07 c/m**2
polarisation of water = 4.25e-05. c/m**2


## Example 4.6,Page No:4.10¶

In :
import math

#variable declaration
er0  = 5.6;              #static dielectric cnstant of NaCl
n    = 1.5;               #optical index of refraction

#calculation
er = er0-n**2;
d  = ((er/float(er0))*100);   #percentage contribution from ionic polaristion in %

#result
print'percentage contribution from ionic polaristion = %3.2f'%d,'%';

percentage contribution from ionic polaristion = 59.82 %


## Example 4.7,Page No:4.10¶

In :
import math

#variable declaration
alpha       = 0.18*10**-40;         #polarisability of He in F *m**2
E           = 3*10**5;               #constant in V/m
N           = 2.6*10**25;            #number of atoms in per m**3
e           = 1.6*10**-19;

#formula
#P=N*p
#charge of He=2*electron charge
#p=2(e*d)
#calculation
P      = N*alpha*E;                   #in coul/m**2
p      = P/float(N);                  #polarisation of He in coul.m
d      = p/float(2*e);                #separation between charges in m

#result
print'separation=%3.2e'%d,'m';

separation=1.69e-17 m


## Example 4.8,Page No:4.10¶

In :
import math

#variable declaration
N      = 10**27;                     #number of HCl molecules in molecules/m**3
E      = 10**5;                      #electric field in V/m
P      = 1.04*3.33*10**-30;          #permanent dipole moment in coul.m
T      = 300;                        #temperature in kelvin
K      = 1.38*10**-23;

#calculation
P0     = (N*(P**2)*E)/float(3*K*T);        #oriental polarisation in coul/m^2

#result
print'oriental polarisation=%3.2e'%P0,'coul/m**2';

oriental polarisation=9.66e-08 coul/m**2


## Example 4.9,Page No:4.11¶

In :
import math

#variable declaration
N      = 6.023*10**26;      #avagadro number  (lb-mol)**-1
alpha  = 3.28*10**-40;      #polarisability in F*m**2
M      = 32;               #molecular weight in kilograms
p      = 2.08*10**3;        #density of sulphur in g/cm**3
e0     = 8.85*10**12;       #permitivity in F/m

#calculation
er = ((2*N*p*alpha)+(3*M*e0))/float((3*M*e0)-(N*p*alpha));

#result

print'relative dielectric constant =%3.1f'%er;
print' Note: calculation mistake in text book in calculating relative dielectric constant';

relative dielectric constant =1.0
Note: calculation mistake in text book in calculating relative dielectric constant


## Example 4.10,Page No:4.12¶

In :
import math

#variable declaration
er         = 4.94;
n          = 1.64;

#calculation
#(alphae)/(alphai) =x
x       = ((er-1)/float(er+2))*(((n**2)+2)/float((n**2)-1));    #ratio of electronic and ionic probabilities

#result
print'ratio of electronic and ionic probabilities =%3.1f'%x;

ratio of electronic and ionic probabilities =1.6


## Example 4.11,Page No:4.17¶

In :
import math

#variable declartion
E      = 1.46*10**-10;             #permitivity in c**2*N**-1*m**-2
E0      = 8.885*10**-12;           #permitivity in c**2*N**-1*m**-2

#calculation
Er         = E/float(E0);
sighe      = E0*(Er-1);         #electrical susceptbility in c**2*N**-1*M**-2

#result
print'dielectric constant=%3.2f'%Er;
print'electrical suseptibility=%3.4e'%sighe,'c**2*N**-1*M**-2';

dielectric constant=16.43
electrical suseptibility=1.3711e-10 c**2*N**-1*M**-2


## Example 4.12,Page No:4.17¶

In :
import math

#variable declaration
r       = 0.1;                    #radius in  m
pw      = 1;                     #density of water in  g/ml
Mw      = 18;                    # molecular mass of water
E       = 6.0*10**-30;           #dipole moment of water in cm
N       = 6.0*10**26;           #avagadro constant in (lb-mol)−1

#calculation
n  = N*(4*(math.pi)*(r**3)*pw)/(Mw*3);      #number of water molecules in a water drop
p  = n*E;                               #polarisation in cm**2

#result
print'polarisation=%3.1e'%p,'cm**2';

polarisation=8.4e-07 cm**2


## Example 4.13,Page No:4.18¶

In :
import math

#variable declaration
Er     = 1.000074;                 #dielectric constant for a gas at 0°C

#calculation
sighe      = Er-1;               #dielectric susceptibility

#result
print'dielectric susceptibility=%3.6f'%sighe;


dielectric susceptibility=0.000074


## Example 4.14,Page No:4.18¶

In :
import math

#variable declaration
E        = 10**6;               #dielectric in volts/s
er       = 3;                  #dielectric in mm
e0       = 8.85*10**-12;

#calculation
E0      = er*E;                  #electric field in V/m
sigma   = e0*E0;                 #free charge in Coul/m^2
P       = e0*(er-1)*E0;          #polarisation in coul/m
D       = e0*er*E0;              #displacement in in dielectric

#result
print'free charge=%3.2e'%sigma,'Coul/m**2';
print'polarisation=%3.2e'%P,'Coul/m';
print'displacement=%3.2e'%D;

free charge=2.65e-05 Coul/m**2
polarisation=5.31e-05 Coul/m
displacement=7.96e-05


## Example 4.15,Page No:4.19¶

In :
import math

#variable declaration
d     = 1.0*10**-3;             #separation between plates in m
A     = 6.45*10**-4;            # surface area in m^2
e0     = 8.85*10**-12;          #permitivity of electron in (m**-3)*(kg**-1)*(s**4)*(A**2)
er    = 6.0;                    #relative permitivity in (m**-3)*(kg**-1)*(s**4)*(A**2)
V     = 10;                     #voltage in V
E     = 10;

#calculation
C      = (e0*er*A)/float(d);             #capacitance in Farad
q      = C*V;                            #charge in coulomb
D      = (e0*er*E)/float(10**-3);        #displacement vector in c/m**2
P      = D-(e0*E/float(10**-3));         #polarisation vector in c/m**2

#result
print'capacitance = %3.2e'%C,'Farad';
print'charge =%3.2e'%q,'coulomb';
print'displacement =%3.2e'%D,'c/m**2';
print'polarisation =%3.2e'%P,'c/m**2';
print'Note:error in calculation of P,E value is taken as 5000 instead of 10**4\n';


capacitance = 3.42e-11 Farad
charge =3.42e-10 coulomb
displacement =5.31e-07 c/m**2
polarisation =4.42e-07 c/m**2
Note:error in calculation of P,E value is taken as 5000 instead of 10**4



## Example 4.16,Page No:4.30¶

In :
import math

#variable declaration
t       = 18*10**-6;            #relaxation time in s
er1     = 1;                   #permitivity in F/m
er      = 1;                   #permitivity in F/m
t       = 18*10**-6;            #relaxation time in s

#calculation
f                = 1/float(2*math.pi*t);               #frequency in Hz
theta_c          = math.atan(er1/float(er));
#theta_c_deg      = (theta_c*180)/float(math.pi);
#phi              = 90-theta_c_deg;             #phase difference in degrees

#result
print'frequency = %3.2f'%(f*10**-3),'KHz';
print'phase difference =%3.0f'%((theta_c*180)/float(math.pi)),'°';


frequency = 8.84 KHz
phase difference = 45 °