# Chapter 13: Dielectric Properties of Materials¶

## Example 13.1, Page 648¶

In [30]:
from math import *
from scipy import integrate

#Variable declaration
q = 1e-006;   # Electric charge on either side of the dipole, C
l = 2e-02;    # Dipole length, m
p = q*l;      # Dipole moment for the pair of opposite charges, C-m
E = 1e+005;   # External electric field, N/C
theta = 90;   # Angle which the dipole makes with the external field, degrees

#Calculations&Results
tau = p*E*sin(theta);   # The maximum torque on dipole placed in external electric field, Nm
print "The maximum torque = %1.0e N-m"%tau
W = p*E*(cos(0)-cos(180*pi/180))
#W = integrate('p*E*sin(thet)', 'thet', 0, pi);    # The work done in rotating the dipole direction = %1.0e J", W
print "The work done in rotating the dipole direction = %1.0e J"%W

The maximum torque = 2e-03 N-m
The work done in rotating the dipole direction = 4e-03 J


## Example 13.2, Page 648¶

In [2]:
from math import *

#Variable declaration
Q = 8e-019;   # Charge of the nucleus, C
p = 3.2e-029;   # Electric dipole moment, C-m
r = 1e-10;    # Distance of dipole relative to the nucleus, m
k = 9e+9;   # Coulomb constant, N-meter-square/C-square

#Calculations&Results
F = k*p*Q*sqrt(3*cos(theta**2)+1)/r**3;    # The force acting on the dipole in the radial direction, N
print "The force acting on the dipole in the radial direction = %3.1e N"%F
theta = pi/2;   # Angle for perpendicular direction, radian
F = k*p*Q*sqrt(3*cos(theta)**2+1)/r**3;
print "The force acting on the dipole in the direction perpendicular to radial direction = %3.1e N"%F

The force acting on the dipole in the radial direction = 4.6e-07 N
The force acting on the dipole in the direction perpendicular to radial direction = 2.3e-07 N


## Example 13.3, Page 649¶

In [3]:
#Variable declaration
chi_e = 35.4e-12;   # Susceptability of the material, C-square/N-meter-square
eps_0 = 8.85e-12;   # Electric permittivity in free space, C-squre/N-meter-square

#Calculations&Results
K = 1 + (chi_e/eps_0);
print "The dielectric constant = %d "%K
eps = (eps_0*K);
print "The electric permittivity  = %5.3e C-square/N-meter square "%eps


The dielectric constant = 5
The electric permittivity  = 4.425e-11 C-square/N-meter square


## Example 13.4, Page 649¶

In [4]:
#Variable declaration
eps = 1.46e-10;   # Electric permittivity, C-square/n-meter-square
eps_0 = 8.85e-12;   # Permittivity in free space, C-squre/N-meter-square

#Calculations&Results
K = (eps/eps_0);
print "The dielectric constant = %4.1f "%K
chi_e = eps_0*(K-1);   # Susceptability,in C-square/N-meter-square
print "The electric susceptability = %4.2e C-square/N-meter square "%chi_e

The dielectric constant = 16.5
The electric susceptability = 1.37e-10 C-square/N-meter square


## Example 13.5, Page 650¶

In [5]:
#Variable declaration
K = 7.0;   # Dielectric constant of the slab
d = 0.01;   # Distance between the two parallel plates, m
V_0 = 100;   # Potential difference across the plates, V
eps_0 = 8.85e-12;   # Electric permability of the free space, C-square/N-meter-square

#Calculations&Results
E_0 = V_0/d;   # Electric intensity in the absence of dielectric slab, V/m
E = E_0/K;    # Electric intensity with dielectric slab introduced between the plates, V/m
print "The electric field intensity in the presence of the dielectric slab = %4.2e V/m "%E
D = (eps_0*K*E);    # Electric displacement, C-square/m-square
print "The electric displacement in the dielectric slab = %4.2e C-square/meter-square "%D
P = eps_0*(K-1)*E;    # Electric polarization in the dielectric slab, C-square/m-square
print "The electric polarization in the dielectric slab = %3.1e C-square/meter-square "%P

The electric field intensity in the presence of the dielectric slab = 1.43e+03 V/m
The electric displacement in the dielectric slab = 8.85e-08 C-square/meter-square
The electric polarization in the dielectric slab = 7.6e-08 C-square/meter-square


## Example 13.6, Page 650¶

In [6]:
#Variable declaration
K = 1.000074;   # Dielectric constant of the He
n = 2.69e+025;   # Atomic density of He, atoms/meter-cube
eps_0 = 8.85e-012;   # Electric permability of the free space, C-square/N-meter-square
E = 1;   # Electric field strength, V/m

#Calculations
p = (eps_0*(K-1)*E)/n;    # Dipole moment induced in He, C-m

#Result
print "The dipole moment induced in each He atom = %4.2e C-m "%p

The dipole moment induced in each He atom = 2.43e-41 C-m


## Example 13.7, Page 650¶

In [7]:
#Variable declaration
K = 1.000134;   # Dielectric constant of the neon
n = 2.69e+25;   # Atomic density of argon,atoms/meter-cube
eps_0 = 8.85e-12;   # Electric Permability in the free space, C-square/N-meter-square
E = 90e+03;    # External electric field, V/m

#Calculations
p = eps_0*(K-1)*E/n;    # Dipole moment induced in each neon atom, C-m
alpha = p/E;    # Atomic polarizability of neon gas, C-metre-square/V

#Results
print "The induced dipole moment of noen atom = %4.2e C-m"%p
print "The electronic polarizability of neon gas = %3.1e C-m-square/V "%alpha

The induced dipole moment of noen atom = 3.97e-36 C-m
The electronic polarizability of neon gas = 4.4e-41 C-m-square/V


## Example 13.8, Page 651¶

In [8]:
#Variable declaration
K = 1.0024;   # Dielectric constant of the argon
n = 2.7e+25;   # Atomic density of argon,atoms/meter-cube
eps_0 = 8.85e-12;   # Electric Permability in the free space, C-square/N-meter-square

#Calculations
alpha = eps_0*(K-1)/n;

#Result
print "The electronic polarizability of argon atom = %4.1e C-m-square/V "%alpha

The electronic polarizability of argon atom = 7.9e-40 C-m-square/V


## Example 13.9, Page 651¶

In [9]:
#Variable declaration
K = 2.24;   # Dielectric constant
eps_0 = 8.85e-12;   # Electric permability in the free space, C-square/N-meter-square
rho = 1.6e+003;   # Density of CCl4, kg/meter-cube
M = 156;   # Molecular weight of CCl4
E = 1e+007;   # External electric field strength, V/m
N_A = 6.02e+26;    # Avogadro's number, per kmol

#Calculations
rho_M = rho*N_A/M;     # Molecular density of CCl4
p = eps_0*(K-1)*E/rho_M;    # Individual dipole moment of CCL4 molecule, C-m

#Result
print "Individual dipole moment of CCL4 molecule = %4.2e C-m "%p

Individual dipole moment of CCL4 molecule = 1.78e-32 C-m


## Example 13.10, Page 652¶

In [11]:
from math import *

#Variable declaration
K = 1.0000684;   # Dielectric constant of He at 1 atm
n = 2.7e+25;   # Density of He at 1 atm and 273 K, atoms/meter-cube

#Calculations
# The atomic polarizibility, alpha = eps_0*(K-1)/n
# In terms of atomic radius, alpha = 4*%pi*eps_0*R^3 so, we have
R = ((K-1)/(4*pi*n))**(1./3);    # Radius of He atom, m

#Result
print "The atomic radius of He = %4.2e m "%R

The atomic radius of He = 5.86e-11 m


## Example 13.11, Page 652¶

In [12]:
#Variable declaration
mu = 1.5;   # Optical index of refraction of NaCl crystal
K = 5.6;   # Static dielectric constant of NaCl crystal

#Calculations
P_IP = (1-((mu**2-1)*(K+2))/((mu**2+2)*(K-1)))*100;

#Result
print "The percentage of ionic polarizibility in NaCl crystal = %4.1f percent "%P_IP

The percentage of ionic polarizibility in NaCl crystal = 51.4 percent


## Example 13.12, Page 653¶

In [31]:
from math import *

#Variable declaration
K_B = 1.38e-23;   # Boltzmann constant, J/mol/K
T = 300;   # Room temperature, K
eps_0 = 8.85e-12;   # Electric permittivity of free space, F/m
N_A = 6.0e+23;    # Avogadro's number

#Calculations
n2 = N_A*1000;     # Number of molecules of non-polar substance in 1000 cc volume
p_0 = sqrt((9*K_B*T*eps_0*0.023)/n2);    # Dipole moment of polar molecules, C-m

#Result
print "The dipole moment of polar molecules = %4.3e C-m"%p_0

The dipole moment of polar molecules = 3.555e-30 C-m