6: Dielectric Properties

Example number 6.1, Page number 6.23

In [1]:
#importing modules
import math
from __future__ import division

#Variable declaration
C = 2     #capacitance(micro-farad)
V = 1000    #voltage applied(V)
epsilon_r = 100     #permitivity

#Calculation
C = C*10**-6      #capacitance(farad)
W = (C*V**2)/2      #energy stored in capacitor(J)
C0 = C/epsilon_r      #capacitance removing the dielectric
W0 = C0*(V**2)/2      #energy stored without dielectric(J)
E = 1-W0       #energy stored in dielectric(J)

#Result
print "energy stored in capacitor is",W,"J"
print "energy stored in the dielectric is",E,"J"

energy stored in capacitor is 1.0 J
energy stored in the dielectric is 0.99 J

Example number 6.2, Page number 6.24

In [3]:
#importing modules
import math
from __future__ import division

#Variable declaration
epsilon_r = 4.94
n_2 = 2.69     #square of index of refraction
alpha_i = 0    #at optical frequencies

#Calculation
#(epsilon_r-1)/(epsilon_r+2) = N*(alpha_e+alpha_i)/(3*epsilon0)
X = (epsilon_r-1)/(epsilon_r+2)
#epsilon_r = n**2. therefore (n**2-1)/(n**2+2) = N*alpha_e/(3*epsilon0)
Y = (n_2-1)/(n_2+2)
#N*(alpha_e+alpha_i)/N*alpha_e = X/Y
#let alpha = alpha_i/alpha_e
alphai_e = (X/Y)-1     #ratio between electronic ionic and electronic polarizability
alphai_e = math.ceil(alphai_e*10**4)/10**4   #rounding off to 4 decimals
alphae_i = 1/alphai_e      #ratio between electronic and ionic polarizability
alphae_i = math.ceil(alphae_i*10**3)/10**3   #rounding off to 3 decimals

#Result
print "ratio between electronic ionic and electronic polarizability is",alphai_e
print "ratio between electronic and ionic polarizability is",alphae_i

ratio between electronic ionic and electronic polarizability is 0.5756
ratio between electronic and ionic polarizability is 1.738

Example number 6.3, Page number 6.25

In [9]:
#importing modules
import math
from __future__ import division

#Variable declaration
epsilon_r = 2.56
tan_delta = 0.7*10**-4
f = 1     #frequency(MHz)
A = 8     #area(cm**2)
d = 0.08     #diameter(mm)
epsilon0 = 8.85*10**-12

#Calculation
A = A*10**-4      #area(m**2)
d = d*10**-3     #diameter(m)
epsilon_rdash = epsilon_r*tan_delta
omega = 2*math.pi*10**6
Rp = d/(omega*epsilon0*epsilon_rdash*A)      #parallel loss resistance(ohm)
Rp = Rp*10**-6      #parallel loss resistance(Mega ohm)
Rp = math.ceil(Rp*10**3)/10**3   #rounding off to 3 decimals
Cp = A*epsilon0*epsilon_r/d     #capacitance(farad)

#Result
print "parallel loss resistance is",Rp,"ohm"
print "capacitance in Farad is",Cp,"Farad"

parallel loss resistance is 10.036 ohm
capacitance in Farad is 2.2656e-10 Farad

Example number 6.4, Page number 6.26

In [10]:
#importing modules
import math
from __future__ import division

#Variable declaration
N = 3*10**28     #density(atoms/m**3)
alpha_e = 10**-40    #electronic polarizability(Farad-m**2)
epsilon0 = 8.854*10**-12

#Calculation
epsilon_r = 1+(N*alpha_e/epsilon0)       #dielectric constant of material
epsilon_r = math.ceil(epsilon_r*10**3)/10**3   #rounding off to 3 decimals

#Result
print "dielectric constant of material is",epsilon_r 

dielectric constant of material is 1.339

Example number 6.5, Page number 6.27

In [12]:
#importing modules
import math
from __future__ import division

#Variable declaration
epsilon0 = 8.854*10**-12
epsilon_r = 1.0000684      #dielectric constant
N = 2.7*10**25     #density(atoms/m**3)

#Calculation
alpha_e = epsilon0*(epsilon_r-1)/N        #electronic polarizability(Fm**2)

#Result
print "electronic polarizability is",round(alpha_e/1e-41,3),"*10^-41 Fm**2"

electronic polarizability is 2.243 *10^-41 Fm**2

Example number 6.6, Page number 6.27

In [13]:
#importing modules
import math
from __future__ import division

#Variable declaration
epsilon0 = 8.85*10**-12
V = 100     #potential(V)
A = 100     #area(cm**2)
d = 1    #plate seperation(cm)

#Calculation
A = A*10**-4      #area(m**2)
d = d*10**-2    #plate seperation(m)
C = epsilon0*A/d     #capacitance(farad)
Q = C*V     #charge on plates

#Result
print "capacitance of capacitor is",C,"F"
print "charge on plates in coulomb is",Q,"coulomb"

capacitance of capacitor is 8.85e-12 F
charge on plates in coulomb is 8.85e-10 coulomb

Example number 6.7, Page number 6.28

In [15]:
#importing modules
import math
from __future__ import division

#Variable declaration
N = 6.02*10**26     #avagadro number
d = 2050      #density(kg/m**3)
AW = 32       #atomic weight of sulphur
epsilon_r = 3.75    #relative dielectric constant
epsilon0 = 8.55*10**-12

#Calculation
n = N*d/AW    #number of atoms(per m**3)
alpha_e = ((epsilon_r-1)/(epsilon_r+2))*3*epsilon0/n     #electronic polarizability(Fm**2) 

#Result
print "electronic polarizability is",round(alpha_e/1e-40,3),"*10^-40 Fm**2" 

electronic polarizability is 3.181 *10^-40 Fm**2

Example number 6.8, Page number 6.29

In [16]:
#importing modules
import math
from __future__ import division

#Variable declaration
Q = 2*10**-10       #charge(coulomb)
d = 4     #plate seperation(mm)
epsilon_r = 3.5    #dielectric constant
A = 650     #area(mm**2)
epsilon0 = 8.85*10**-12

#Calculation
d = d*10**-3      #plate seperation(m)
A = A*10**-6     #area(m**2)
V = Q*d/(epsilon0*epsilon_r*A)       #voltage across capacitors(V)
V = math.ceil(V*10**3)/10**3   #rounding off to 3 decimals

#Result
print "resultant voltage across capacitors is",V,"V"

resultant voltage across capacitors is 39.735 V

Example number 6.9, Page number 6.30

In [18]:
#importing modules
import math
from __future__ import division

#Variable declaration
V = 10      #potential(V)
d = 2*10**-3      #plate seperation(m)
epsilon_r = 6
epsilon0 = 8.85*10**-12

#Calculation
E = V/d     #electric field(V/m)
D = epsilon0*epsilon_r*E       #dielectric displacement(C/m**2)

#Result
print "dielectric displacement is",D,"Cm^-2"

dielectric displacement is 2.655e-07 Cm^-2