# Chapter 10 - Dielectric and Magnetic Materials¶

## Example 1 - pg 312¶

In [1]:
#pg 312
#Given:
import math
er = 1.0000684; # relative dielectric constant
N = 2.7*10**25; # atoms/m^3
#We know, er - 1 = 4*pi*N*R^3
#calculations
R = ((er-1)/(4*math.pi*N))**(1./3) ; # in m
#results
print"R :",round(R*10**10,1)," x 10^-10 m"

R : 0.6  x 10^-10 m


## Example 2 - pg 313¶

In [2]:
#pg 313
#calculate the Susceptibility
#Given :
import math
R = 1; # radius in A
N = 5*10**28 ; #  atoms/m**3
mu_0 = 4*math.pi*10**-7; # permeability of free space in H/m
mu_r = 1;#relative permiability
m = 9.1*10**-31 # electron mass in kg
e = 1.6*10**-19 ; # charge of an electron in C
# R = 1*10**-10 m  because 1 A = 1.0*10**-10 m
#calculations
chi = -(N*e**2*(R*10**-10)**2*mu_0*mu_r)/(4*m); #Susceptibility of diamagnetic material
#results
print "Susceptibility of diamagnetic materials  is",math.floor(chi*10**5), "x 10^-5"

Susceptibility of diamagnetic materials  is -1.0 x 10^-5


## Example 3 - pg 320¶

In [3]:
#pg 320
#calculate the dipole moment and Sum
#Given :
import math
e0 = 8.85*10**-12 ; # dielectric constant in farad/m
er1 = 1.006715 ; #relative dielectric constant
er2 = 1.005970;# relative dielectric constant
T1 = 300. ; # Temperature in K  (273+27 = 300 K)
T2 = 450.; # Temperature in K  (273 + 177 = 450 K)
k = 1.38*10**-23; # in J/K
N = 2.44*10**25 ; # molecules/m**3
#calculations
#e0*(er1 - er2)= ((N*mu_p**2)/(3*k))*((1/T1)- (1/T2))
mu_p = math.sqrt((e0*(er1 - er2)*3*k)/(((1/T1)-(1/T2))* N)); #dipole moment in C m
D = 3.3*10**-30; # dipole of 1 Debye is equal to 3.33 x 10**-30 C m
#e0*(er1 - 1) = N*(alpha_e + alpha_i + (mu_p**2/3*k*T1))
Sum = ((e0*(er1 - 1))/N) - ((mu_p)**2/(3*k*T1)); # alpha_e + alpha_i   in farad m**2
#results
print "Dipole moment (debye) = ",round(mu_p/D,2)
print "Sum =",round(Sum*10**39,1),"x 10**-39 farad m^2"

Dipole moment (debye) =  0.96
Sum = 1.6 x 10**-39 farad m^2


## Example 4 - pg 321¶

In [4]:
#pg 321
#calculate the value of E
#Given :
mu_p = 1.2 ;# dipole moment in debye units
T = 300 ; # Temperature in Kelvin ( 273+27 = 300 K)
k = 1.38*10**-23 ; #  in J/K
per = 0.5/100 ; # percentage of saturated polarisation
# 0.05*N*mu_p = (N*(mu_p)**2*E/(3*k*T))
#calculations
E = (3*k*T*per)/(mu_p*3.33*10**-30); # External field in V/m
#results
print " E =",round(E*10**-7,2),"x 10^7 V/m"

 E = 1.55 x 10^7 V/m


## Example 5 - pg 321¶

In [5]:
#pg 321
#calculate the Susceptibility
import math
#Given :
N = 5*10**28 ;# number of dipoles per m**3
betaa = 1;# Bohr magneton
T = 300 ; # Room temperature in k
k = 1.38*10**-23 ; #  in J/K
mu_0 = 4*math.pi*10**-7 ; #Magnetic permeability in H/m
#calculations
chi = (N*mu_0*betaa*(1*9.27*10**-24)**2)/(k*T);
#results
print "Susceptibility =",round(chi*10**3,0),"x 10**-3"
print 'Result obtained differs from that in textbook, because in textbook only the order is considered'

Susceptibility = 1.0 x 10**-3
Result obtained differs from that in textbook, because in textbook only the order is considered


## Example 6 - pg 324¶

In [7]:
#pg 324
#calculate the Relative dielectric constant
#Given :
M = 32.; # Atomic weight in kg/kmole
Na =6.023*10**26 ; # Avogadro constant in atoms/kmole
alpha_e = 3.28*10**-40; # electronic polarisability in farad/m**2
rho = 2.08; #density in gm/cm**3
e0 = 8.85*10**-12 ; # dielectric constant in farad/m
# (er - 1)/(er + 2)  = (N*alpha_e/3*e0)
#1 gm = 1.0*10**-3 kg , 1 cm**3 = 1.0*10**-6 m**3
#calculations
N = (Na*(rho*10**3))/M; #  atoms/m**3
er =( 2*((N*alpha_e)/(3*e0)) + 1 )/(1 - ((N*alpha_e)/(3*e0)));
#results
print "Relative dielectric constant = ",round(er,2)

Relative dielectric constant =  3.81


## Example 7 - pg 326¶

In [8]:
#pg 326
#calculate the Power loss
#Given :
area = 50000.; # area of hysteresis on a graph
axis1 = 10**-4 ; # units of scale in Wb/m**2
axis2 = 10**2; # units of scale in A/m
vol = 0.01; # volume in m**3
F = 50; #frequency in Hz
#calculations
E1 = area*axis1*axis2; # Energy lost per cycle in J/m**3
E2 = E1*vol ; # Energy lost in core per cycle in J
P = E2*F; # Power loss in W
#results
print "Power loss (W) =  ",P

Power loss (W) =   250.0


## Example 8 - pg 328¶

In [9]:
#pg 328
#calculate the value of energy
import math
#Given :
mu_d = 9.27*10**-24; # Bhor magneton in Am**2
mu_0 = 4*math.pi*10**-7; # Magnetic permiability in H/m
r = 2; # dipoles distance in A
#U = mu_d*B = -( mu_0*mu_d**2)/(2*pi*r)
#r = 2*10**-10 m , 1 A = 1.0*10**-10 m
#calculations
U = ( mu_0*mu_d**2)/(2*math.pi*(r*10**-10)**3); # Energy
print "U =",round(U*10**25,1),"x 10^-25 "

U = 21.5 x 10^-25


## Example 9 - pg 329¶

In [10]:
#pg 329
#calculate the Saturation magnetisation
#Given :
a = 2.87; # lattice constant in A
mu = 4.; # 4 Bohr magnetons/atom
# BCC = 2 atoms/unit cell , 1 A = 1.0*10**-10 m
N = 2./(2.87*10**-10)**3; # atoms/m**3
#calculations
#1 Bohr magneton = 9.27*10**-24 Am**2
Msat = N*mu*9.27*10**-24;# Saturation in magnetisation in A/m
#results
print " Saturation Magnetisation =",round(Msat*10**-6,2),"x 10^6 A/m"

 Saturation Magnetisation = 3.14 x 10^6 A/m


## Example 10 - pg 338¶

In [11]:
#pg 338
#calculate the percentage attributed to polarisability
#Given :
er = 6.75 ; # relative dielectric constant for glass
f = 10**9 ;# frequency in Hz
n = 1.5;# refractive index of glass
e0 = 8.85*10**-12; # dielectric constant in farad/m
#Pe = e0*(n**2 - 1)*E  , Pi = e0*(er - n**2)*E ,  P = Pi + Pe = e0*(er - 1)*E
#Percentage = [(e0*(er - n**2)*E)/(e0*(er -1)*E)]*100 , both the E's cancel each other
#calculations
per = (e0*(er - n**2))/(e0*(er -1))*100;# percentage
print "Percentage = ",round(per,1)

Percentage =  78.3