# Chapter 5:Conductivity of Metals and Superconductivity¶

## Example 5.1,Page No:5.5¶

In :
import math

#variable declaration
d       = 2*10**-3;                 #diameter in m
I       = 5*10**-3;                 #current in A
e       = 1.6*10**-19;              #charge of electron in coulombs
a       = 3.61*10**-10;             #side of cube in m
N       = 4;                       #number of atoms in per unit cell

#formula
#J=n*v*e

#calculation
r       = d/float(2);                #radius in m
n       = N/float(a**3);             #number of atoms per unit volume in atoms/m**3
A       = math.pi*(r**2);            #area in m**2
J       = I/float(A);                #current density in Amp/m**2
v       = J/float(n*e);              #average drift velocity in m/s

#result
print'velocity=%3.2e'%v,'m/s';

velocity=1.17e-07 m/s


## Example 5.2,Page No:5.6¶

In :
import math

#variable declaration
I       = 6;                    #current in A
d       = 1*10**-3;             #diameter in m
n       = 4.5*10**28;           #electrons available in electron/m**3
e      = 1.6*10**-19;           #charge of  electron  in coulombs

#calculation
r     = d/float(2);               #radius in m
A     = math.pi*(r**2);           #area in m**2
J     = I/float(A);               #current density in A/m**3
vd    = J/float(n*e);             #density in m/s

#result
print'velocity=%3.2e'%vd,'m/s';

velocity=1.06e-03 m/s


## Example 5.3,Page No:5.6¶

In :
import math

#variable declaration
V       = 63.5;             #atomic weight in kg
d       = 8.92*10**3;        #density of copper in kg/m**3
r       = 0.7*10**-3;        #radius in m
I       = 10;               #current in A
e       = 1.6*10**-19;          #charge of electronin coulomb
h       = 6.02*10**28;          #planck's constant in (m**2)*kg/s

#calculation
A        = math.pi*(r**2);          # area in m**2
N        = h*d;
n        = N/float(V);
J        = I/float(A);              #current density in m/s
vd       = J/float(n*e);            #drift velocity in m/s

#result
print'velocity=%2.2e'%vd,'m/s';

velocity=4.80e-06 m/s


## Example 5.4,Page No:5.7¶

In :
import math

#variable declaration
R        = 0.182;             #resistance in ohm
l       = 1;                  #length in m
A       = 0.1*10**-6;         #area in m**2

#formula
#R=(p*l)/A

#calculation
p      = (R*A)/float(l);              #resistivity in ohm m

#result
print'restivity=%3.2e'%p,'ohm m';

restivity=1.82e-08 ohm m


## Example 5.5,Page No:5.7¶

In :
import math

#variable declaration
n        = 5.8*10**28;                 #number of silver electrons in electrond/m**3
p        = 1.45*10**-8;                #resistivity in ohm m
E        = 10**2;                      #electric field in V/m
e        = 1.6*10**-19;

#formula
#sigma  = n*e*u
#sigma=p
#calculation
u       = 1/float(n*e*p);
vd      = u*E;                    #drift velocity in m/s

#result
print'velocity=%3.1f'%vd,'m/s';

velocity=0.7 m/s


## Example 5.6,Page No:5.8¶

In :
import math

#variable declaration
W           = 107.9;               #atomic weight in amu(atomic mass unit)
p           = 10.5*10**3;           #density in kg/m**3
sigma       =6.8*10**7;            #conductivity in ohm**-1.m**-1
e           =1.6*10**-19;          #charge of electron in coulombs
N           = 6.02*10**26;         #avagadro number in mol**-1

#calculation
n       = (N*p)/float(W);              #number of atoms per unit volume
u      = sigma/float(n*e);          #density of electron in m**2.V**-1.s**-1

#result
print'density=%3.2e'%u,'m**2.V**-1.s**-1';

density=7.25e-03 m**2.V**-1.s**-1


## Example 5.7,Page No:5.8¶

In :
import math

#variable declaration
#for common metal copper
n           = 8.5*10**28;               #number of atoms in m**-3
sigma       = 6*10**7;                  #sigma  in ohm**-1 m**-1
m           = 9.1*10**-31;              #mass of electron in kilogram
e           = 1.6*10**-19;              #charge of electron in coulombs

#calculation
t  = (m*sigma)/float(n*(e**2));       #relaxation time in s

#result
print'time=%3.2e'%t,'s';

time=2.51e-14 s


## Example 5.9,Page No:5.14¶

In :
import math

#variable declaration
t       = 3.0*10**-14;              #time in s
n       = 2.5*10**22;               #in electrons per m**3
m       = 9.1*10**-31;              #mass of electron in kilograms
e       = 1.6*10**-19;              #charge of electron in coulombs
T       = 3.25;                     #temperature in K

#formula
#K/(sigma*T)=2.44*10**-8   from wiedemann Franz law
#calculation
sigma    = (n*(e**2)*t)/float(m*10**-6);             #conductivity in m**3
K        = (2.44*10**-8)*sigma*T;                    #thermalconductivity in W/m-K

#result
print'thermal conductivity=%3.4f '%K,'W/m-K';
print' Note: calculation mistake in textbook in calculating K as T value is taken 325 instead of 3.25';

thermal conductivity=1.6731  W/m-K
Note: calculation mistake in textbook in calculating K as T value is taken 325 instead of 3.25


## Example 5.10,Page No:5.20¶

In :
import math

#variable declaration
a       = 10**-10;                   #one dimension in m
m       = 9.1*10**-31;               #mass of kg
h       = 6.62*10**-34;              #planck's constant in joule-s

#formula
#En   = ((n**2)*(h**2))/float(8*m*(a**2))
#calculation
E1      = (h**2)/float(8*m*(a**2));           #energy in J
E2      = (4*(h**2))/float(8*m*(a**2));       #energy in J
dE       = (3*(h**2))/float(8*m*(a**2));      #energy diefference in J
x        = dE/float(1.6*10**-19);      #energy diefference in eV

#result
print'energy diefference=%3.2e'%x,'eV';

energy diefference=1.13e+02 eV


## Example 5.11,Page No:5.20¶

In :
import math

#variable declaration
N         =6.02*10**23;            #avagadro number in atoms /mole
h        = 6.63*10**-34;            #planck's constant in joule-s
m        = 9.11*10**-31;            #mass in kg
M        = 23;                      #atomic weight in grams /mole
p        = 0.971;                   #density in gram/cm**3

#formula
#x=N/V=(N*p)/M
#calculation
x       = (N*p)/float(M);
x1      = x*10**6;
eF      = (((h**2)/float(2*m)))*(((3*x1)/(8*math.pi))**(2/float(3)));       #Fermi energy
eF1     = (eF)/float(1.6*10**-19);

#result
print'fermi energy=%3.2f'%eF1,'eV';

fermi energy=3.16 eV


## Example 5.12,Page No:5.21¶

In :
import math

#variable declaration
x       = 2.54*10**28;           #number of electrons in per m**2
h       = 6.63*10**-34;          # planck's constant in joule-s
m       = 9.11*10**-31;          # mass in kg
p       = 0.971;                 #density in grams/cm**3
k       = 1.38*10**-23;

#calculation
#x       = (N*p)/float(M);
eF      = (((h**2)/(2*m)))*(((3*x)/float(8*math.pi))**(2/float(3)));
eF1     = (eF)/float(1.6*10**-19);                                  #Fermi energy in eV
vF      = math.sqrt((2*eF)/float(m));                               #fermi velocity in m/s
TF      = eF/float(k);                                             #fermi temperature in K

#result
print'fermi energy =%3.2f'%eF1,'eV';
print'fermi velocity =%3.2e'%vF,'m/s';
print'femi temperature =%3.2e'%TF,'K';

fermi energy =3.16 eV
fermi velocity =1.05e+06 m/s
femi temperature =3.66e+04 K


## Example 5.13,Page No:5.21¶

In :
import math

#variable declaration
M      = 65.4;              #atomic weight
p      = 7.13;              #density in g/cm**3
h      = 6.62*10**-34;      # planck's constant in joules-s
m      = 7.7*10**-31;       # mass
v      = 6.02*10**23;       #avagadros number in atoms/gram-atom

#calculation
#x =N/V
V      = M/float(p);                                                        #volume of one atom in cm**3
n      = v/float(V);                                                        # number of Zn atoms in volume v
x      = 2*n*(10**6);                                                        #number of free electrons in unit volume iper m**2
eF     = ((h**2)/float(2*m))*(((3*x)/float(8*math.pi))**(2/float(3)));        # fermi energy in J
eF1    = eF/float(1.6*(10**-19));

#result
print'fermi energy =%3.2d'%eF1,'eV';


fermi energy = 11 eV


## Example 5.14,Page No:5.22¶

In :
import math

#variable declaration
eF     = 4.27;             #fermi energy in eV
m      = 9.11*10**-31;      # mass of electron in kg
h      = 6.63*10**-34;      # planck's constant J.s

#formula
#x= N/V
#calculation
eF1    = eF*1.6*10**-19;                                                   #fermi energy in eV
x      = (((2*m*eF1)/float(h**2))**(3/float(2)))*((8*math.pi)/float(3));    #number of electrons per unit volume

#result
print'number of electrons per unit volume =%4.00e'%x,'m**-3';


number of electrons per unit volume =4e+28 m**-3


## Example 5.15,Page No:5.23¶

In :
import math

#variable declaration
eF1       = 4.70;             # fermi energy in eV
eF2       = 2.20;             #fermi energy in eV
x1        = 4.6*10**28;        # electron density of lithium per m**3

#formula
#N/V = (((2*m*eF1)/(h**2))**(3/2))*((8*math.pi)/3);
#N/V = k*(eF**3/2)
#N/V = x
#calculation
x2      = x1*((eF2/float(eF1))**(3/float(2)));            #electron density for metal in per m**3

#result
print'electron density  for a metal =%4.2e'%x2,'m**-3';

electron density  for a metal =1.47e+28 m**-3


## Example 5.16,Page No:5.24¶

In :
import math

#variable declaration
eF      = 5.4;                         #fermi energy in eV
k       = 1.38*10**-23;                 # k in joule/K

#calculation
e0      = (3*eF)/float(5);                           #average energy in eV
T       = (e0*(1.6*10**-19)*2)/float(3*k);           #temperature in K

#result
print'average energy =%3.2f'%e0,'eV';
print'temperature =%3.2e'%T,'K';

average energy =3.24 eV
temperature =2.50e+04 K


## Example 5.17,Page No:5.25¶

In :
import math

#variable declaration
EF        = 15;                     #fermi energy  in eV
m         = 9.1*10**-31;            #mass of electron in kilogarams

#calculation
E0        = (3*EF)/float(5);                                     #average energy en eV
v         = math.sqrt((2*E0*1.6*10**-19)/float(m));              #speed of electron in m/s

#result
print'average energy =%3.1f'%E0,'eV';
print'speed =%3.2e'%v,'m/s';

average energy =9.0 eV
speed =1.78e+06 m/s


## Example 5.18,Page No:5.25¶

In :
import math

#variable declaration
EF        = 7.5;                     #fermi energy  in eV
m         = 9.1*10**-31;            #mass of electron in kilograms

#calculation
E0        = (3*EF)/float(5);             #average energy en eV
v         = math.sqrt((2*E0*1.6*10**-19)/float(m));       #speed in m

#result
print'average energy =%3.2f'%E0,'eV';
print' speed =%3.2e'%v,'m/s';


average energy =4.50 eV
speed =1.26e+06 m/s


## Example 5.19,Page No:5.25¶

In :
import math

#variable declaration
m     = 9.1*10**-31;       #mass of electron in kg
h     = 6.62*10**-34;      #planck's constant in (m**2)*kg/s
#formula
#x=N/V
x      = 2.5*10**28;

#calculation
EF       = ((h**2)/float(8*(math.pi**2)*m))*((3*(math.pi**2)*x)**(2/float(3)));      #fermi energy in J
EF1      = EF/float(1.6*10**-19);                                                    #fermi energy in eV
vF       = (h/float(2*m*math.pi))*((3*(math.pi**2)*x)**(1/float(3)));                #fermi velocity in m/s

#result
print'energy=%3.2f'%EF1,'eV';
print' speed= =%3.2e'%vF,'m/s';

energy=3.12 eV
speed= =1.05e+06 m/s


## Example 5.20,Page No:5.29¶

In :
import math

#variable declaration
Ps      = 10**7;                 #power in W
V      = 33*10**3;               #power transmitted in W
R      = 2;                      #resistance in ohm

#calculation
I      = Ps/float(V);                    #current in A
Pd     = (I**2*R)/float(1000);           #power lost in feeder in kW
n      = ((Ps-Pd)/float(Ps))*100;        #efficiency in %
v      = I*R;                            #voltage drop in V
Vd     = (v/float(V))*100;              #percentage voltage drop

#result
print'efficiency =%0f '%n,'%';
print'voltage drop =%3.1f'%Vd,'%';

efficiency =99.998163  %
voltage drop =1.8 %


## Example 5.21,Page No:5.36¶

In :
import math

#variable declaration
a1   = 2.76;               #a1 in uv/°C
a2   = 16.6;              #a2 in uv/°C
b1   = 0.012;              #b1 in uv/°C
b2   = -0.03;              #b2 in uv/°C

#calculation
#aFe,Pb   =a1
#aCu,Pb   = a2
#bCu,Fe   = b1
#bFe,Pb   = b2

#calculation
a3    = a1-a2;              #a3 in uv/°C
b3    = b1-b2;              #b3 in uv/(°C)**2

#result
print'aCu,Fe = %3.1f'%a3,'uV/°C';
print' bCu,Fe = %3.3f'%b3,'uV/(°C)**2';

aCu,Fe = -13.8 uV/°C
bCu,Fe = 0.042 uV/(°C)**2


## Example 5.23,Page No:5.37¶

In :
import math

#variable declaration
a       = 15;              #a in uv/°C
b       = -1/float(30);              #b in uv/°C

#E = at+bt^2
#dE/dT =a+2*b*t
#t=tn
#dE/dT =0
#calculation
tn     = -(a/float(2*(b)))               #neutral temperature in °C
#t1+t2  = 2*t2;
t2    = 2*tn               #inversion temperature in °C

#result
print'neutral temperature =%3.2d '%tn,'°C';
print'temperature of inversin = %3.2d '%t2,'°C';

neutral temperature =225  °C
temperature of inversin = 450  °C


## Example 5.24,Page No:5.37¶

In :
import math

#variable declaration
p2        =  2.75;         #resistivity of alloy 1 percent of Ni in uΩ-cm
p1        =  1.42;         #resistivity of pure copper in uΩ-cm
p3        = 1.98;          #resistivity of alloy 3 percent of silver in uΩ-cm

#p(Ni+Cu) =p1
#pCu =p2
#p(Cu+silver)=p3
#calculation
pNi        = p2-p1;
p4         = (p3-p1)/float(3);
palloy     = p1+(2*pNi)+(2*p4);        #resistivity of alloy 2 percent of silver and 2 percent of nickel in uΩ-cm

#result
print'resistivity of alloy =%3.4f'%palloy,'uΩ-cm';

resistivity of alloy =4.4533 uΩ-cm


## Example 5.25,Page No:5.41¶

In :
import math

#variable declaration
M1       = 202;           #mass number
M2       = 200;          # mass number
Tc1      = 4.153;       # temperature in K
alpha    = 0.5;

#formula
#m**alpha*(Tc)= conatant
#calculation
Tc2    = ((M1**alpha)*Tc1)/float(M2**alpha);     #transition temperature in K

#result
print'transition temperature =%3.3f'%Tc2,'K';

transition temperature =4.174 K


## Example 5.26,Page No:5.41¶

In :
import math

#variable declaraion
Tc1      = 2.1;                    #temperature in K
M1       = 26.91;                  #mass number
M2       = 32.13;                  #mass number

#formula
#Tc*(M1**2) = constant
#calculation
Tc2        = (Tc1*(M1**(1/float(2))))/float(M2**(1/float(2)));       #critical temperature in K

#result
print'critical temperature =%3.2f'%Tc2,'K';

critical temperature =1.92 K


## Example 5.27,Page No:5.42¶

In :
import math

#variable declaration
Hc1        =  1.41*10**5;       #critical fields in amp/m
Hc2        = 4.205*10**5;       # critical fields in amp/m
T1         = 14.1;              #temperature in K
T2         = 12.9;              # temperature in K
T3         = 4.2;               #temperature in K

#formula
#Hcn =Hc*((1-((T/Tc)**4)))
#calculation
Tc        =(((((Hc2*(T1**2))-(Hc1*(T2**2)))/float(Hc2-Hc1)))**(1/float(2)));          #temperature in K
Hc0       = Hc1/float(1-((T1/float(Tc))**2));                                         #critical field in A/m
Hc2       = Hc0*(1-(T3/float(Tc))**2);                                                #critical field in A/m

#result
print'transition temperature =%3.2f'%Tc,'K';
print'critical field  =%3.2e'%Hc2,'A/m';

transition temperature =14.67 K
critical field  =1.70e+06 A/m


## Example 5.28,Page No:5.43¶

In :
import math

#variable declaration
Hc0        = 700000;       #critical field at 0 K
T          = 4;            #temperature in K
Tc         = 7.26;         #temperature in K

#calculation
Hc    = Hc0*(1-(T/float(Tc))**2);    #critical field n A/m

#result
print'critical field =%3.4e'%Hc,'A/m';
print' Note: calculation mistake in texttbook in calculating Hc';

critical field =4.8751e+05 A/m
Note: calculation mistake in texttbook in calculating Hc


## Example 5.29,Page No:5.44¶

In :
import math

#variable declaration
Hc0        = 8*10**4;       #critical field
T          = 4.5;          #temperature in K
Tc         = 7.2;          #temperature in K
D          = 1*10**-3;      #diameter in m

#calculation
Hc    = Hc0*(1-(T/float(Tc))**2);
r     = D/float(2);                #radius in m
Ic     = 2*math.pi*r*Hc;           #critical current in A

#result
print'critical current =%3.2f'%Ic,'A';

critical current =153.15 A


## Example 5.30,Page No:5.44¶

In :
import math

#variable declaration
Hc0        = 0.0306;       #critical field at 0 K
T          = 2;           #temperature in K
Tc         = 3.7;         #temperature in K

#calculation
Hc    = Hc0*(1-(T/float(Tc))**2);     #critical field in tesla

#result
print'critical field =%3.4f'%Hc,'tesla';

critical field =0.0217 tesla


## Example 5.31,Page No:5.44¶

In :
import math

#variable declaration
HcT        = 1.5*10**5;     # critical field for niobium at 0 K
Hc0        = 2*10**5;       # critical field for nobium at 0 K
T          = 8;             # temperature in K

#calculation
Tc     = T/((1-(HcT/float(Hc0)))**0.5);     #transition temperature in K

#result
print'transition temperature =%3.2f'%Tc,'K';

transition temperature =16.00 K


## Example 5.32,Page No:5.45¶

In :
import math

#variable declaration
Hc1        =  0.176;       #critical fields
Hc2        = 0.528;        #critical fields
T1         = 14;           #temperature in K
T2         = 13;           #temperature in K
T3         = 4.2;          #temperature in K

#formula
#Hcn =Hc*((1-((T/Tc)**4)))
#calculation
Tc     =(((((Hc2*(T1**2))-(Hc1*(T2**2)))/float(Hc2-Hc1)))**(1/float(2)));  #transition temperature in K
Hc0    = Hc1/(1-((T1/float(Tc))**2));          #critical field in T
Hc2    = Hc0*(1-((T3/float(Tc))**2));          #critical field in T

#result
print'transition temperature =%3.2f '%Tc,'K';
print' critical field  =%3.2f '%Hc2,'T';

transition temperature =14.47  K
critical field  =2.50  T


## Example 5.33,Page No:5.46¶

In :
import math

#variable declaration
Hc         = 7900;                                  #magnetic field in A/m
r          = 2.0*10**-3;                            #radius of super condutor in m

#calculation
I          = 2*math.pi*r*Hc;                          #critical current in A

#result
print'critical current =%4f'%I,'A';
print'Note: calculation mistake in textbook in calculation of I';

critical current =99.274328 A
Note: calculation mistake in textbook in calculation of I


## Example 5.34,Page No:5.46¶

In :
import math

#variable declaration
d           = 10**-3;           #diameter in m
Bc          = 0.0548;           # Bc in T

#calculation
u0          = 4*math.pi*10**-7;                 #permiability m**2
r           = d/float(2);                       #radius in m
Ic          = (2*math.pi*r*Bc)/float(u0);       #current in Amp

#result
print'current =%3.2d '%Ic,'Amp';

current =137  Amp


## Example 5.35,Page No:5.52¶

In :
import math

#variable declaration
D          =8.5*10**3;               #density in kg/m**3
W          =93;                      #atomic weight
m          =9.1*10**-31;             #mass of electron in kilograms
e          =2*1.6*10**-19;           #charge of electron in coulombs
N          =6.023*10**26;            #avagadro number in (lb-mol)−1

#calculation
u0         =4*math.pi*10**-7;
ns         =(D*N)/float(W);                         #in per m**3
lamdaL     =(m/float(u0*ns*e**2))**(1/float(2));    #London's penetration depth in nm

#result
print'penetration depth=%3.2f'%(lamdaL*10**9),'nm';

penetration depth=11.33 nm


## Example 5.36,Page No:5.52¶

In :
import math

#variable declaration
Tc     =7.2;                  #temperature in K
lamda  =380;                  #penetration depth in Å
T      =5.5;                  #temperature in K

#calculation

lamdaT=lamda*((1-((T/float(Tc))**4))**(-1/float(2)));     #penetration depth in Å

#result
print'penetration depth=%3.1f'%lamdaT,'Å';
print' Note: calculation mistake in textbook in calculating lamdaT';

penetration depth=467.9 Å
Note: calculation mistake in textbook in calculating lamdaT


## Example 5.37,Page No:5.53¶

In :
import math

#variable declaration
lamda1      = 16;         #penetration depth in nm
lamda2      = 96;         #penetration depth in nm
T1          = 2.18;       #temperature in K
T2          = 8.1;        # temperature in K

#formula
#lamdaT =lamda0*((1-((T/Tc)**4))**(-1/4))
#calculation
Tc          = ((((lamda2*(T2**4))-(lamda1*(T1**4)))/float(lamda2-lamda1))**(1/float(4)));   #critical temperature in K

#result
print'critical temperature =%3.2f '%Tc,'K';


critical temperature =8.48  K


## Example 5.38,Page No:5.55¶

In :
import math

#variable declaration
Eg      =30.5*1.6*10**-23;       #energy gap in eV
h       =6.6*10**-34;           #planck's constant in (m**2)*kg/s
c       =3.0*10**8;             #velocity of light in m

#formula
#Eg=h*v
#calculation
v       = Eg/float(h);                #velocity in m
lamda   = c/float(v);               #wavelength in m

#result
print'wavelength=%2.2f'%(lamda*10**3),'mm';


wavelength=0.41 mm


## Example 5.39,Page No:5.55¶

In :
import math

#variable declaration
k  =1.38*10**-23;
Tc  =4.2;                  #tempetrature in K
h  =6.6*10**-34;           #planck's constant in (m**2)*kg/s
c  =3*10**8;               # velocity of light in m

#calculation
Eg      = (3*k*Tc);            #energy gap in eV
lamda   = h*c/float(Eg);       #wavelngth in m

#result
print'region of electromagnetic spectrum=%3.2e'%lamda,'m';

region of electromagnetic spectrum=1.14e-03 m