7: Conducting Materials

Example number 7.1, Page number 178

In [4]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=5.9*10**28;   #electron concentration of conductor(m^-3)
v=0.625;    #drift velocity of a conductor(ms^-1)
x=6.22*10**7;   #electrical conductivity(ohm^-1 m^-1)
e=1.6*10**-19;   #charge of electron(c)

#Calculation
J=n*e*v;      #current density in the conductor corresponds to drift velocity(Am^-1)
z=x/(n*e);    #mobility of the charge(m^2V^-1s^-1)
 
#Result
print "The current density in the conductor corresponds to a drift velocity is",J/10**9,"*10**9 A m^-1"
print "Mobility of the charge carrires is",round(z*10**3,5),"*10**-3 m^2 V^-1 s^-1"

The current density in the conductor corresponds to a drift velocity is 5.9 *10**9 A m^-1
Mobility of the charge carrires is 6.58898 *10**-3 m^2 V^-1 s^-1

Example number 7.2, Page number 179

In [6]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=8.5*10**28;   #density of free electrons in copper(m^-3)
A=1.05*10**-6;   #sectional area of copper(m^2)
e=1.6*10**-19;   #charge of electron(c)
I=1;  #copper wire carries a current(A)

#Calculation
V=1/(A*n*e);    #drift velocity of free electrons in copper wire(ms^-1)

#Result
print "The drift velocity of free electron in a copper wire is",round(V*10**5,4),"*10**-5 ms^-1"

The drift velocity of free electron in a copper wire is 7.0028 *10**-5 ms^-1

Example number 7.3, Page number 179

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

#Variable declaration
X=3.5*10**-3;  #mobility of free electrons in copper(m^2 V^-1 s^-1)
E=0.5;    #elactric field strength of copper(V m^-1)

#Calculation
V=X*E;   #drift velocity of free electrons in copper(m s^-1)

#Result
print "The drift velocity of free electrons in copper is",V*10**3,"*10**-3 ms^-1"

The drift velocity of free electrons in copper is 1.75 *10**-3 ms^-1

Example number 7.4, Page number 179

In [5]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=6.5*10**28;    #conduction electron(m^-3)
r=1.435*10**-8;  #metal resistivity(ohm-metre)
e=1.6*10**-19;   #charge of electron(c)
m=9.11*10**-31;   #mass of a electron(kg)

#Calculation
T=m/(r*n*e**2);    #relaxation time of conduction electrons(s)

#Result
print "The relaxation time of conduction electrons is",round(T*10**14,3),"*10**-14 s"

The relaxation time of conduction electrons is 3.815 *10**-14 s

Example number 7.5, Page number 180

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

#Variable declaration
r=1.72*10**-8;   #resistivity of copper(ohm metre)
T=293;     #temperature of copper(K)
n=8.48*10**28;   #density of free electron(m^-3)
e=1.6*10**-19;   #charge of electron(c)
m=9.11*10**-31;   #mass of a electron(kg)
k=1.38*10**-23;   #boltzmann constant(m^2 Kg s^-2 k^-1)

#Calculation
t=m/(r*n*(e**2));   #relaxation time(s)
v=math.sqrt(3*k*T/m);  #thermal velocity(ms^-1)
Lamda=t*v;     #mean free path between collision of free electrons in copper(m)

#Result
print "The mean free path between collision of free electrons in copper is",round(Lamda*10**9,4),"*10**-9 m"
print "answer varies due to rounding off errors"

The mean free path between collision of free electrons in copper is 2.8153 *10**-9 m
answer varies due to rounding off errors

Example number 7.6, Page number 180

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

#Variable declaration
t=1*10**-3;   #thickness of metal(m)
V=1;     #potential difference applied in volts(V)
T=300;   #temperature(K)
m=0.04;  #mobility(m^2 V^-1 s^-1)
k=1.38*10**-23;   #boltzmann constant(m^2 Kg s^-2 k^-1)
m1=9.11*10**-31;   #mass of a electron(kg)

#Calculation
v=math.sqrt(3*k*T/m1);  #thermal velocity(ms^-1)
E=V/t;    #unit potenyial voltage gradient(V m^-1)
vd=E*m;   #drift velocity of electrons(m s^-1)

#Result
print "The thermal velocity is",round(v/10**3,2),"*10**3 m s^-1"
print "Drift velocity of electrons is",vd,"m s^-1"
print "Thus the terminal velocity is high compared to the drift velocity"

The thermal velocity is 116.76 *10**3 m s^-1
Drift velocity of electrons is 40.0 m s^-1
Thus the terminal velocity is high compared to the drift velocity

Example number 7.7, Page number 181

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

#Variable declaration
AW=63.5;   #atomic weight of copper
D=8.93*10**3;   #density of copper(kg m^-3)
t=2.48*10**-14;   #relaxation time of copper(s)
AV=6.023*10**26;   #avagadro no(mole^-1)
e=1.6*10**-19;     #charge of electron(c)
m=9.11*10**-31;    #mass of a electron(kg)

#Calculation
n=AV*D/AW;    #density of electrons per unit volume(m^-3)
sigma=n*e**2*t/m;   #electrical conductivity of copper(Sm^-1)

#Result
print "The electrical conductivity of copper is",round(sigma/10**7,1),"*10**7 S m^-1"
print "answer varies due to rounding off errors"

The electrical conductivity of copper is 5.9 *10**7 S m^-1
answer varies due to rounding off errors

Example number 7.8, Page number 181

In [19]:
#importing modules
import math
from __future__ import division

#Variable declaration
I=10;    #current(A)
r=0.8*10**-2;   #radius of wire(m)
n=8.48*10**28;  #density of free electron(m^-3)
e=1.6*10**-19;  #charge of electron(c)

#Calculation
J=I/(math.pi*r**2);  #current density of copper(Am^-2)
v=J/(n*e);     #drift velocity of copper(ms^-1)

#Result
print "The drift velocity in copper is",round(v*10**6,4),"*10**-6 ms^-1"
print "The current density in copper is",round(J/10**4,4),"*10**4 Am^-2"
print "answer varies due to rounding off errors"

The drift velocity in copper is 3.6657 *10**-6 ms^-1
The current density in copper is 4.9736 *10**4 Am^-2
answer varies due to rounding off errors

Example number 7.9, Page number 182

In [22]:
#importing modules
import math
from __future__ import division

#Variable declaration
r=1.54*10**-8;   #resistivity of silver wire at room temperature(ohm metre)
E=100;    #Electric field along the wire(V/m)
n=5.8*10**28;   #n is assuming of conduction electrons(m^-3)
e=1.6*10**-19;   #charge of electron(c)

#Calculation
mew=1/(r*n*e);    #mobility of charge(m^2 V^-1 s^-1)
vd=mew*E;    #drift velocity of electrons(m s^-1)

#Result
print "The mobility of charge is",round(mew*10**3,3),"*10**-3 m^2 V^-1 s^-1"
print "The drift velocity of electrons is",round(vd,4),"m s^-1"

The mobility of charge is 6.997 *10**-3 m^2 V^-1 s^-1
The drift velocity of electrons is 0.6997 m s^-1

Example number 7.10, Page number 182

In [5]:
#importing modules
import math
from __future__ import division

#Variable declaration
D=8.92*10**3;    #density of copper(kg m^-3)
AW=63.5;     #atomic weight of copper
r=1.73*10**-8;    #resistivity of copper(ohm metre)
AV=6.023*10**26;   #avagadro no(mole^-1)
e=1.6*10**-19;     #charge of electron(c)
m=9.11*10**-31;    #mass of a electron(kg)

#Calculation
n=AV*D/AW;    #density of electrons per unit volume(m^-3)
tow=m/(r*n*e**2);     #average time collision of electrons in copper(s)
mew=1/(r*n*e);      #mobility of charge(m^2 V^-1 s^-1)

#Result
print "The relaxation time collision of electrons in copper obeying classical laws is",round(tow*10**14,2),"*10**-14 s"
print "The mobility charge of copper obeying classical laws is",round(mew*10**2,3),"*10**-2 m^2 V^-1 s^-1"

The relaxation time collision of electrons in copper obeying classical laws is 2.43 *10**-14 s
The mobility charge of copper obeying classical laws is 0.427 *10**-2 m^2 V^-1 s^-1

Example number 7.11, Page number 183

In [6]:
#importing modules
import math
from __future__ import division

#Variable declaration
r=1.85*10**-10;    #the radius of sodium atom(m)
t=3*10**-14;     #the classic value of mean free time(sec)
temp=0;      #temperature(centigrade)
na=2;        #number of atoms in a unit cell
ne=1;        #number of electrons per unit cell
e=1.6*10**-19;    #charge of electron(c)
m=9.11*10**-31;   #mass of a electron(kg)

#Calculation 
a=4*r/math.sqrt(3);    #a is one side in bcc structure unit cell(m)
v=a**3;   #volume of bcc structure unit cell(m^3)
n=na*ne/v;    #density of electrons per unit volume(m^-3)
rho=m/(n*e**2*t);    #The electrical resistivity(ohm metre)

#Result
print "The electrical resistivity is",round(rho*10**8,2),"*10**-8 ohm metre"
print "answer varies due to rounding off errors"

The electrical resistivity is 4.63 *10**-8 ohm metre
answer varies due to rounding off errors

Example number 7.12, Page number 184

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

#Variable declaration
rho=2.7*10**-8;    #electrical resistivity of aluminium(ohm metre)
AW=26.98;     #atomic weight of aluminium
d=2.7*10**3;  #density of volume(Kg/m^3)
R=60*10**-3;   #resistance(W)
l=5;    #length of aluminium wire(m)
i=15;     #aluminuim wire carries a current(A)
fe=3;     #number of free electrons 
AV=6.023*10**26;    #avagadro no(mole^-1)
e=1.6*10**-19;      #charge of electron(c)

#Calculation
n=AV*d*fe/AW;     #density of electrons per unit volume(electrons/m^-3)
mew=1/(n*e*rho);   #mobility of the charge(m^2 V^-1 S^-1)
E=i*R/l;     #free electron concentration(V/m)
vd=mew*E;      #drift velocity(m s^-1)

#Result
print "Free electron concentration in aluminium is",E,"V/m"
print "Mobility of the charge is",round(mew*10**3,2),"*10**-3 m^2 V^-1 S^-1"
print "The drift velocity of electrons is",round(vd*10**4,3),"*10**-4 m s^-1"
print "answer varies due to rounding off errors"

Free electron concentration in aluminium is 0.18 V/m
Mobility of the charge is 1.28 *10**-3 m^2 V^-1 S^-1
The drift velocity of electrons is 2.304 *10**-4 m s^-1
answer varies due to rounding off errors

Example number 7.13, Page number 184

In [21]:
#importing modules
import math
from __future__ import division

#Variable declaration
l=1*10**-2;    #length of intrinsic Ge rod(m)
b=1*10**-3;    #breadth of intrinsic Ge rod(m)
t=1*10**-3;    #thickness of intrinsic Ge rod(m)
temp=300;    #temperature(K)
d=2.5*10**19;   #intrinsic carrier density(Kg/m^3)
z=0.39;     #mobility of electron(m^2 V^-1 S^-1)
zh=0.19;   #mobility of hole(m^2 V^-1 S^-1) 
e=1.6*10**-19;   #charge of electron(c)

#Calculation
x=d*e*(z+zh);   #electrical conductivity(ohm^-1 m^-1)
r=1/x;     #electrical resistivity(ohm metre)
A=b*t;     #area(m^2)
R=r*l/A;   #resistance of an intrinsic Ge rod(ohm)

#Result
print "The resistance of an intrinsic Ge rod is",int(R),"ohm"

The resistance of an intrinsic Ge rod is 4310 ohm

Example number 7.14, Page number 188

In [19]:
#importing modules
import math
from __future__ import division

#Variable declaration
d=8.48*10**28;    #free electron density of copper(m^-3)
y=2.8138*10**-9;  #mean free path(m)
v=1.1536*10**5;   #velocity of copper(m s^-1)
t=20;    #temperature of copper(C)
Kb=1.38*10**-23;    #Boltzmann's constant(m^2 Kg s^-2 k^-1)

#Calculation
K=1/2*(d*v*y*Kb);    #thermal conductivity of copper(W m^-1 K^-1)

#Result
print "The thermal conductivity of copper is",round(K,4),"W m^-1 K^-1"

The thermal conductivity of copper is 189.9299 W m^-1 K^-1

Example number 7.15, Page number 189

In [22]:
#importing modules
import math
from __future__ import division

#Variable declaration
er=50*10**-8;    #electrical resistivity(ohm metre)
t=300;    #temperature(K)
r=13*10**-3;   #radius of brass(m)
th=35*10**-3;   #thickness of brass(m)
L=2.44*10**-8;   #Lorentz number(W ohm K^-2)

#Calculation
K=L*t/er;    #thermal conductivity of brass(W m^-1 K^-1)
A=math.pi*r**2;   #area of brass disk(m^2)
Rt=th/(K*A);      #thermal resistance of brass(K W^-1)

#Result
print "The thermal conductivity of brass is",K,"W m^-1 K^-1"
print "The thermal resistance of brass is",round(Rt,3),"K W^-1"
print "answer varies due to rounding off errors"

The thermal conductivity of brass is 14.64 W m^-1 K^-1
The thermal resistance of brass is 4.503 K W^-1
answer varies due to rounding off errors

Example number 7.16, Page number 189

In [24]:
#importing modules
import math
from __future__ import division

#Variable declaration
x=5.87*10**7;    #electrical conductivity(ohm^-1 m^-1)
k=380;     #thermal conductivity of copper(W m-1 K^-1)
t=293;     #temperature of copper(K)

#Calculation
L=k/(x*t);     #Lorentz number(W ohm K^-2)

#Result
print "Lorentz number is",round(L*10**8,4),"*10**-8 W ohm K^-2"

Lorentz number is 2.2094 *10**-8 W ohm K^-2

Example number 7.17, Page number 189

In [25]:
#importing modules
import math
from __future__ import division

#Variable declaration
x=6.40*10**7;   #electrical conductivity(mho m^-1)
t=300;    #temperature of copper(K)
L=2.44*10**-8;   #Lorentz number(W ohm K^-2)

#Calculation
K=x*t*L;      #thermal conductivity of copper(W m^-1 K^-1)

#Result
print "The thermal conductivity of copper is",K,"W m^-1 K^-1"

The thermal conductivity of copper is 468.48 W m^-1 K^-1