Chapter 2 : Electrical and thermal conduction in solids

Example 2.2 Page No : 43

In [1]:
# Given
sigma = 5.9*10**5   # in ohm**-1*cm**-1
e = 1.6*10**-19     # Coulombs
d = 8.93            # g/cm**3
Mat = 63.5          # g/mol
NA = 6.02*10**23    # mol**-1

# Calculations and Results
n = d*NA/Mat
u_d = sigma/(e*n)   # electron drift mobility
print("Drift mobility of electrons in copper at room temperature in cm2/V/s is {0:.4f}".format(u_d))
Drift mobility of electrons in copper at room temperature in cm2/V/s is 43.5570

Example 2.3 Page No : 49

In [2]:
# Given
u_d = 3.2*10**-3    # in m**2/V/s
u = 1.2*10**6       # m/s
v_dx = 0.1*u

# Calculations and Results
# drift velocity of conduction electrons is v_dx=u_d*E
E = v_dx/u_d
print("Applied electric field in V/m is {0:.4g}".format(E))
Applied electric field in V/m is 3.75e+07

Example 2.4 Page No : 50

In [3]:
# Given
T_summer = 20.0     # in degree celsius
T_summer += 273.0   # in kelvin
T_winter = -30.0    # in degree celsius
T_winter += 273.0   # in kelvin

# Calculations and Results
# we have R is proportional to A*T
# Hence
R = (T_summer-T_winter)/T_summer
R *= 100
print(" Percentage change in the resistance of a pure metal wire from Saskatchewans summer "
      "too winter in % is {0:.4f}".format(R))
 Percentage change in the resistance of a pure metal wire from Saskatchewans summer too winter in % is 17.0648

Example 2.5 Page No : 52

In [4]:
import math 

# Given
d = 8.96*10**3      # in Kg/m3
NA = 6.02*10**23    # mol**-1
Mat = 63.56*10**-3  # Kg/mol
k = 1.38*10**-23    # J/K
T = 300.0             # kelvin
e = 1.6*10**-19     # in coulombs
m_e = 9.1*10**-31   # in Kg
u = 1.25*10**6      # m/s
f = 4*10**12        # frequency in s**-1

# Calculations and Results
Ns = d*NA/Mat       # atomic concentration in m**-3
M = Mat/NA
w = 2*math.pi*f     # angular frequency of the vibration
# by virtue of Equipartition of energy  theorem
a = math.sqrt((2*k*T)/(M*w**2))
S = math.pi*a**2    # cross sectional area
t = 1.0/(S*u*Ns)      # mean free time
u_d = e*t/m_e       # drift velocity
u_d *= 10**4        # change in units
Ns /= 10**6         # in cm**-3
sigma = e*Ns*u_d    # conductivity
print("drift velocity of electrons in m2/V/s is {0:.4f}".format(u_d))
print("conductivity of copper in ohm**-1/cm is {0:.4g}".format(sigma))
# slight change in the answer is due to the computation method, otherwise answer is matching with textbook
drift velocity of electrons in m2/V/s is 42.4947
conductivity of copper in ohm**-1/cm is 5.77e+05

Example 2.7 Page No : 53

In [5]:
# Given
n = 1.2
To = 293.0              # in kelvin

# Calculations and Results
alpha_o = n/To
print("Theoretical value of TCR at 293K is {0:.4f} which is in "
      "well agreement with experimental value".format(alpha_o))
alpha_o = 0.00393       # experimental value
n = alpha_o*To
print("Theoretical value of n at 293K (n = {0:.4f}) is in well agreement with experimental value".format(n))
Theoretical value of TCR at 293K is 0.0041 which is in well agreement with experimental value
Theoretical value of n at 293K (n = 1.1515) is in well agreement with experimental value

Example 2.9 Page No : 58

In [6]:
import math

# Given
P = 40.0                # in Watt
V = 120.0               # in Volts
D = 33*10**-6         # in meter
L = 0.381             # in meter
To = 293.0              # in kelvin
P_radiated = 40.0       # in watt
epsilon = 0.35
sigma_s = 5.6*10**-8  # in W/m2/K4

# Calculations and Results
I = P/V
A = math.pi*D**2/4.0
R = V/I               # resistance of the filament
p_t = R*A/L           # resistivity of tungsten
p_o = 5.51*10**-8     # resistivity at room temperature in ohm*m
# p_t=p_o*(T/To)**1.2
T = To*(p_t/p_o)**(1.0/1.2)
print("Temperature of the bulb when it is operated at the rated voltage in Kelvin is {0:.4f}".format(T))
A = L*math.pi*D
# Stefans Law
T = (P_radiated/(epsilon*sigma_s*A))**(1.0/4.0)
print("Temperature of the filament in kelvin is {0:.4f}".format(T))
Temperature of the bulb when it is operated at the rated voltage in Kelvin is 2746.7599
Temperature of the filament in kelvin is 2681.0430

Example 2.10 Page No : 63

In [7]:
# Given
M_Au = 197.0
w = 0.1
M_Cu = 63.55
p_exp = 108.0         # n*ohm*m

# Calculations and Results
X = M_Au*w/((1-w)*M_Cu+(w*M_Au))
C = 450.0             # n*ohm*m
p_Au = 22.8         # resistivity in n*ohm*m
p = p_Au+C*X*(1-X)  # Nordheim rule
x = ((p-p_exp)/p)*100
print("resistivity of the alloy in n*ohm*m is {0:.4f}".format(p))
print("The difference in the value from experimental value in % is {0:.4f}".format(x))
resistivity of the alloy in n*ohm*m is 108.5513
The difference in the value from experimental value in % is 0.5079

Example 2.11 Page No : 68

In [8]:
# Given
u = 1.58*10**6    # in m/s
N = 8.5*10**28    # m**-3
e = 1.6*10**-19   # in coulombs
me = 9.1*10**-31  # in Kg
N_I = 0.01*N

# Calculations and Results
l_I = N_I**(-1/3)
t_I = l_I/u
p = me/(e**2*N*t_I)
print(" worst case resistivity in ohm*m {0:.4g}".format(p))
# slight change in answer due to computational method
 worst case resistivity in ohm*m 5.616e+11

Example 2.13 Page No : 74

In [9]:
# Given
Xd = 0.15
p_c = 1*10**-7          # ohm*m

# Calculations and Results
p_eff = p_c*((1+0.5*Xd)/(1-Xd))
print("Effective resistivity in ohm m is {0:.4g}".format(p_eff))
# slight change in the answer due to printing the answer
Effective resistivity in ohm m is 1.265e-07

Example 2.14 Page No : 75

In [10]:
# Given
Xd = 0.15
p_c = 4*10**-8      # ohm*m

# Calculations and Results
p_eff = p_c*((1+0.5*Xd)/(1-Xd))
print("Effective resistivity in ohm m is {0:.4g}".format(p_eff))
# change in the answer due to coding
Effective resistivity in ohm m is 5.059e-08

Example 2.16 Page No : 76

In [11]:
import math 

# Given
# at f=10MHz
a = 10**-3              # in m
f = 10*10**6            # in Hz
w = 2*math.pi*f
sigma_dc = 5.9*10**7    # in m**-1
u = 1.257*10**-6        # in Wb/A/m

# Calculations and Results
delta = 1.0/math.sqrt(0.5*w*sigma_dc*u)
# let r=r_ac/r_dc=a/(2*delta)
r = a/(2*delta)
print("Change in dc resistance of a copper wire at 10MHz  is {0:.4f}".format(r))
# part(b)
f = 1*10**9  # in Hz
w = 2*math.pi*f
delta = 1.0/math.sqrt(0.5*w*sigma_dc*u)
# let r=r_ac/r_dc=a/(2*delta)
r = a/(2*delta)
print("Change in dc resistance of a copper wire at 1GHz  is {0:.4f}".format(r))
Change in dc resistance of a copper wire at 10MHz  is 24.1345
Change in dc resistance of a copper wire at 1GHz  is 241.3452

Example 2.18 Page No : 79

In [12]:
# Given
sigma = 5.9*10**7       # ohm**-1*m**-2
RH = -0.55*10**-10      # m**3/A/s

# Calculations and Results
u_d = -RH*sigma
print("drift mobility of electrons in copper in m2/V/s is {0:.4g}".format(u_d))
drift mobility of electrons in copper in m2/V/s is 0.003245

Example 2.19 Page No : 80

In [13]:
# Given
no = 8.5*10**28     # in m3
e = 1.6*10**-19     # in coulombs
u_d = 3.2*10**-3    # m2/V/s
sigma = 5.9*10**7   # in ohm**-1*m**-1

# Calculations and Results
n = sigma/(e*u_d)
print("concentration of conduction electrons in copper in m**-3 is {0:.4g}".format(n))
A = n/no
print("Average number of electrons contributed per atom is {0:.4f}".format(A))
concentration of conduction electrons in copper in m**-3 is 1.152e+29
Average number of electrons contributed per atom is 1.3557

Example 2.20 Page No : 85

In [14]:
# Given
sigma = 1*10**7         # ohm**-1*m**-1
T = 300.0               # kelvin
C_WFL = 2.44*10**-8     # W*ohm/K2
X_d = 0.15

# Calculations and Results
K_c = sigma*T*C_WFL
K_eff = K_c*((1-X_d)/(1+0.5*X_d))
print("Thermal Conductivity at room temperature in W/m/K is {0:.4f}".format(K_eff))
Thermal Conductivity at room temperature in W/m/K is 57.8791

Example 2.21 Page No : 89

In [15]:
import math 
# Chapter2
# Ex_21
# Given
sigma = 50*10**-9       # in ohm
T = 300.0               # kelvin
C_WFL = 2.45*10**-8     # in W*ohm/K2
L = 30*10**-3           # in m
d = 20*10**-3           # in m
Q = 10.0                # in W

# Calculations and Results
# Wiedemann-Franz Lorenz Law
k = sigma**-1*T*C_WFL  # thermal conductivity
A = math.pi*(d**2)/4.0
theta = L/(k*A)  # thermal resistance
delta_T = theta*Q
print("Temperature drop across the disk in degree celsius is {0:.4f}".format(delta_T))
Temperature drop across the disk in degree celsius is 6.4961