Chapter 3: Polarisation

Example 3.1, Page 3.23

In [3]:
from math import pi, atan

# Given 
mu = 1.5 # refractive index of glass

#Calculations
Ip = atan(mu) * (180 / pi) # by brewster's law
r = 90 - Ip # calculation for angle of refraction

#Result
print "Brewster angle = %.f degree\nAngle of refraction = %.f degree"%(Ip,r)

Brewster angle = 56 degree
Angle of refraction = 34 degree

Example 3.2, Page 3.24

In [2]:
from math import pi, atan

# Given 
mu = 1.33 # refractive index of glass

#Calculations
Ip = atan(mu) * (180 / pi) # by Brewster's law

#Result
print "Angle of brewster = %.2f degree"%Ip

Angle of brewster = 53.06 degree

Example 3.3, Page 3.24

In [3]:
from math import pi, atan

# Given 
mu_w = 1.33 # refractive index of water
mu_g = 1.54 # refractive index of glass

#Calculations
Ip_1 = atan(mu_g / mu_w) * (180 / pi)#calculation for polarizing angle for water
Ip_2 = atan(mu_w / mu_g) * (180 / pi) # calculation for polarizing angle for glass

#Result
print "Polarizing angle for water to glass = %.2f degree,\n Polarizing angle for glass to water = %.2f degree"%(Ip_1,Ip_2)
print "So polarizing angle is greater for a beam incident from water to glass"

Polarizing angle for water to glass = 49.18 degree,
 Polarizing angle for glass to water = 40.82 degree
So polarizing angle is greater for a beam incident from water to glass

Example 3.4, Page 24

In [47]:
from math import pi, asin, tan, sin

# Given 
Ip = pi / 3 # polarizing angle of piece of glass for green light in radian
a = pi / 3 # angle of prism in radian 

#Calculations
mu = tan(Ip) # calculation for refractive index
delta_m = 2 * (asin(mu * sin(a / 2)) - (a / 2)) * (180 / pi) # calculation for angle of minimum deviation

#Result
print "Angle of minimum deviation = %.f degree"%delta_m

Angle of minimum deviation = 60 degree

Example 3.5, Page 3.25

In [4]:
from math import pi, atan

# Given 
mu_w = 1.33 # refractive index of water
mu_g = 1.5 # refractive index of glass

#Calculations
Ip = atan(mu_g / mu_w) * (180 / pi) # calculation for Brewster angle

#Result
print "Brewster angle = %.1f degree"%Ip

Brewster angle = 48.4 degree

Example 3.6, Page 3.25

In [5]:
from math import pi, atan

# Given 
mu = 1.732 # refractive index of glass

#Calculations
Ip = atan(mu) * (180 / pi) # by Brewster's law
r = 90 - Ip# calculation for angle of refraction

#Result
print "Angle of incidence = %.f degree\nAngle of refraction = %.f degree"%(Ip,r)

Angle of incidence = 60 degree
Angle of refraction = 30 degree

Example 3.7, Page 3.25

In [9]:
from math import pi, cos

# Given 
alpha = pi / 3 # angle between polarizer and analyzer

#Calculation
r = (cos(alpha))**2 # where r = transmitted intensity / incident intensity

#Result
print "Ratio between transmitted intensity to incident intensity = %.2f "%r

Ratio between transmitted intensity to incident intensity = 0.25

Example 3.8, Page 3.25

In [15]:
from math import sqrt,acos,degrees

#Given 
r1 = 1./3  #ratio of intensity of transmitted light to the intensity of transmitted beam in first case
r2 = 1./3  #ratio of intensity of transmitted light to the intensity of incident beam in second case
p = 50    #percentage reduction in intensity of unpolarized light by the sheet 

#Calculations
theta1 = degrees(acos(sqrt(r1)))   #calculation for the angle between characteristics directions of the sheet in first case
theta2 = degrees(acos(sqrt(2*r2))) #calculation for the angle between characteristics directions of the sheet in second case

#Result
print "The angle between characteristics directions of the sheet in 1st case = %.2f degrees."%(theta1)
print "The angle between characteristics directions of the sheet in 2nd case = %.2f degrees."%(theta2)

The angle between characteristics directions of the sheet in 1st case = 54.74 degrees.
The angle between characteristics directions of the sheet in 2nd case = 35.26 degrees.

Example 3.9, Page 3.26

In [45]:
from math import acos, sqrt, pi

# Given 
r = 3. / 4 # ratio of intensity of transmitted light to the intensity of incident light

#Calculation
theta = acos(sqrt(r)) * (180 / pi) # calculation for angle between the nicol prisms

#Result
print "Angle between the nicol prisms = %.f degree"%theta

Angle between the nicol prisms = 30 degree

Example 3.10, Page 3.26

In [18]:
from math import pi, cos

# Given 
theta1 = pi / 6 # angle between Nicole prisms in first case in radian
theta2 = pi / 4 # angle between Nicole prisms in second case in radian
theta3 = pi / 3 # angle between Nicole prisms in third case in radian
theta4 = pi / 2 # angle between Nicole prisms in fourth case in radian

#Calculations
I1 = (1 - (cos(theta1))**2) * 100
I2 = (1 - (cos(theta2))**2) * 100
I3 = (1 - (cos(theta3))**2) * 100
I4 = (1 - (cos(theta4))**2) * 100

#Result
print "Percentage reduction in intensity of light-\n(i)%.f %%\n(ii)%.f %%\n(iii)%.f %%\n(iv)%.f %%"%(I1,I2,I3,I4)

Percentage reduction in intensity of light-
(i)25 %
(ii)50 %
(iii)75 %
(iv)100 %

Example 3.11, Page 3.27

In [44]:
from math import pi, acos, sqrt 

# Given 
i1 = 1. / 2 # reduced intensity ratio in first case
i2 = 1. / 4 # reduced intensity ratio in second case

#Calculations
theta1 = acos(sqrt(i1)) * (180 / pi)# calculation for angle between nicols in first case 
theta2 = acos(sqrt(i2)) * (180 / pi)# calculation for angle between nicols in second case

#Result
print "Angle between the Nicols in first case = %.f degree\nAnd in second case = %.f degree"%(theta1,theta2)

Angle between the Nicols in first case = 45 degree
And in second case = 60 degree

Example 3.12, Page 3.27

In [20]:
# Given 
l = 5e-7 # wavelength of light in meter
mu_e = 1.553 # refractive index for extraordinary light
mu_o = 1.544 # refractive index for ordinary light

#Calculations
t = l / (2 * (mu_e - mu_o)) # calculation for thickness of half-wave plate of quartz

#Result
print "Thickness of half-wave plate of quartz = %.2e meter"%t

Thickness of half-wave plate of quartz = 2.78e-05 meter

Example 3.13, Page 3.27

In [21]:
# Given 
l = 5.893e-7 # wavelength of light in meter
mu_e = 1.533 # refractive index for extraordinary light
mu_o = 1.554 # refractive index for ordinary light

#Calculation
t = l / (4 * (mu_o - mu_e)) # calculation for thickness of quartz plate

#Result
print "Thickness of quartz plate  = %.2e meter"%t

Thickness of quartz plate  = 7.02e-06 meter

Example 3.14, Page 3.28

In [22]:
# Given 
l = 5.89e-7 # wavelength of light in meter
mu_e1 = 1.5 # refractive index for extraordinary light in first case
mu_o1 = 1.55 # refractive index for ordinary light in first case
mu_e2 = 1.57 # refractive index for extraordinary light in second case
mu_o2 = 1.55 # refractive index for ordinary light in second case

#Calculations
t1 = l / (4 * (mu_o1 - mu_e1))
t2 = l / (4 * (mu_e2 - mu_o2))
 # calculation for thickness of plate of quartz

#Result
print "Thickness of plate of quartz in first case = %.3e meter,\nAnd thickness of plate of quartz in second case = %.2e meter"%(t1,t2)

Thickness of plate of quartz in first case = 2.945e-06 meter,
And thickness of plate of quartz in second case = 7.36e-06 meter

Example 3.15, Page 3.28

In [23]:
# Given 
l = 5.89e-7 # wavelength of light in meter
mu_e = 1.486 # refractive index for extraordinary light
mu_o = 1.658 # refractive index for ordinary light

#Calculation
t = l / (4 * (mu_o - mu_e)) # calculation for thickness of calcite plate 

#Result
print "Thickness of calcite plate  = %.2e meter"%t

Thickness of calcite plate  = 8.56e-07 meter

Example 3.16, Page 3.28

In [24]:
# Given 
l = 5e-7 # wavelength of light in meter
mu_e = 1.5533 # refractive index for extraordinary light
mu_o = 1.5442 # refractive index for ordinary light

#Calculation
t = l / (4 * (mu_e - mu_o)) # calculation for thickness of quartz plate

#Result
print "Thickness of quartz plate  = %.2e meter"%t

Thickness of quartz plate  = 1.37e-05 meter

Example 3.17, Page 3.28

In [25]:
# Given 
l = 5.89e-7 # wavelength of light in meter
mu_e = 1.54 # refractive index for extraordinary light
mu_o = 1.55 # refractive index for ordinary light

#Calculation
t = l / (4 * (mu_o - mu_e)) # calculation for thickness of quartz plate

#Result
print "Thickness of quartz plate  = %.2e meter"%t

Thickness of quartz plate  = 1.47e-05 meter

Example 3.18, Page 3.28

In [26]:
# Given 
l = 5.89e-7 # wavelength of light in meter
mu_e = 1.553 # refractive index for extraordinary light
mu_o = 1.544 # refractive index for ordinary light

#Calculation
t = l / (4 * (mu_e - mu_o)) # calculation for thickness of quartz plate

#Result
print "Thickness of quartz plate  = %.2e meter"%t

Thickness of quartz plate  = 1.64e-05 meter

Example 3.19, Page 3.29

In [27]:
# Given 
mu_e = 1.5442 # refractive index for extraordinary light
mu_o = 1.5533 # refractive index for ordinary light
l = 5e-7 # wavelength of plane polarized light in meter

#Calculation
t = l / (2 * (mu_o - mu_e))# calculation for thickness of quartz plate

#Result
print "Thickness of quartz plate  = %.2e meter"%t

Thickness of quartz plate  = 2.75e-05 meter

Example 3.20, Page 3.29

In [28]:
# Given 
theta = 10 # rotation of plane of polarization in degree
s = 60 # specific rotation of sugar solution in degree per decimeter per unit concentration
l = 2.5 # length of Polari meter in decimeter

#Calculation
c = theta / (s * l) # calculation for concentration of sugar solution

#Result
print "Concentration of sugar solution = %.3f gm/cc"%c

Concentration of sugar solution = 0.067 gm/cc

Example 3.21, Page 3.29

In [43]:
# Given 
theta = 26.4 # rotation of plane of polarization in degree
c = 0.2 # concentration of sugar solution in gm/cc
l = 2 # length of polarizing tube in decimeter

#Calculation
s = theta / (l * c)# calculation for specific rotation of sugar solution

#Result
print "Specific rotation of sugar solution = %.f degree/(dm-cc)"%s

Specific rotation of sugar solution = 66 degree/(dm-cc)

Example 3.22, Page 3.29

In [42]:
# Given 
theta = 6.5 # rotation of plane of polarization in degree
c = 0.05 # concentration of sugar solution in gm/cc
l = 2 # length of polarizing tube in decimeter

#Calculation
s = theta / (l * c) # calculation for specific rotation of sugar solution

#Result
print "Specific rotation of sugar solution = %.f degree/(dm-cc)"%s

Specific rotation of sugar solution = 65 degree/(dm-cc)

Example 3.23, Page 3.30

In [29]:
# Given 
w = 80 # weight of impure sugar in gm
theta = 9.9 # rotation of plane of polarization in degree
s = 66 # specific rotation of sugar solution in degree per decimeter per unit concentration
l = 2 # length of Polari meter in decimeter

#Calculations
c = theta / (s * l) * (1000) # in gm/l
per_c = (c * 100) / w # calculation for concentration of sugar solution

#Result
print "Concentration of sugar solution = %.2f percent"%per_c

Concentration of sugar solution = 93.75 percent

Example 3.24, Page 3.30

In [30]:
# Given 
theta = 11. # rotation of plane of polarization in degree
s = 66 # specific rotation of sugar solution in degree per decimeter per unit concentration
l = 2 # length of Polari meter in decimeter

#Calculation
c = theta / (s * l) # calculation for concentration of sugar solution

#Result
print "Concentration of sugar solution = %.4f gm/cc"%c

Concentration of sugar solution = 0.0833 gm/cc

Example 3.25, Page 3.30

In [40]:
# Given 
theta = 26.4 # rotation of plane of polarization in degree
c = 0.2 # concentration of sugar solution in gm/cc
l = 2 # length of polarizing tube in decimeter

#calculation
s = theta / (l * c) # calculation for specific rotation of sugar solution

#Result
print "Specific rotation of sugar solution = %.f degree/(dm-cc)"%s

Specific rotation of sugar solution = 66 degree/(dm-cc)

Example 3.26, Page 3.30

In [39]:
# Given 
theta = 13 # rotation of plane of polarization in degree
r = (1. / 3) # ratio of the final concentration to the initial solution
l = 2 # length of Polari meter in decimeter
l_ = 3 # length of second polarizing tube in decimeter 

#Calculation
theta_ = (l_ * r * theta) / l# calculation for optical rotation of diluted solution

#Result
print "Optical rotation of diluted solution = %.1f degree"%theta_

Optical rotation of diluted solution = 6.5 degree