{
"metadata": {
"name": "",
"signature": "sha256:08d693988f55ded7946a3910c4b750b4a1419d79f72e5b56719c0d924a66fd5a"
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"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h1>Chapter 5: Electric Fields in Material Space<h1>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 5.1, Page number: 167<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"import scipy.integrate\n",
"\n",
"#Variable Declaration\n",
"\n",
"r1= 0.2 # radius of hemispherical shell in metres\n",
"r2= 0.1 # radius of spherical shell in metres\n",
"\n",
"#Calculations\n",
"\n",
"#Calculation of current through hemispherical shell \n",
"\n",
"def J1(phi,theta):\n",
"\ts1=(1/r1)*(2* scipy.cos(theta)* scipy.sin(theta))\n",
"\treturn s1\n",
"\n",
"if __name__ == '__main__':\n",
"\n",
" I1, error = scipy.integrate.dblquad(lambda theta , phi: J1(phi,theta), \n",
" 0, 2*scipy.pi, lambda theta: 0, lambda theta: scipy.pi/2) \n",
"\t \n",
"#Calculation of current through spherical shell \n",
"\n",
"def J2(phi,theta):\n",
"\ts2=(1/r2)*(2* scipy.cos(theta)* scipy.sin(theta))\n",
"\treturn s2\n",
"\n",
"if __name__ == '__main__':\n",
"\n",
" I2, error = scipy.integrate.dblquad(lambda theta , phi: J1(phi,theta), \n",
" 0, 2*scipy.pi, lambda theta: 0, lambda theta: scipy.pi) \n",
"\t \n",
"#Results\n",
"\n",
"print 'Current through hemispherical shell=',round(I1,1),'A' \n",
"print 'Current through spherical shell=',round(I2,0),'A'\n",
"\t"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Current through hemispherical shell= 31.4 A\n",
"Current through spherical shell= 0.0 A\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 5.2, Page number: 168<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"#Variable Declaration\n",
"\n",
"ps=10**-7 #Surface charge density of the belt in Couloumb/metre^2\n",
"u=2 #Speed of the belt in metres/sec\n",
"w=0.1 #Width of the belt in metres\n",
"t=5 #Time taken in seconds \n",
"\n",
"#Calculations\n",
"\n",
"I=ps*u*w #Current in amperes\n",
"Q=I*t*10**9 #Charge collected in 5 seconds in nano Coloumbs\n",
"\n",
"#Result\n",
"\n",
"print \"The charge collected in 5 seconds is \",Q,\"nC\" "
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The charge collected in 5 seconds is 100.0 nC\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 5.3, Page number: 169<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"#Variable Declaration\n",
"\n",
"n=10**29 #Number density of electrons in m^-3\n",
"e=-1.6*10**-19 #Electronic charge in Coloumbs\n",
"sigma=5*10**7 #Current density in S/m\n",
"E=10**-2 #Electric Field in V/m\n",
"S=(3.14*10**-6)/4 #Cross sectional area of the wire in m^2\n",
"\n",
"#Calculations\n",
"\n",
"pv=n*e #Charge density of free electrons in C/m^3\n",
"J=sigma*E*10**-3 #Current density in kA/m^2\n",
"I=J*S*10**3 #Current in amperes\n",
"u=J*10**3/pv #Drift velocity in m/s\n",
"\n",
"#Results\n",
"\n",
"print \"The charge density is \",pv,\"C/m^3\" \n",
"print \"The current density is \",J,\"kA/m^2\" \n",
"print \"The current is \",round(I,3), \"A\"\n",
"print \"The drift velocity is \",-u,\"m/s\" "
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The charge density is -16000000000.0 C/m^3\n",
"The current density is 500.0 kA/m^2\n",
"The current is 0.393 A\n",
"The drift velocity is 3.125e-05 m/s\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 5.4, Page number: 170<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"\n",
"#Variable Declaration\n",
"\n",
"l=4 #Length of the lead bar in m\n",
"d=3 #Width of the lead bar in cm\n",
"r=0.5 #Radius of the hole drilled in cm\n",
"sigma=5*10**6 #Conductivity of the bar in S/m\n",
"\n",
"#Calculation\n",
"\n",
"S=(d**2-(scipy.pi*r**2)) #Cross sectional area in cm^2\n",
"R=l/(S*sigma*10**-4) #Resistance in ohms\n",
"\n",
"#Result\n",
"\n",
"print 'The resistance between the square ends is',round(R*10**6),'micro ohms'\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The resistance between the square ends is 974.0 micro ohms\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 5.6, Page number: 177<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"import scipy\n",
"\n",
"#Variable Declaration\n",
"\n",
"e0=10**-9/(36*scipy.pi) #permittivity of free space in Farad/m\n",
"er=2.55 #relative permittivity (dimensionless)\n",
"E=10*10**3 #Electric field in V/m\n",
"chi=er-1.0 #Electric susceptibility (dimensionless)\n",
"d=1.5 #Distance between plates in mm\n",
"\n",
"#Calculations\n",
"\n",
"D=e0*er*E*10**9 #D in nC/m^2\n",
"\n",
"P=chi*e0*E*10**9 #P in nC/m^2\n",
"\n",
"ps=D #The surface charge density of \n",
" #free charge in nC/m^2\n",
" \n",
"pps =P #The surface charge density of\n",
" #polarization charge in nC/m^2\n",
" \n",
"V=E*d*10**-3 #The potential difference between \n",
" #the plates in volts\n",
"\n",
"#Results\n",
"\n",
"print 'D =',round(D,2),'nC/m^2'\n",
"print 'P =',round(P,0),'nC/m^2'\n",
"print 'Surface charge density of free charge =',round(ps,2),'nC/m^2'\n",
"print 'Surface charge density of polarization charge =',round(pps,0),'nC/m^2'\n",
"print 'The potential difference between the plates =',V,'V'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"D = 225.47 nC/m^2\n",
"P = 137.0 nC/m^2\n",
"Surface charge density of free charge = 225.47 nC/m^2\n",
"Surface charge density of polarization charge = 137.0 nC/m^2\n",
"The potential difference between the plates = 15.0 V\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 5.7, Page number: 178<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"\n",
"import scipy\n",
"\n",
"#Variable Declaration\n",
"\n",
"e0=10**-9/(36*scipy.pi) #permittivity of free space\n",
" #in Farad/m\n",
" \n",
"er=5.7 #relative permittivity\n",
" #(dimensionless)\n",
" \n",
"chi=er-1 #Electric susceptibility\n",
" #(dimensionless)\n",
" \n",
"r=0.1 #radius of sphere in m\n",
"\n",
"q1=2 #charge on sphere in pC\n",
"\n",
"q2=-4 #value of point charge in pC\n",
"\n",
"#Calculations\n",
"\n",
"E=q1/(4*scipy.pi*e0*er*r**2) #Electric field on the\n",
" #sphere in pV/m\n",
" \n",
"P=chi*e0*E #Polarisation in pC/m^2\n",
"\n",
"pps=P #The surface density of polarization \n",
" #charge in pC/m^2\n",
" \n",
"F=(q1*q2*10**-12)/(4*scipy.pi*e0*er*r**2) #Force exerted on point charge in pN\n",
"\n",
"#Results\n",
"\n",
"print 'The surface density of polarization'\n",
"print 'charge on the surface of the sphere =',round(pps,2),'pC/m^2'\n",
"print 'Force exerted on -4 pC charge =',round(F,3),'pN in the radial direction'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The surface density of polarization\n",
"charge on the surface of the sphere = 13.12 pC/m^2\n",
"Force exerted on -4 pC charge = -1.263 pN in the radial direction\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 5.9, Page number: 188<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"#Variable Declarartion\n",
"\n",
"import scipy\n",
"from numpy import *\n",
"\n",
"an=array([0,0,1]) #Unit vector normal to the interface\n",
"E1=array([5,-2,3]) #Electric field for z >=0 in kV/m\n",
"e_r1=4 #Relative permittivity for z >=0 (dimensionless)\n",
"e_r2=3 #Relative permittivity for z <=0 (dimensionless)\n",
"e0=(10**-9)/(36*scipy.pi) #Permittivity of free space in Farad/m\n",
"V=2*2*2 #Volume of cube placed in region 2 in m^3\n",
"\n",
"#Calculations\n",
"\n",
"E1n=array([0,0,dot(E1,an)]) #The normal component of E1 in kV/m\n",
"E1t=E1-E1n #Transverse component of E1 in kV/m\n",
"E2t=E1t #Transverse component of E2 in kV/m\n",
"E2n=e_r1*E1n/e_r2 #Normal Component of E2 in kV/m\n",
"E2=E2n+E2t #The total field E2 in kV/m\n",
"\n",
"theta1= 90- 180*scipy.arccos(dot(E1,an)/ #Angle between E1 and \n",
" scipy.sqrt(dot(E1,E1)))/scipy.pi #interface in degrees\n",
" \n",
"theta2= 90- 180*scipy.arccos(dot(E2,an)/ #Angle between E2 and \n",
" scipy.sqrt(dot(E2,E2)))/scipy.pi #interface in degrees\n",
"\n",
"\n",
"We1= 0.5*e0*e_r1*dot(E1,E1)*10**6 # The energy density of E1 in J/m^3\n",
"We2= 0.5*e0*e_r2*dot(E2,E2)*10**6 # The energy density of E2 in J/m^3\n",
"W= We2*V # The energy within the cube in J\n",
"\n",
"#Results\n",
"\n",
"print 'The electric field for the region z <=0 is',E2,'kV/m'\n",
"print 'The angle E1 makes with the boundary is',round(theta1,1),'degrees'\n",
"print 'The angle E2 makes with the boundary is',round(theta2,1),'degrees'\n",
"print 'The energy density in dielectric 1 is',round(We1*10**6,0),'J/m^3'\n",
"print 'The energy density in dielectric 2 is',round(We2*10**6,0),'J/m^3'\n",
"print 'The energy within the cube is',round(W*1000,3),'mJ'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The electric field for the region z <=0 is [ 5 -2 4] kV/m\n",
"The angle E1 makes with the boundary is 29.1 degrees\n",
"The angle E2 makes with the boundary is 36.6 degrees\n",
"The energy density in dielectric 1 is 672.0 J/m^3\n",
"The energy density in dielectric 2 is 597.0 J/m^3\n",
"The energy within the cube is 4.775 mJ\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 5.10, Page number: 190<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"\n",
"#Variable Declaration\n",
"\n",
"e=(10**-9)/(36*scipy.pi) #Permittivity of free space in Farad/m\n",
"er=2 #Relative permittivity (dimensionless)\n",
"ps=2 #Surface charge in nC/m^2\n",
"\n",
"#Calculations\n",
"\n",
"#Point A is in the region y <=0. Hence E=D=0\n",
"#For point B which is in the region y >=0,\n",
"\n",
"Dn=ps #Displacement current in nC/m^2\n",
"En=Dn*10**-9/(e*er) #Electric Field\n",
"\n",
"#Result\n",
"\n",
"print 'E at point A= 0'\n",
"print 'D at point A= 0'\n",
"print 'E at point B=',round(En,2),'V/m along positive y direction'\n",
"print 'D at point B=',Dn,'nC/m^2 along positive y direction'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"E at point A= 0\n",
"D at point A= 0\n",
"E at point B= 113.1 V/m along positive y direction\n",
"D at point B= 2 nC/m^2 along positive y direction\n"
]
}
],
"prompt_number": 8
}
],
"metadata": {}
}
]
}