{
"metadata": {
"name": "",
"signature": "sha256:891c986a46f113e35878dc6d6a0b8d702286ad7b4ea91bae03f716286f9bdeaa"
},
"nbformat": 3,
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"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h1>Chapter 4: Electrostatic Fields<h1>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 4.1, Page number: 107<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"Q1=1*10**-3 #charge 1 at (-1,-1,4) in C\n",
"Q2=-2*10**-3 #charge 2 at (3,2,-1) in C\n",
"Q=10*10**-9 #charge 3 at (0,3,1) in C\n",
"P1=array([0,3,1])-array([3,2,-1]) #distance vector from charge 3 to 1\n",
"P2=array([0,3,1])-array([-1,-1,4]) #distance vector from charge 3 to 2\n",
"e=10**-9/(36*scipy.pi) #permittivity in Farad/m \n",
"\n",
"#Calculations\n",
"\n",
"modP1=scipy.sqrt(dot(P1,P1))\n",
"modP2=scipy.sqrt(dot(P2,P2))\n",
"F1=(Q*Q1)*P1*10**3/(4*scipy.pi*e*modP1**3) #force on charge 3 by 1 in mN\n",
"F2=(Q*Q2)*P2*10**3/(4*scipy.pi*e*modP2**3) #force on charge 3 by 2 in mN\n",
"\n",
" #Total force on charge 3\n",
" \n",
"Fx=round(dot(F1+F2,ax),3)\n",
"Fy=round(dot(F1+F2,ay),3)\n",
"Fz=round(dot(F1+F2,az),3)\n",
"F=array([Fx,Fy,Fz]) #Total force in mN\n",
"E=(10**-6)*(F/Q) #Electric field in kV/m\n",
"\n",
"#Results \n",
"\n",
"print 'Total force on charge at (0,3,1) =',F,'in mN'\n",
"print 'Electric field at (0,3,1) =',E,'kV/m'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Total force on charge at (0,3,1) = [-6.512 -3.713 7.509] in mN\n",
"Electric field at (0,3,1) = [-651.2 -371.3 750.9] kV/m\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Example 4.3, Page number: 109"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"#Variable Declaration\n",
"\n",
"E=500*10**3 #electric field in V/m\n",
"Qm=9*10**-6 #Q/m in C/kg\n",
"y=0.8 #distance fallen in m\n",
"g=9.8 #acceleration due to gravity in m/s^2\n",
"\n",
"#Calculations\n",
"\n",
"t=scipy.sqrt(2*y/g) #time taken to fall in seconds\n",
"x=Qm*E*t**2/2 #half the separation between particles in m\n",
"sep=2*x #separation between particles in m\n",
"\n",
"#Result\n",
"\n",
"print 'The separation between particles is',round(sep*100,2),'cm'\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The separation between particles is 73.47 cm\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Example 4.5, Page number: 120"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"import scipy.integrate\n",
"\n",
"#Variable Declaration\n",
"\n",
"Eo=10**-9/(36*scipy.pi) #permittivity of free space\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"q=-1 #charge in mC\n",
"\n",
"#Calculations\n",
"\n",
"def charge(x,y): \n",
" return x*y*(x**2+y**2+25)**(1.5)\n",
"Q, errq = scipy.integrate.dblquad(lambda y , x: charge(x,y), #total charge in nC\n",
" 0, 1, lambda y: 0, lambda y: 1) \n",
"\n",
"d=(4*scipy.pi*Eo*(x**2+y**2+25)**(1.5))\n",
"\n",
"def elecx(x,y): \n",
" return 10**-9*x*y*(x**2+y**2+25)**(1.5)*(-x)/d #x component of electric field\n",
"Ex, errx = scipy.integrate.dblquad(lambda y , x: elecx(x,y), \n",
" 0, 1, lambda y: 0, lambda y: 1) \n",
"\n",
"def elecy(x,y): \n",
" return 10**-9*x*y*(x**2+y**2+25)**(1.5)*(-y)/d #y component of electric field\n",
"Ey, erry = scipy.integrate.dblquad(lambda y , x: elecy(x,y), \n",
" 0, 1, lambda y: 0, lambda y: 1) \n",
"\n",
"def elecz(x,y): \n",
" return 10**-9*x*y*(5)/(4*scipy.pi*Eo) #z component of electric field\n",
"Ez, errz = scipy.integrate.dblquad(lambda y , x: elecz(x,y), \n",
" 0, 1, lambda y: 0, lambda y: 1) \n",
"\n",
"E=array([round(Ex,1),round(Ey,1),round(Ez,2)]) #electric field in V/m\n",
"\n",
"F=q*E #force in mN \n",
"\n",
"#Results\n",
"\n",
"print 'Total charge =',round(Q,2),'nC'\n",
"print 'Electric field at (0,0,5) =',E,'V/m'\n",
"print 'Force experienced by -1mC =',F,'mN'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Total charge = 33.15 nC\n",
"Electric field at (0,0,5) = [ -1.5 -1.5 11.25] V/m\n",
"Force experienced by -1mC = [ 1.5 1.5 -11.25] mN\n"
]
}
],
"prompt_number": 10
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 4.6, Page number: 121<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"ps1=10*10**-9 #Surface charge density of plane 1\n",
"ps2=15*10**-9 #Surface charge density of plane 2\n",
"pl=10*scipy.pi*10**-9 #charge density of line\n",
"e=(10**-9)/(36*scipy.pi) #permittivity of free space in Farad/m\n",
"\n",
"#Calculations\n",
"\n",
"E1=(ps1/(2*e))*-ax/scipy.pi #field due to plane 1 in multiples of pi in V/m\n",
"E2=(ps2/(2*e))*ay/scipy.pi #field due to plane 2 in multiples of pi in V/m\n",
"\n",
" #field due to line charge in multiples of pi in V/m\n",
" \n",
"a=(ax-3*az) \n",
"moda=scipy.sqrt(dot((ax-3*az),(ax-3*az)))\n",
"e3=(pl/(2*scipy.pi*e*moda**2))*a\n",
"E3=array([dot(e3,ax)/scipy.pi,0,dot(e3,az)/scipy.pi])\n",
"\n",
" #total field in multiples of pi in V/m\n",
" \n",
"E=E1+E2+E3 \n",
"\n",
"#Result\n",
"\n",
"print 'The total electric field at (1,1,-1) =',E,'Pi V/m'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The total electric field at (1,1,-1) = [-162. 270. -54.] Pi V/m\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 4.7, Page number: 123<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"Q=-5*scipy.pi*10**-3 #charge at (4,0,0) in C\n",
"pl=3*scipy.pi*10**-3 #charge density of line charge in C/m\n",
"r=array([4,0,3]) #point where D is to be found \n",
"rp=array([4,0,0]) #position of point charge\n",
"\n",
"#Calculations \n",
"\n",
"R=r-rp \n",
"modR=scipy.sqrt(dot(R,R)) \n",
"Dq=(Q*R)/(4*scipy.pi*modR**3) #flux density due to point charge in C/m^2\n",
"p=scipy.sqrt(dot(r,r))\n",
"ap=r/p \n",
"Dl=(pl/(2*scipy.pi*p))*ap #flux density due to line charge in C/m^2\n",
"D=(Dq+Dl)*10**6 #total flux density in micro C/m^2\n",
"Dz=round(dot(D,az),0)\n",
"Dx=round(dot(D,ax),0)\n",
"Dy=round(dot(D,ay),0)\n",
"Dround=array([Dx,Dy,Dz]) #value of D rounded to 0 decimal points\n",
"\n",
"#Result\n",
"\n",
"print 'D at (4,0,0) due to point charge and line charge =',Dround,'micro C/m^2'\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"D at (4,0,0) due to point charge and line charge = [ 240. 0. 41.] micro C/m^2\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 4.8, Page number: 130<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"\n",
"import scipy\n",
"from numpy import *\n",
"import scipy.integrate\n",
"from fractions import Fraction\n",
"\n",
"#Variable Declaration\n",
"\n",
"ap=array([1,0,0]) #Unit vector along rho direction\n",
"aph=array([0,1,0]) #Unit vector along phi direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"point=array([1,scipy.pi/4,3])\n",
"p1=0\n",
"p2=1\n",
"ph1=0\n",
"ph2=2*scipy.pi\n",
"\n",
"#Calculations\n",
"\n",
"pointp=dot(point,ap)\n",
"pointph=dot(point,aph)\n",
"pv=pointp*scipy.cos(pointph)**2 #charge density at (1,pi/4,3) in C/m^3\n",
"\n",
"def ctop(phi,p): \n",
" return 2*p**2*(scipy.cos(phi)**2)\n",
"psya, erra = scipy.integrate.dblquad(lambda p , phi: ctop(phi,p), \n",
" ph1, ph2, lambda p: p1, lambda p: p2)\n",
"\n",
"def cbot(phi,p): \n",
" return 2*p**2*(scipy.cos(phi)**2)\n",
"psyb, errb = scipy.integrate.dblquad(lambda p , phi: cbot(phi,p), \n",
" ph1, ph2, lambda p: p1, lambda p: p2)\n",
" \n",
"psy=psya+psyb #Charge in C\n",
"psyp=psy/scipy.pi #Charge in multiples of Pi in C\n",
"psyf=Fraction(psyp).limit_denominator(100) #converting to fraction\n",
"\n",
"\n",
"#Results\n",
"\n",
"print 'Charge density at (1,pi/4,3) =',pv,'C/m^3'\n",
"print 'Total charge enclosed by the cylinder =',psyf,'Pi C'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Charge density at (1,pi/4,3) = 0.5 C/m^3\n",
"Total charge enclosed by the cylinder = 4/3 Pi C\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 4.10, Page number: 136<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"\n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"Q1=-4 #charge 1 in micro C\n",
"Q2=5 #charge 2 in micro C\n",
"e=10**-9/(36*scipy.pi) #permittivity of free space in Farad/m \n",
"\n",
"#Calculations\n",
"\n",
"R1=array([1,0,1])-array([2,-1,3]) #distance vector from (1,0,1) to charge 1\n",
"R2=array([1,0,1])-array([0,4,-2]) #distance vector from (1,0,1) to charge 2\n",
"modR1=scipy.sqrt(dot(R1,R1))\n",
"modR2=scipy.sqrt(dot(R2,R2)) \n",
"V=10**-9*((Q1/modR1)+(Q2/modR2))/(4*scipy.pi*e) #potential in kV\n",
"\n",
"#Result\n",
"\n",
"print 'The potential at (1, 0, 1) =',round(V,3),'kV'\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The potential at (1, 0, 1) = -5.872 kV\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "heading",
"level": 3,
"metadata": {},
"source": [
"Example 4.11, Page number: 136"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"\n",
"#Variable Declaration\n",
"\n",
"Eo=10**-9/(36*scipy.pi) #permittivity of free space\n",
"Vo=0 #potential at O in V\n",
"Vb=100 #potential at B in V\n",
"po=scipy.sqrt(2)\n",
"ro=5\n",
"pa=1\n",
"ra=9\n",
"pb=1\n",
"rb=scipy.sqrt(21)\n",
"pc=scipy.sqrt(20)\n",
"rc=scipy.sqrt(11)\n",
"pl=2*10**-9 #charge density of the line in C/m\n",
"Q=5*10**-9 #point charge at (-3,4,0) in C\n",
"\n",
"#Calculations\n",
"\n",
"Va=Vo-(-pl*scipy.log(po/pa)/(2*scipy.pi*Eo)+Q*(ra-ro)/(4*scipy.pi*Eo*ra*ro))\n",
"Vc=Vb+(-pl*scipy.log(pc/pb)/(2*scipy.pi*Eo)+Q*(rb-rc)/(4*scipy.pi*Eo*rb*rc))\n",
"Vbc=Vc-Vb\n",
"\n",
"#Results\n",
"\n",
"print 'Va =',round(Va,3),'V'\n",
"print 'Vc =',round(Vc,3),'V'\n",
"print 'Vbc =',round(Vbc,3),'V'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Va = 8.477 V\n",
"Vc = 49.825 V\n",
"Vbc = -50.175 V\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 4.12, Page number: 140<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"\n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"ar=array([1,0,0]) #Unit vector along radial direction\n",
"ath=array([0,1,0]) #Unit vector along theta direction\n",
"aph=array([0,0,1]) #Unit vector along phi direction\n",
"e=(10**-9)/(36*scipy.pi) #permittivity of free space in Farad/m\n",
"\n",
" #The point (2, pi/2, 0)\n",
"r=2\n",
"th=scipy.pi/2\n",
"ph=0\n",
" #Point A\n",
"ra=1\n",
"tha=scipy.pi*30/180\n",
"pha=scipy.pi*120/180\n",
" #Point B\n",
"rb=4\n",
"thb=scipy.pi/2\n",
"phb=scipy.pi*60/180\n",
"\n",
"q=10*10**-6 \n",
"\n",
"#Calculations\n",
"\n",
"Er=(20.0/r**3)*scipy.sin(th)*scipy.cos(ph) #Radial component of E in V/m\n",
"Eth=-(10/r**3)*scipy.cos(th)*scipy.cos(ph) #Theta component of E in V/m\n",
"Eph=(10/r**3)*scipy.sin(ph) #Phi component of E in V/m\n",
"E=array([Er,Eth,Eph])\n",
"D=E*e*10**12 #Electric flux density D in pC/m^2\n",
"Dr=round(dot(D,ar),1) #Radial component of D in V/m rounded to 1 decimal\n",
"Dth=round(dot(D,ath),0) #Theta component of D in pC/m^2 rounded to 0 decimal\n",
"Dph=round(dot(D,aph),0) #Phi component of D in pC/m^2 rounded to 0 decimal\n",
"Dc=array([Dr,Dth,Dph]) #Rounded D in pC/m^2\n",
"\n",
"Va=10*scipy.sin(tha)*cos(pha)/ra**2 #potential at point A in V\n",
"Vb=10*scipy.sin(thb)*cos(phb)/rb**2 #potential at point B in V\n",
"W=q*(Vb-Va)*10**6 #work done in micro J\n",
"\n",
"#Results\n",
"\n",
"print 'The electric flux density D at (2, pi/2, 0) =',Dc,'pC/m^2'\n",
"print 'Work done in moving the charge =',W,'micro J'\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The electric flux density D at (2, pi/2, 0) = [ 22.1 -0. 0. ] pC/m^2\n",
"Work done in moving the charge = 28.125 micro J\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 4.13, Page number: 145<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"\n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"p1=-5*10**-9 #dipole moment of dipole 1 in C/m\n",
"p2=9*10**-9 #dipole moment of dipole 2 in C/m\n",
"z1=2 #z component of position vector of dipole 1\n",
"z2=-3 #z component of position vector of dipole 2\n",
"e=10**-9/(36*scipy.pi) #permittivity of free space in Farad/m\n",
"\n",
"#Calculation\n",
"\n",
"V=(1/(4*scipy.pi*e))*((p1*abs(z1)/z1**3)+(p2*abs(z2)/z2**3))\n",
"\n",
"#Result\n",
"\n",
"print 'Potential at origin =',V, 'V'\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Potential at origin = -20.25 V\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 4.14, Page number: 148<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"Q1=-1*10**-9 #Charge 1 in C\n",
"Q2=4*10**-9 #Charge 2 in C\n",
"Q3=3*10**-9 #Charge 3 in C\n",
"e=10**-9/(36*scipy.pi) #permittivity of free space in farad/m\n",
"\n",
"#Calculations\n",
"\n",
"V1=(1/(4*scipy.pi*e)*(Q2+Q3))\n",
"V2=(1/(4*scipy.pi*e)*(Q1+Q3/(2**.5)))\n",
"V3=(1/(4*scipy.pi*e)*(Q1+Q2/(2**.5)))\n",
"W=0.5*((V1*Q1)+(V2*Q2)+(V3*Q3))*10**9 #Energy in nJ\n",
"\n",
"#Result\n",
"\n",
"print 'Energy in the system =',round(W,2),'nJ'"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Energy in the system = 13.37 nJ\n"
]
}
],
"prompt_number": 8
}
],
"metadata": {}
}
]
}