Electromagnetism

cf. Robert G. Jahn. Physics of Electric Propulsion (Dover Books on Physics) Dover Publications. May 26, 2006



In [1]:

    
import sympy



In [2]:

    
import os, sys

Get the current directory:



In [3]:

    
currentdir = os.getcwd(); currentdir









    Out[3]:





'/home/topolo/PropD/Propulsion/EM'



In [4]:

    
sys.path.append(currentdir)
sys.path.append('/home/topolo/PropD/Propulsion') # you are going to type in manually the parent directory of the 
# current directory, i.e. the directory "above" the current directory, in absolute path form

Get the physical constants and unit conversions you need straight from NIST



In [5]:

    
import Physique









    



/home/topolo/PropD/Propulsion/Physique
/home/topolo/PropD/Propulsion/Physique/rawdata/



In [9]:

    
from Physique import FundConst
g_0pd = FundConst[ FundConst["Quantity"].str.contains("gravity") ]
g_0pd









    Out[9]:






  
    
      
      Quantity
      Value
      Uncertainty
      Unit
    
  
  
    
      303
      standard acceleration of gravity
      9.80665
      None
      m s^-2



In [7]:

    
g_0pd.Value.get_values()









    Out[7]:





array([Decimal('9.80665')], dtype=object)

electric constant, electric (vacuum) permittivity $\epsilon_0$



In [18]:

    
epsilon_0pd = FundConst[ FundConst['Quantity'].str.contains('electric constant')]
epsilon_0pd









    Out[18]:






  
    
      
      Quantity
      Value
      Uncertainty
      Unit
    
  
  
    
      76
      electric constant
      8.854187817E-12
      None
      F m^-1



In [24]:

    
1./(4. * sympy.pi.n() * epsilon_0pd.Value)









    Out[24]:





8987551787.99791



In [14]:

    
from Physique import conv as convDF
convDF.columns









    Out[14]:





Index([u'Toconvertfrom', u'to', u'Multiplyby'], dtype='object')



In [37]:

    
convDF[convDF['to'].str.contains("coulomb ")]
esu_to_Cpd = convDF[convDF['Toconvertfrom'].str.contains("statcoulomb")]
esu_to_Cpd









    Out[37]:






  
    
      
      Toconvertfrom
      to
      Multiplyby
    
  
  
    
      405
      statcoulomb
      coulomb (C)
      3.335641E-10



In [43]:

    
convDF[convDF['Toconvertfrom'].str.contains("dyne ")]
dyne_to_Npd = convDF.loc[137,:]
dyne_to_Npd









    Out[43]:





Toconvertfrom    dyne (dyn)
to               newton (N)
Multiplyby         0.000010
Name: 137, dtype: object



In [15]:

    
convDF[convDF['Toconvertfrom'].str.contains("farad")]









    Out[15]:






  
    
      
      Toconvertfrom
      to
      Multiplyby
    
  
  
    
      2
      abfarad
      farad (F)
      1.0E+9
    
    
      141
      EMU of capacitance (abfarad)
      farad (F)
      1.0E+9
    
    
      149
      ESU of capacitance (statfarad)
      farad (F)
      1.112650E-12
    
    
      154
      faraday (based on carbon 12)
      coulomb (C)
      96485.31
    
    
      406
      statfarad
      farad (F)
      1.112650E-12

Implementation, using Python `sympy`, of Electromagnetism in terms of differential geometry

The (sympy documentation for the differential geometry module)[http://docs.sympy.org/dev/modules/diffgeom.html] is poor: there wasn't any explanation that connects that mathematics and physics to the commands - this could be attributed to the medium, in that it's difficult to display LaTeX with the code, but with jupyter notebooks, this could be alleviated.



In [59]:

    
from sympy import symbols, sin, cos, pi, Function
from sympy.diffgeom import Differential, Manifold, Patch, CoordSystem, WedgeProduct
t, r, phi, z = symbols('t, r, phi, z')
M = Manifold('M',4)
Mchart = Patch('P',M)
Cart = CoordSystem('Cart',Mchart)
cylin = CoordSystem('cylin',Mchart)



In [60]:

    
cylin.connect_to(Cart,[t,r,phi,z],[t,r*cos(phi),r*sin(phi),z])
cylin.jacobian(Cart,[t,r,phi,z])









    Out[60]:





Matrix([
[1,        0,           0, 0],
[0, cos(phi), -r*sin(phi), 0],
[0, sin(phi),  r*cos(phi), 0],
[0,        0,           0, 1]])

Define the Electric Field 1-form.



In [61]:

    
E_1 = Function('E_1'); E_2 = Function('E_2'); E_3 = Function('E_3')
E_1Cart = E_1(*Cart.coord_functions()); E_2Cart = E_2(*Cart.coord_functions()); E_3Cart = E_3(*Cart.coord_functions())
EformCart = E_1Cart*Cart.base_oneforms()[1]+E_2Cart*Cart.base_oneforms()[2]+E_3Cart*Cart.base_oneforms()[3]

Define the magnetic field 2-form.



In [62]:

    
B_12 = Function('B_12'); B_23 = Function('B_23'); B_31 = Function('B_31')
B_12Cart = B_12(*Cart.coord_functions()); B_23Cart = B_23(*Cart.coord_functions()); 
B_31Cart = B_31(*Cart.coord_functions());



In [63]:

    
BformCart = B_12Cart*WedgeProduct(Cart.base_oneform(1), Cart.base_oneform(2)) + B_23Cart*WedgeProduct(Cart.base_oneform(2), Cart.base_oneform(3)) + B_31Cart*WedgeProduct(Cart.base_oneform(3), Cart.base_oneform(1))



In [64]:

    
BformCart









    Out[64]:





B_12(Cart_0, Cart_1, Cart_2, Cart_3)*WedgeProduct(dCart_1, dCart_2) + B_23(Cart_0, Cart_1, Cart_2, Cart_3)*WedgeProduct(dCart_2, dCart_3) + B_31(Cart_0, Cart_1, Cart_2, Cart_3)*WedgeProduct(dCart_3, dCart_1)



In [65]:

    
F2formCart = BformCart + WedgeProduct(EformCart,Cart.base_oneform(0))



In [68]:

    
# dir(Differential(F2formCart))



In [46]:

    
sympy.sin( sympy.pi/4) == sympy.cos(sympy.pi/4)









    Out[46]:





True



In [ ]:

	Toconvertfrom	to	Multiplyby
2	abfarad	farad (F)	1.0E+9
141	EMU of capacitance (abfarad)	farad (F)	1.0E+9
149	ESU of capacitance (statfarad)	farad (F)	1.112650E-12
154	faraday (based on carbon 12)	coulomb (C)	96485.31
406	statfarad	farad (F)	1.112650E-12

Electromagnetism

cf. Robert G. Jahn. Physics of Electric Propulsion (Dover Books on Physics) Dover Publications. May 26, 2006

Get the physical constants and unit conversions you need straight from NIST

Implementation, using Python sympy, of Electromagnetism in terms of differential geometry

Implementation, using Python `sympy`, of Electromagnetism in terms of differential geometry