Paraxial field of a solenoid

File: solenoid-pedge.biot

BiotSavart version 4.0.14 introduced sophisticated paraxial calculations from the paper by R.H. Jackson. To validate the implementation in the case of a solenoid we use the example provided by Jackson and calculate the magnetic field along the inner edge.

R.H. Jackson, "Off-axis expansion solution of Laplace's equation: Application to accurate and rapid calculation of coil magnetic fields", IEEE Transactions on Electron Devices 46 (1999) 1050.

The specifications of the coil from the article are

Coil inner radius 4 cm
Coil outer radius 6 cm
Coil length 20 cm
Current density 100 A/cm^2
Current 4000 A

The coil (cutaway view) and probes:

solid model

Results

The calculated magnetic field is accurate right up to the inner surface of the solenoid. At r=4cm and z=0cm we have the following comparison between paraxial solenoid field and field obtained by discretizing with indicated resolution and using non-paraxial loop field as integration kernel:

Paraxial field 22.853 mT
Discrete (1mm resolution) 22.801 mT
Discrete (0.5mm resolution) 22.827 mT
curve

For paraxial calculation the exported field is found in file export-field.txt which has the following contents:

    4.000000e-02	    0.000000e+00	   -7.500000e-01	   -1.256201e-06	    0.000000e+00	    1.538942e-05	    1.544061e-05
    4.000000e-02	    0.000000e+00	   -7.400000e-01	   -1.326900e-06	    0.000000e+00	    1.603011e-05	    1.608494e-05
    4.000000e-02	    0.000000e+00	   -7.300000e-01	   -1.402661e-06	    0.000000e+00	    1.670698e-05	    1.676576e-05
    4.000000e-02	    0.000000e+00	   -7.200000e-01	   -1.483929e-06	    0.000000e+00	    1.742261e-05	    1.748569e-05
    4.000000e-02	    0.000000e+00	   -7.100000e-01	   -1.571192e-06	    0.000000e+00	    1.817984e-05	    1.824761e-05
    4.000000e-02	    0.000000e+00	   -7.000000e-01	   -1.664990e-06	    0.000000e+00	    1.898173e-05	    1.905461e-05
    4.000000e-02	    0.000000e+00	   -6.900000e-01	   -1.765924e-06	    0.000000e+00	    1.983165e-05	    1.991012e-05
    4.000000e-02	    0.000000e+00	   -6.800000e-01	   -1.874656e-06	    0.000000e+00	    2.073325e-05	    2.081782e-05
    4.000000e-02	    0.000000e+00	   -6.700000e-01	   -1.991923e-06	    0.000000e+00	    2.169053e-05	    2.178180e-05
    4.000000e-02	    0.000000e+00	   -6.600000e-01	   -2.118547e-06	    0.000000e+00	    2.270787e-05	    2.280648e-05
    4.000000e-02	    0.000000e+00	   -6.500000e-01	   -2.255440e-06	    0.000000e+00	    2.379008e-05	    2.389676e-05
    4.000000e-02	    0.000000e+00	   -6.400000e-01	   -2.403623e-06	    0.000000e+00	    2.494243e-05	    2.505798e-05
    4.000000e-02	    0.000000e+00	   -6.300000e-01	   -2.564238e-06	    0.000000e+00	    2.617071e-05	    2.629603e-05
    4.000000e-02	    0.000000e+00	   -6.200000e-01	   -2.738562e-06	    0.000000e+00	    2.748129e-05	    2.761741e-05
    4.000000e-02	    0.000000e+00	   -6.100000e-01	   -2.928030e-06	    0.000000e+00	    2.888123e-05	    2.902927e-05
    4.000000e-02	    0.000000e+00	   -6.000000e-01	   -3.134257e-06	    0.000000e+00	    3.037828e-05	    3.053954e-05
    4.000000e-02	    0.000000e+00	   -5.900000e-01	   -3.359062e-06	    0.000000e+00	    3.198106e-05	    3.215699e-05
    4.000000e-02	    0.000000e+00	   -5.800000e-01	   -3.604499e-06	    0.000000e+00	    3.369910e-05	    3.389133e-05
    4.000000e-02	    0.000000e+00	   -5.700000e-01	   -3.872896e-06	    0.000000e+00	    3.554299e-05	    3.575337e-05
    4.000000e-02	    0.000000e+00	   -5.600000e-01	   -4.166891e-06	    0.000000e+00	    3.752451e-05	    3.775516e-05
    4.000000e-02	    0.000000e+00	   -5.500000e-01	   -4.489486e-06	    0.000000e+00	    3.965680e-05	    3.991011e-05
    4.000000e-02	    0.000000e+00	   -5.400000e-01	   -4.844104e-06	    0.000000e+00	    4.195453e-05	    4.223326e-05
    4.000000e-02	    0.000000e+00	   -5.300000e-01	   -5.234658e-06	    0.000000e+00	    4.443414e-05	    4.474142e-05
    4.000000e-02	    0.000000e+00	   -5.200000e-01	   -5.665631e-06	    0.000000e+00	    4.711405e-05	    4.745349e-05
    4.000000e-02	    0.000000e+00	   -5.100000e-01	   -6.142178e-06	    0.000000e+00	    5.001500e-05	    5.039074e-05
    4.000000e-02	    0.000000e+00	   -5.000000e-01	   -6.670239e-06	    0.000000e+00	    5.316035e-05	    5.357718e-05
    4.000000e-02	    0.000000e+00	   -4.900000e-01	   -7.256685e-06	    0.000000e+00	    5.657650e-05	    5.703998e-05
    4.000000e-02	    0.000000e+00	   -4.800000e-01	   -7.909481e-06	    0.000000e+00	    6.029336e-05	    6.080995e-05
    4.000000e-02	    0.000000e+00	   -4.700000e-01	   -8.637900e-06	    0.000000e+00	    6.434492e-05	    6.492213e-05
    4.000000e-02	    0.000000e+00	   -4.600000e-01	   -9.452766e-06	    0.000000e+00	    6.876988e-05	    6.941650e-05
    4.000000e-02	    0.000000e+00	   -4.500000e-01	   -1.036677e-05	    0.000000e+00	    7.361245e-05	    7.433884e-05
    4.000000e-02	    0.000000e+00	   -4.400000e-01	   -1.139483e-05	    0.000000e+00	    7.892332e-05	    7.974166e-05
    4.000000e-02	    0.000000e+00	   -4.300000e-01	   -1.255457e-05	    0.000000e+00	    8.476073e-05	    8.568546e-05
    4.000000e-02	    0.000000e+00	   -4.200000e-01	   -1.386689e-05	    0.000000e+00	    9.119188e-05	    9.224017e-05
    4.000000e-02	    0.000000e+00	   -4.100000e-01	   -1.535669e-05	    0.000000e+00	    9.829455e-05	    9.948691e-05
    4.000000e-02	    0.000000e+00	   -4.000000e-01	   -1.705374e-05	    0.000000e+00	    1.061591e-04	    1.075201e-04
    4.000000e-02	    0.000000e+00	   -3.900000e-01	   -1.899386e-05	    0.000000e+00	    1.148909e-04	    1.164503e-04
    4.000000e-02	    0.000000e+00	   -3.800000e-01	   -2.122032e-05	    0.000000e+00	    1.246133e-04	    1.264072e-04
    4.000000e-02	    0.000000e+00	   -3.700000e-01	   -2.378566e-05	    0.000000e+00	    1.354714e-04	    1.375436e-04
    4.000000e-02	    0.000000e+00	   -3.600000e-01	   -2.675410e-05	    0.000000e+00	    1.476365e-04	    1.500410e-04
    4.000000e-02	    0.000000e+00	   -3.500000e-01	   -3.020453e-05	    0.000000e+00	    1.613119e-04	    1.641153e-04
    4.000000e-02	    0.000000e+00	   -3.400000e-01	   -3.423450e-05	    0.000000e+00	    1.767400e-04	    1.800251e-04
    4.000000e-02	    0.000000e+00	   -3.300000e-01	   -3.896547e-05	    0.000000e+00	    1.942114e-04	    1.980818e-04
    4.000000e-02	    0.000000e+00	   -3.200000e-01	   -4.454964e-05	    0.000000e+00	    2.140763e-04	    2.186626e-04
    4.000000e-02	    0.000000e+00	   -3.100000e-01	   -5.117923e-05	    0.000000e+00	    2.367588e-04	    2.422273e-04
    4.000000e-02	    0.000000e+00	   -3.000000e-01	   -5.909885e-05	    0.000000e+00	    2.627764e-04	    2.693402e-04
    4.000000e-02	    0.000000e+00	   -2.900000e-01	   -6.862244e-05	    0.000000e+00	    2.927640e-04	    3.006988e-04
    4.000000e-02	    0.000000e+00	   -2.800000e-01	   -8.015637e-05	    0.000000e+00	    3.275054e-04	    3.371719e-04
    4.000000e-02	    0.000000e+00	   -2.700000e-01	   -9.423166e-05	    0.000000e+00	    3.679760e-04	    3.798499e-04
    4.000000e-02	    0.000000e+00	   -2.600000e-01	   -1.115490e-04	    0.000000e+00	    4.153976e-04	    4.301143e-04
    4.000000e-02	    0.000000e+00	   -2.500000e-01	   -1.330426e-04	    0.000000e+00	    4.713128e-04	    4.897306e-04
    4.000000e-02	    0.000000e+00	   -2.400000e-01	   -1.599719e-04	    0.000000e+00	    5.376851e-04	    5.609779e-04
    4.000000e-02	    0.000000e+00	   -2.300000e-01	   -1.940542e-04	    0.000000e+00	    6.170348e-04	    6.468299e-04
    4.000000e-02	    0.000000e+00	   -2.200000e-01	   -2.376597e-04	    0.000000e+00	    7.126262e-04	    7.512112e-04
    4.000000e-02	    0.000000e+00	   -2.100000e-01	   -2.941020e-04	    0.000000e+00	    8.287274e-04	    8.793663e-04
    4.000000e-02	    0.000000e+00	   -2.000000e-01	   -3.680750e-04	    0.000000e+00	    9.709769e-04	    1.038400e-03
    4.000000e-02	    0.000000e+00	   -1.900000e-01	   -4.663210e-04	    0.000000e+00	    1.146911e-03	    1.238087e-03
    4.000000e-02	    0.000000e+00	   -1.800000e-01	   -5.986665e-04	    0.000000e+00	    1.366741e-03	    1.492107e-03
    4.000000e-02	    0.000000e+00	   -1.700000e-01	   -7.796556e-04	    0.000000e+00	    1.644547e-03	    1.819999e-03
    4.000000e-02	    0.000000e+00	   -1.600000e-01	   -1.031187e-03	    0.000000e+00	    2.000210e-03	    2.250375e-03
    4.000000e-02	    0.000000e+00	   -1.500000e-01	   -1.387022e-03	    0.000000e+00	    2.463009e-03	    2.826702e-03
    4.000000e-02	    0.000000e+00	   -1.400000e-01	   -1.901483e-03	    0.000000e+00	    3.080536e-03	    3.620130e-03
    4.000000e-02	    0.000000e+00	   -1.300000e-01	   -2.661732e-03	    0.000000e+00	    3.941775e-03	    4.756302e-03
    4.000000e-02	    0.000000e+00	   -1.200000e-01	   -3.774696e-03	    0.000000e+00	    5.170975e-03	    6.402133e-03
    4.000000e-02	    0.000000e+00	   -1.100000e-01	   -5.681430e-03	    0.000000e+00	    7.205905e-03	    9.176258e-03
    4.000000e-02	    0.000000e+00	   -1.000000e-01	   -7.593188e-03	    0.000000e+00	    1.220675e-02	    1.437572e-02
    4.000000e-02	    0.000000e+00	   -9.000000e-02	   -5.662585e-03	    0.000000e+00	    1.720328e-02	    1.811126e-02
    4.000000e-02	    0.000000e+00	   -8.000000e-02	   -3.736270e-03	    0.000000e+00	    1.922513e-02	    1.958482e-02
    4.000000e-02	    0.000000e+00	   -7.000000e-02	   -2.602197e-03	    0.000000e+00	    2.043199e-02	    2.059703e-02
    4.000000e-02	    0.000000e+00	   -6.000000e-02	   -1.818399e-03	    0.000000e+00	    2.126084e-02	    2.133846e-02
    4.000000e-02	    0.000000e+00	   -5.000000e-02	   -1.276842e-03	    0.000000e+00	    2.183468e-02	    2.187198e-02
    4.000000e-02	    0.000000e+00	   -4.000000e-02	   -8.889669e-04	    0.000000e+00	    2.224081e-02	    2.225856e-02
    4.000000e-02	    0.000000e+00	   -3.000000e-02	   -5.986111e-04	    0.000000e+00	    2.252447e-02	    2.253243e-02
    4.000000e-02	    0.000000e+00	   -2.000000e-02	   -3.695384e-04	    0.000000e+00	    2.271184e-02	    2.271485e-02
    4.000000e-02	    0.000000e+00	   -1.000000e-02	   -1.764429e-04	    0.000000e+00	    2.281862e-02	    2.281931e-02
    4.000000e-02	    0.000000e+00	    5.308254e-16	    9.106824e-18	    0.000000e+00	    2.285332e-02	    2.285332e-02
    4.000000e-02	    0.000000e+00	    1.000000e-02	    1.764429e-04	    0.000000e+00	    2.281862e-02	    2.281931e-02
    4.000000e-02	    0.000000e+00	    2.000000e-02	    3.695384e-04	    0.000000e+00	    2.271184e-02	    2.271485e-02
    4.000000e-02	    0.000000e+00	    3.000000e-02	    5.986111e-04	    0.000000e+00	    2.252447e-02	    2.253243e-02
    4.000000e-02	    0.000000e+00	    4.000000e-02	    8.889669e-04	    0.000000e+00	    2.224081e-02	    2.225856e-02
    4.000000e-02	    0.000000e+00	    5.000000e-02	    1.276842e-03	    0.000000e+00	    2.183468e-02	    2.187198e-02
    4.000000e-02	    0.000000e+00	    6.000000e-02	    1.818399e-03	    0.000000e+00	    2.126084e-02	    2.133846e-02
    4.000000e-02	    0.000000e+00	    7.000000e-02	    2.602197e-03	    0.000000e+00	    2.043199e-02	    2.059703e-02
    4.000000e-02	    0.000000e+00	    8.000000e-02	    3.736270e-03	    0.000000e+00	    1.922513e-02	    1.958482e-02
    4.000000e-02	    0.000000e+00	    9.000000e-02	    5.662585e-03	    0.000000e+00	    1.720328e-02	    1.811126e-02
    4.000000e-02	    0.000000e+00	    1.000000e-01	    7.593188e-03	    0.000000e+00	    1.220675e-02	    1.437572e-02
    4.000000e-02	    0.000000e+00	    1.100000e-01	    5.681430e-03	    0.000000e+00	    7.205905e-03	    9.176258e-03
    4.000000e-02	    0.000000e+00	    1.200000e-01	    3.774696e-03	    0.000000e+00	    5.170975e-03	    6.402133e-03
    4.000000e-02	    0.000000e+00	    1.300000e-01	    2.661732e-03	    0.000000e+00	    3.941775e-03	    4.756302e-03
    4.000000e-02	    0.000000e+00	    1.400000e-01	    1.901483e-03	    0.000000e+00	    3.080536e-03	    3.620130e-03
    4.000000e-02	    0.000000e+00	    1.500000e-01	    1.387022e-03	    0.000000e+00	    2.463009e-03	    2.826702e-03
    4.000000e-02	    0.000000e+00	    1.600000e-01	    1.031187e-03	    0.000000e+00	    2.000210e-03	    2.250375e-03
    4.000000e-02	    0.000000e+00	    1.700000e-01	    7.796556e-04	    0.000000e+00	    1.644547e-03	    1.819999e-03
    4.000000e-02	    0.000000e+00	    1.800000e-01	    5.986665e-04	    0.000000e+00	    1.366741e-03	    1.492107e-03
    4.000000e-02	    0.000000e+00	    1.900000e-01	    4.663210e-04	    0.000000e+00	    1.146911e-03	    1.238087e-03
    4.000000e-02	    0.000000e+00	    2.000000e-01	    3.680750e-04	    0.000000e+00	    9.709769e-04	    1.038400e-03
    4.000000e-02	    0.000000e+00	    2.100000e-01	    2.941020e-04	    0.000000e+00	    8.287274e-04	    8.793663e-04
    4.000000e-02	    0.000000e+00	    2.200000e-01	    2.376597e-04	    0.000000e+00	    7.126262e-04	    7.512112e-04
    4.000000e-02	    0.000000e+00	    2.300000e-01	    1.940542e-04	    0.000000e+00	    6.170348e-04	    6.468299e-04
    4.000000e-02	    0.000000e+00	    2.400000e-01	    1.599719e-04	    0.000000e+00	    5.376851e-04	    5.609779e-04
    4.000000e-02	    0.000000e+00	    2.500000e-01	    1.330426e-04	    0.000000e+00	    4.713128e-04	    4.897306e-04
    4.000000e-02	    0.000000e+00	    2.600000e-01	    1.115490e-04	    0.000000e+00	    4.153976e-04	    4.301143e-04
    4.000000e-02	    0.000000e+00	    2.700000e-01	    9.423166e-05	    0.000000e+00	    3.679760e-04	    3.798499e-04
    4.000000e-02	    0.000000e+00	    2.800000e-01	    8.015637e-05	    0.000000e+00	    3.275054e-04	    3.371719e-04
    4.000000e-02	    0.000000e+00	    2.900000e-01	    6.862244e-05	    0.000000e+00	    2.927640e-04	    3.006988e-04
    4.000000e-02	    0.000000e+00	    3.000000e-01	    5.909885e-05	    0.000000e+00	    2.627764e-04	    2.693402e-04
    4.000000e-02	    0.000000e+00	    3.100000e-01	    5.117923e-05	    0.000000e+00	    2.367588e-04	    2.422273e-04
    4.000000e-02	    0.000000e+00	    3.200000e-01	    4.454964e-05	    0.000000e+00	    2.140763e-04	    2.186626e-04
    4.000000e-02	    0.000000e+00	    3.300000e-01	    3.896547e-05	    0.000000e+00	    1.942114e-04	    1.980818e-04
    4.000000e-02	    0.000000e+00	    3.400000e-01	    3.423450e-05	    0.000000e+00	    1.767400e-04	    1.800251e-04
    4.000000e-02	    0.000000e+00	    3.500000e-01	    3.020453e-05	    0.000000e+00	    1.613119e-04	    1.641153e-04
    4.000000e-02	    0.000000e+00	    3.600000e-01	    2.675410e-05	    0.000000e+00	    1.476365e-04	    1.500410e-04
    4.000000e-02	    0.000000e+00	    3.700000e-01	    2.378566e-05	    0.000000e+00	    1.354714e-04	    1.375436e-04
    4.000000e-02	    0.000000e+00	    3.800000e-01	    2.122032e-05	    0.000000e+00	    1.246133e-04	    1.264072e-04
    4.000000e-02	    0.000000e+00	    3.900000e-01	    1.899386e-05	    0.000000e+00	    1.148909e-04	    1.164503e-04
    4.000000e-02	    0.000000e+00	    4.000000e-01	    1.705374e-05	    0.000000e+00	    1.061591e-04	    1.075201e-04
    4.000000e-02	    0.000000e+00	    4.100000e-01	    1.535669e-05	    0.000000e+00	    9.829455e-05	    9.948691e-05
    4.000000e-02	    0.000000e+00	    4.200000e-01	    1.386689e-05	    0.000000e+00	    9.119188e-05	    9.224017e-05
    4.000000e-02	    0.000000e+00	    4.300000e-01	    1.255457e-05	    0.000000e+00	    8.476073e-05	    8.568546e-05
    4.000000e-02	    0.000000e+00	    4.400000e-01	    1.139483e-05	    0.000000e+00	    7.892332e-05	    7.974166e-05
    4.000000e-02	    0.000000e+00	    4.500000e-01	    1.036677e-05	    0.000000e+00	    7.361245e-05	    7.433884e-05
    4.000000e-02	    0.000000e+00	    4.600000e-01	    9.452766e-06	    0.000000e+00	    6.876988e-05	    6.941650e-05
    4.000000e-02	    0.000000e+00	    4.700000e-01	    8.637900e-06	    0.000000e+00	    6.434492e-05	    6.492213e-05
    4.000000e-02	    0.000000e+00	    4.800000e-01	    7.909481e-06	    0.000000e+00	    6.029336e-05	    6.080995e-05
    4.000000e-02	    0.000000e+00	    4.900000e-01	    7.256685e-06	    0.000000e+00	    5.657650e-05	    5.703998e-05
    4.000000e-02	    0.000000e+00	    5.000000e-01	    6.670239e-06	    0.000000e+00	    5.316035e-05	    5.357718e-05
    4.000000e-02	    0.000000e+00	    5.100000e-01	    6.142178e-06	    0.000000e+00	    5.001500e-05	    5.039074e-05
    4.000000e-02	    0.000000e+00	    5.200000e-01	    5.665631e-06	    0.000000e+00	    4.711405e-05	    4.745349e-05
    4.000000e-02	    0.000000e+00	    5.300000e-01	    5.234658e-06	    0.000000e+00	    4.443414e-05	    4.474142e-05
    4.000000e-02	    0.000000e+00	    5.400000e-01	    4.844104e-06	    0.000000e+00	    4.195453e-05	    4.223326e-05
    4.000000e-02	    0.000000e+00	    5.500000e-01	    4.489486e-06	    0.000000e+00	    3.965680e-05	    3.991011e-05
    4.000000e-02	    0.000000e+00	    5.600000e-01	    4.166891e-06	    0.000000e+00	    3.752451e-05	    3.775516e-05
    4.000000e-02	    0.000000e+00	    5.700000e-01	    3.872896e-06	    0.000000e+00	    3.554299e-05	    3.575337e-05
    4.000000e-02	    0.000000e+00	    5.800000e-01	    3.604499e-06	    0.000000e+00	    3.369910e-05	    3.389133e-05
    4.000000e-02	    0.000000e+00	    5.900000e-01	    3.359062e-06	    0.000000e+00	    3.198106e-05	    3.215699e-05
    4.000000e-02	    0.000000e+00	    6.000000e-01	    3.134257e-06	    0.000000e+00	    3.037828e-05	    3.053954e-05
    4.000000e-02	    0.000000e+00	    6.100000e-01	    2.928030e-06	    0.000000e+00	    2.888123e-05	    2.902927e-05
    4.000000e-02	    0.000000e+00	    6.200000e-01	    2.738562e-06	    0.000000e+00	    2.748129e-05	    2.761741e-05
    4.000000e-02	    0.000000e+00	    6.300000e-01	    2.564238e-06	    0.000000e+00	    2.617071e-05	    2.629603e-05
    4.000000e-02	    0.000000e+00	    6.400000e-01	    2.403623e-06	    0.000000e+00	    2.494243e-05	    2.505798e-05
    4.000000e-02	    0.000000e+00	    6.500000e-01	    2.255440e-06	    0.000000e+00	    2.379008e-05	    2.389676e-05
    4.000000e-02	    0.000000e+00	    6.600000e-01	    2.118547e-06	    0.000000e+00	    2.270787e-05	    2.280648e-05
    4.000000e-02	    0.000000e+00	    6.700000e-01	    1.991923e-06	    0.000000e+00	    2.169053e-05	    2.178180e-05
    4.000000e-02	    0.000000e+00	    6.800000e-01	    1.874656e-06	    0.000000e+00	    2.073325e-05	    2.081782e-05
    4.000000e-02	    0.000000e+00	    6.900000e-01	    1.765924e-06	    0.000000e+00	    1.983165e-05	    1.991012e-05
    4.000000e-02	    0.000000e+00	    7.000000e-01	    1.664990e-06	    0.000000e+00	    1.898173e-05	    1.905461e-05
    4.000000e-02	    0.000000e+00	    7.100000e-01	    1.571192e-06	    0.000000e+00	    1.817984e-05	    1.824761e-05
    4.000000e-02	    0.000000e+00	    7.200000e-01	    1.483929e-06	    0.000000e+00	    1.742261e-05	    1.748569e-05
    4.000000e-02	    0.000000e+00	    7.300000e-01	    1.402661e-06	    0.000000e+00	    1.670698e-05	    1.676576e-05
    4.000000e-02	    0.000000e+00	    7.400000e-01	    1.326900e-06	    0.000000e+00	    1.603011e-05	    1.608494e-05
    4.000000e-02	    0.000000e+00	    7.500000e-01	    1.256201e-06	    0.000000e+00	    1.538942e-05	    1.544061e-05