Test3d25.dat, 25th 'test' file for CPO3DS

Field inside a dielectric sphere

Care must be taken in setting up dielectric simulations, so it is very important to read the guidance.

See also the converse system, field in a cavity inside a dielectric.

A uniform field of 1V/mm is created by applying a linear field to a sphere.

A sphere of radius 0.05mm is put at the centre of the system. It has a dielectric constant K = 2.

The inaccuracy used for evaluation of the fields at the dielectric interfaces is 1.E-7 (but see below).

As the text-books say, the field inside the cavity should be uniform, of strength

E_cavity = E_ext*3/(2 + K)

In the present simulation, with the present number of segments in the outer sphere, E_ext is 0.998664, slightly different from the 1V/mm. Therefore we should have E_cavity = 0.75*0.998664 = 0.748998. The program gives 0.749031, which is in error by 0.000038, which is 0.0051%.

Here are some results for other values of K and other values of N, the number of subdivisions (in a 1/8 sector of the sphere). The last column shows the error with respect to the external field.

K |
N |
program |
correct |
%rel error |
%error wrt E_ext |

2 |
6 |
0.749163 |
0.748998 |
0.0220 |
0.0165 |

2 |
12 |
0.749048 |
0.748998 |
0.0067 |
0.0050 |

2 |
48 |
0.749036 |
0.748998 |
0.0051 |
0.0038 |

2 |
192 |
0.748985 |
0.748998 |
0.0017 |
0.0013 |

2 |
48 |
0.749036 |
0.748998 |
0.0051 |
0.0038 |

3.5 |
48 |
0.544680 |
0.544726 |
0.0082 |
0.0046 |

10 |
48 |
0.249989 |
0.249666 |
0.1292 |
0.0323 |

1E6 |
48 |
-0.000146 |
0.000003 |
- |
0.0149 |

conductor |
48 |
0.00227 |
0 |
- |
0.23 |

For an infinite value of K the field inside the sphere should be 0, as in a conducting sphere, so for comparison the last row of the table shows the field obtained by the program for a conducting sphere.

The errors do not change monotonically with K or N because an empirical correction is used in the program to try to minimise the dependence on these parameters.

Here are some results for different values of the user-selected inaccuracy for the dielectric field evaluations. This inaccuracy is used for the evaluation of the fields at the dielectric interfaces can be selected by the user (at the bottom of the 'tracing accuracy' sheet). The default value is 0.001 (which can be changed, see the bottom of the 'tracing accuracy' sheet). Note that this inaccuracy is not the same as the inaccuracy used in calculating the surface charges of the conducting electrodes, which is always 1.E-7 and which cannot usually be changed by the user.

In all cases here K = 2, N = 48.

inacc |
program |
correct |
%rel error |
time, sec |

1.E-7 |
0.749036 |
0.748998 |
0.005 |
10.8 |

1.E-6 |
0.749036 |
0.748998 |
0.005 |
10.8 |

1.E-5 |
0.749144 |
0.748998 |
0.020 |
5.5 |

1.E-4 |
0.749061 |
0.748998 |
0.008 |
2.8 |

1.E-3 |
0.748808 |
0.748998 |
0.025 |
2.5 |

1.E-2 |
0.748559 |
0.748998 |
0.059 |
1.7 |

1.E-1 |
0.746855 |
0.748998 |
0.286 |
1.5 |

The default value therefore seems to be adequate for most purposes. The actual inaccuracy is only loosely related to the obtained inaccuracy.

Note that dielectric calculations with CPO3D are always much slower than the analogous non-dielectric calculations.

Note that, apart from water, most materials have K < 2.

Acknowledgment to David Manura, Scientific Instrument Services, Inc., www.simion.com/cpo for initiating this.

See also:

test3d22 Spherical capacitor with dielectric in gap.

test3d23 Parallel plate capacitor with dielectric in gap.

test3d24 Field in a cavity inside a dielectric.

test3d26 Field inside a dielectric cylinder.

test3d27 Cylindrical capacitor with dielectric in gap.