Test3d24.dat, 24th 'test' file for CPO3DS

Field in a cavity 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 inside a dielectric sphere.

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

of radius 5mm.

The sphere is nearly filled with a medium of dielectric constant K = 2.

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

A cavity of radius 0.05mm exists at the centre of the dielectric cylinder.

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

E_cavity = E_ext*3*K/(2*K + 1)

In the present example this gives E_cavity = 1.2V/mm.

More exactly, the gaps at the end of the dielectric cylinder cause a reduction. Also there are errors due to the limited number of segments. In the present simulation the internal field is 0.994946 (found by removing the cavity). Therefore the expected internal field is 1.19394.

It can be seen from the output file that the calculated internal field is 1.19397 at the centre of the cavity, giving an error of 0.003%.

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 |
12 |
1.19395 |
1.19394 |
0.001 |
0.001 |

2 |
48 |
1.19397 |
1.19394 |
0.003 |
0.003 |

2 |
192 |
1.19394 |
1.19394 |
0.0 |
0.0 |

2 |
48 |
1.19397 |
1.19394 |
0.003 |
0.003 |

3.5 |
48 |
1.30582 |
1.30584 |
0.002 |
0.002 |

10 |
48 |
1.42132 |
1.42125 |
0.005 |
0.007 |

1000 |
48 |
1.49135 |
1.49142 |
0.005 |
0.007 |

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 |
%abs error |
time, sec |

1.E-7 |
1.19397 |
1.19394 |
0.003 |
11.4 |

1.E-6 |
1.19397 |
1.19394 |
0.003 |
11.4 |

1.E-5 |
1.19396 |
1.19394 |
0.002 |
7.0 |

1.E-4 |
1.19364 |
1.19394 |
0.030 |
6.2 |

1.E-3 |
1.19382 |
1.19394 |
0.012 |
5.3 |

1.E-2 |
1.19634 |
1.19394 |
0.240 |
1.7 |

1.E-1 |
1.19531 |
1.19394 |
0.137 |
1.2 |

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.

See also:

test3d22 Spherical capacitor with dielectric in gap.

test3d23 Parallel plate capacitor with dielectric in gap.

test3d25 Field inside a dielectric sphere.

test3d26 Field inside a dielectric cylinder.

test3d27 Cylindrical capacitor with dielectric in gap.