test3d21, 21st benchmark test, field of point charges.

This test was superceeded in May 2007 by the addition of the option to import charges -see test3d27, mirror image force on a charged toner particle.

2 point charges are situated at distances of 0.25 and 0.75 mm from a flat sheet.

In fact the charges are not points but are rays of length L = 2E-3.

The magnitude of each charge is required to be the electronic charge Q = 1.602189E-19 Coulomb.

This is achieved by giving each ray an energy of 1 eV and a current I, where Q = I*L/V and where V is the electron velocity.

Therefore I = 4.751246E-5 mA (using electron mass 9.109534E-31).

In fact this current has to divided by 4 because the rays are reflected in the zx and zy planes, which effectively increases the point charge by a factor of 4, so I = 1.187811E-5 mA.

No space-charge iterations are used (because if they were to be used then in the second and later iterations the electrons would start in the potential field of the previous run, and this field is singular at the point charges).

To calculate the correct potentials, we use the fact there are effectively 2 'mirror' charges of magnitude -Q and at z = -0.25 and -0.75.

The potential in the xz plane is therefore the sum of 4 components +/-Q/(4*pi*epsilon0*r), where r is the distance from the point to a charge

or mirror charge. 1/(4*pi*epsilon0) = 8.987552E+9.

Therefore Q/(4*pi*epsilon0*1mm) = 1.439975E-6.

Using contour\potential\grid, the following results were obtained for a grid of points. A final column has been added here manually, giving the correct potentials. The agreement is within approximately 0.01% Higher accuracy would be obtained with more segments, a larger plate, shorter rays, etc.

x, y, z and potential: exact value

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0

0.00000E+00 0.00000E+00 2.50000E-01 -6.08596E-03 singular

0.00000E+00 0.00000E+00 5.00000E-01 -8.44832E-06 -8.44786E-06

0.00000E+00 0.00000E+00 7.50000E-01 -6.08788E-03 singular

0.00000E+00 0.00000E+00 1.00000E+00 -5.70554E-06 -5.70505E-06

2.50000E-01 0.00000E+00 0.00000E+00 0.00000E+00 0

2.50000E-01 0.00000E+00 2.50000E-01 -4.36330E-06 -4.36292E-06

2.50000E-01 0.00000E+00 5.00000E-01 -5.19514E-06 -5.19468E-06

2.50000E-01 0.00000E+00 7.50000E-01 -5.99241E-06 -5.99191E-06

2.50000E-01 0.00000E+00 1.00000E+00 -3.95064E-06 -3.95012E-06