Test3d29.dat, 29th 'test' file for CPO3DS

Mirror image force on a charged toner particle

Here two new options are used, the importing charge option to set up a charged particle of finite size and the force option to calculate the net electrostatic force on the particle (or any defined set of segments).

The particle is a sphere if radius 5 microns that has its centre at 10 microns from a grounded plate, inside a grounded box.

This is the procedure for setting up the sphere as a charged particle:

(1) Copy the present data file.

(2) Edit the copy to remove all the electrodes except the sphere.

(3) Put the sphere at V = 1 and put the printing level at 'all'.

(4) Make sure that the 'zero total charge' option is disabled.

(5) Disable (temporarily) the option to import charges, which can be accessed from the bottom of the sheet /databuilder/segments/advanced options/.

(6) Run and then copy the ray output file tmp27a.dat to tempin.dat

(7) Run prog30, which will read tempin.dat and put the charges into tempout.dat.

(8) Rename tempout.dat as desired -we have used test27.dat, which is supplied with the CPO3D package.

(9) Go back to the present data file (which already names test27.dat as the file that holds the charges on the sphere) and run it. The force on the sphere will then be at the end of the ray output file.

(Note that the check for overlapping segments has been disabled, because the sphere touches the base plate, but if the geometry is changed the check should be restored, if possible.)

By zooming and doing a potential contour plot you will see that the contours around the sphere are modified by the presence of the induced charges on the plate.

In fact the field is the same as that between the sphere and a mirror image behind the plate (see any textbook).

The total charge Q on an isolated sphere of radius R and potential V is

Q = k*R*V,

where k = 4*pi*epsilon0 = 1.112650E-10.

The force between this sphere and a mirror image at a distance S away (measured between the centres of the spheres) is

F = (Q/S)**2/k = k*(R*V/S)**2

In the present case V = 1, S = 2*R, so F = 2.78162E-11 (SI units, Newtons).

The present simulation gives, using the force option, 2.7868E-11, in error by 0.2%. Higher accuracy would of course be obtained by using more segments.

The force can be made zero by adjusting the applied field E (that is, by adjusting V3). The distance between the base and top plates is 0.1mm, so E = V3*1.E4 V/m and the force due to this field is Q*E = k*R*V*E. Therefore this force cancels the mirror force when the magnitude of E

is V/(4*R), that is when V3 = -5. In fact the program gives cancellation when V3 = -5.00717, in error by 0.2%.

When V3 is more negative than this the toner particle will be drawn towards V3 (ignoring non-electrostatic forces).

The toner particle can be placed away from the plate (by using the 'transform' option to shift the first 5 electrodes in the -z direction or by shifting the sphere in the opposite direction (since the read-in charges move with their parent segments).

The 'scale' factor can be used to multiply the charge on the sphere, by x say, and then the required field is also multiplied by x.

There is an option to include the mirror image force in ray tracing.

Miscellaneous notes:

(1) A toner particle is of course made of dielectric material, but this does not affect the present simulation, because CPO3D is only aware of the surface charges and is not awar of how they were produced nor of the type of surface on which they are situated.

(2) The present sphere is only a crude approximation to a toner particle, since these are usually not spherical and not uniformly charged (see for example J. Q. Feng and D. A. Hays, Theory of electric field detachment of charged toner particles in electrophotography, J. of Imaging Science and Technology, vol 44 (2000) 19-25). Also there are non-electrostatic forces present.

(3) The charges on the individual segments of the sphere could be replaced by Q/N, where N is the total number of segments, assuming that all the segments have the same area. In fact in the CPO3D simulation they do not have exactly the same area, so the procedure given above is preferable (and is necessary for an accurate simulation of the mirror problem).

(4) When the particle is not uniformly charged then the segment charges would have to be obtained without using CPO3D (except that a dipole could be simulated by using applying a linear potential distribution).