test3d30, mirror image force 2

Test3d30.dat, 30th 'test' file for CPO3DS

Including mirror image force in ray tracing

The geometry is the same as in test3d03, for convenience, but the applied voltages are zero.

The only forces are therefore those due to the attraction between a charged particles and its mirror image in a conducting electrode.

This option is accessed in /databuilder/sources of rays/advanced options/, at the bottom of the sheet.

When a particle of charge q is at a distance x from a conducting surface the attractive force is

F = -k/x**2,

where k = q**2/(4*pi*epsilon*(2*x)**2).

For example, when q = e, x = 1 micron, then F = E*q, where E = 0.34 V/mm.

This force is therefore usually negligible, except for systems that involve highly charged small composite particles that are near to surfaces (or touching them), such as carbon toner particles.

The trajectory starts parallel to a surface, at a distance d = 0.05mm away. It has charge 10000*e, mass 1000 AU and energy 100keV.

For a particle at a distance x from a surface the equation of motion is

d2x/dt2 = -f/x**2, where f = k/m.

Using dimensionless parameters, put s = x/x0, then

d2s/dt2 = -g/s**2, where g = f/x0**3.

Use the initial conditions s = 1, ds/dt = 0, t = 0.

Then the solution of the equation of motion is

h*t = pi/2 -arcsin(sqrt(s)) + sqrt(s*(1 - s)), where h = sqrt(2*g).

Therefore the time to move from x = x0 to x = 0 is tf = 0.5*pi/h.

For the present conditions, tf = 6.6633E-9 s.

The CPO3D program gives 6.67595E-06 ms, in error by 0.2%. A high energy has been used to ensure that ds/dt = 0 initially (for reasons not yet explored it seems that a high energy is necessary for this, although in principle tf should be independent of this energy).

As s reaches 0, ds/dt becomes infinite. The program has some difficulty in coping with this, which is why the program starts to use very short step lengths in this region (and it helps to have a high particle energy).