Help finding volume please?

[Constant]

I have a point charge in a conducting sphere. The charge can be placed anywhere in the sphere & is acted on by the image charge effect & gravity. If the charge touches the side of the sphere within time t it is considered deposited.

The image charge is given by: F(x) = -q divΦ(x)

 

Φ = - q 1

4πε0r0 1-R2

 

r0 is the radius of the sphere. r is the radius at which the charge is at. And R= r/ r0

 

I’m trying to find the volume of the sphere within which the charge will definitely deposit in time t.

I have modelled this situation using excel. Due to symmetry I’m only modelling a quarter of the sphere, (the quarter encompassing the positive & negative z axes, & the positive x & y axis).

In my model I put in the initial position of the charge using x, y & z co-ordinates.

I’m stuck on which initial positions I need to put the charge in in order to find this volume? I know that this should be quite a simple thing to do but I have really confused myself with this!

Thanks

Edited March 28, 2010 by Constant
Trying to make equation clearer

[Amr Morsi]

First, I think that F(x) = -q gradΦ(x), not F(x) = -q divΦ(x), is the correct formula.

 

Second, you have the potential energy of the charge, which when added to the kinetic energy, gives the total energy E, which is a constant. The last equation is function of dr/dt, d(theta)/dt and r. Another equation for theta is

 

r^2 * d(theta)/dt=constant=a,

this equation is analogous to conservation of angular momentum.

 

Both differential equations can be solved to find r in terms of t. Try it yourself.