Re: [Phys-l] Photocopier toner

[John Denker <jsd@av8n.com>]

On 03/27/2009 09:34 AM, I wrote:

> [A dielectric particle] 
> would pick up an induced dipole in proportion to the
> electric field _gradient_ and would therefore go to the
> _edges_ of the image on the drum.  The result would be a 
> weird silhouette.


The conclusion is correct but I garbled the physics.  The
induced dipole is (as always) proportional to the field 
(not the gradient thereof).  The _force_ on the induced 
dipole goes roughly like the field gradient.  In fact it 
goes like the gradient of |field|^2.  

It is an amusing exercise to work this out.  Hint: electrostatic 
field energy + principle of virtual work.  

> This is not entirely hypothetical;  there is always some
> uncharged toner floating around, and it is a well-known
> imperfection of old and/or broken copiers that they tend
> to exaggerate edges in the image, making copies look
> "sharper" than the original.

The same method and the same result are valid for induced 
_magnetic_ dipoles, i.e. the force on a piece of non-permanent 
magnetic material in the field of a strong magnet.

As a specific sub-case, is useful and interesting to work out 
the force on an induced dipole in the field of a strong permanent
dipole.  The result is a power law, with an impressively large
exponent.  The student is unlikely to get the right answer by
guessing.  Also the answer clearly cannot be obtained by
dimensional analysis, so this is a fine example of non-dimensional
scaling.