Re: [Phys-l] Vector Field Diagram Conventions

[John Denker <jsd@av8n.com>]

On 07/12/2007 04:30 PM, Jeffrey Schnick wrote:
> [1] In Volume II section 1-2 of the Feynman Lectures on Physics, Feynman
> gives a couple of different schemes for representing a vector field
> graphically. 

OK.

> [2] In figure 1-2 he provides an example of a vector field
> diagram in which he shows field lines starting up in empty space (other
> than at the edges of the diagram).  He says that ..."it will require, in
> general, that new lines sometimes start up in order to keep the number
> up to the strength of the field." 

OK.

> [3] It is my understanding, in terms of
> the electric field, that new lines should start up only at positively
> charged particles/objects and/or at infinity. 

Yes.  J.C. Maxwell had something to say about that.

> Has the convention
> changed since the Feynman Lectures came out? 

No.  

One way out of the trap is to note that paragraph [2] refers to 
/some/ vector field, in all generality ... while paragraph [3] 
refers to _the_ electric field.

Another perfectly good way out of the trap is to note that
Feynman never said that the region in which the field lines
were observed was charge-free.

Thirdly, note that Feynman's figure 1-1 can be interpreted as a 
D=2 slice through a D=3 vector field, so even if the field in all 
its D=3 glory happens to be divergenceless (e.g. electric field 
in the absence of charges), the _projection_ of the field onto 
the plane of the slice need not be divergenceless.

As a concrete example, consider the projection onto the XY plane
of the field lines due to a point charge situated on the Z axis
at Z=-1, i.e. at the point (0,0,-1).

> Of those of you that teach
> students how to draw and/or interpret vector field diagrams that use
> field lines, do any of you adhere to the convention that field lines
> begin at empty points in space "in order to keep the number up to the
> strength of the field." ?


No, I don't quite adhere to that.  If you implicitly (or explicitly) 
assume that all vector fields are divergencless, then it robs two of 
the Maxwell equations of all physical significance.  The assertion 
that the electric field (in empty space) is divergenceless is not 
trivial.  

There are lots of vector fields that do have nonzero divergence.
Consider for example the flow of any non-conserved quantity.  
Specifically, consider the flow of gasoline molecules into the
cylinder of a piston engine.  Gasoline molecules flow in (via
the intake valve) and vastly fewer gasoline molecules flow out.