Re: visualizing a non-potential

["John S. Denker" <jsd@MONMOUTH.COM>]

On 04/27/2003 05:07 PM, Bob Sciamanda wrote:

> It just occurred to me that there are cases where the representation of a
> CONSERVATIVE E field by continuous lines tangent to the field does not
> allow the interpretation of the line density (per unit perpendicular area)
> as a  proportional measure of the magnitude of E.  This occurs inside
> continuous charge distributions, where the divergence of E is non-zero.
> E.g.:  inside an isolated sphere of constant charge density (rho):
>
> E(r) = (rho*r)/(3*epsilon) so that the total "E-flux" piercing an interior
> sphere is (4*PI*r^2)*E(r) = (4*PI*rho)/(3*epsilon) * r^3 (just Gauss'
> law).
> So the E-flux is a function of r and cannot be represented by continuous
> lines.  To properly represent this flux, the NUMBER of lines must
> continuously increase with r, because of the continuous source rho.

I'm confused here.

Perhaps there is a confusion between
 a) electrical field lines, and
 b) equipotential lines.

Flux has to do with field lines, not equipotential lines.

a) Given a potential, equipotential lines can always be
constructed.  They represent the exterior derivative
of the potential.  They cannot start or end, according
to the theorem "the boundary of a boundary is zero".

  http://www.monmouth.com/~jsd/physics/thermo-forms.htm#item-bdybdy

b) Electrical field lines start and end on charges.  In
the case of a smoothly distributed charge density, this
is slightly (but only slightly) dodgy.