Re: [Phys-l] About drinkable water resistivity

[John Denker <jsd@av8n.com>]

On 07/04/2008 05:13 AM, CARABAJAL PEREZ, MARCIAL ROBERTO wrote:

> We observe that inside tank limits, resistivity practically didn't vary
> with distance among probes. What could be an acceptable model to explain
> this ?. 

It might be acceptable to treat it as a _physics_ problem.  
 A) That includes Ohm's law.
 B) That includes conservation of charge.
 C) That includes drawing the electric field lines, and treating the 
  voltage as the integral of the electric field.
 D) That also includes the use of scaling laws.
 *) et cetera.

Starting with item (D), we believe that _resistivity_ (not resistance
per se) will be an intrinsic and _intensive_ property of the liquid.
Intensive means that it scales like system volume to the zeroth power.
Resistivity has dimensions of resistance times length (for example
ohm-cm in the conventional not-quite-SI units).

When you measure the resistance of the sample, what do you expect?
Did you expect the the resistance to be proportional to the distance
between probes?  How are you going to make that work dimensionally?
You can't multiply ohm-cm by cm to get ohms.  That is, you can't
multiply the resistivity by the distance between probes to get the
resistance.

To make this more quantitative, consider the diagram at
  http://www.av8n.com/physics/img48/dipole-scaling.png

There are three parts.  The top part shows the paths taken by some of
the current.  We ignore the other current paths, because if we understand
the scaling of these paths, we also understand the scaling of the others.
At the end of the day, the voltage between the two electrods is the same,
independent of path.  These paths are, for practical purposes, electric
field lines, because of Ohm's law.

The middle part shows an approximation to the current paths i.e. field
lines.  This is less accurate, but more convenient, as we shall see in
a moment.  The inaccuracy is small, and does not affect the scaling.

The third pat can be viewed in two ways.  
 *) On the one hand, it is a copy of the middle part, with all the distances 
  scaled up by 20% (not including the size of the electrodes).
 *) On the other hand, it is an /unscaled/ copy of the middle part, with
  the magenta-shaded part just spliced in.

So the question is, what is the added resistance (dR) due to the spliced-in
piece?  To say the same thing another way, given a constant current, what is 
the voltage drop (dV) across the spliced-in piece?   Most importantly, when
the overall length-scale of the diagram is X, how does the spliced-in resistance 
scale with X?  

I find that the scaling is

                          dX
       dR  scales like  ------               [1]
                          X^2

where the denominator accounts for the fact that the spliced-in piece extends 
in the Y and Z directions by an amount proportional to X.

You can make this all fancy and quantitative by integrating equation [1], but
you don't really need to;  equation [1] as it stands is IMHO a sufficient
explanation of why dR is small when X is large.

===========

We now return to the fundamental scaling question:  Given a fixed resistivity,
if the system resistance does not scale like system size X, what do you think
it *does* scale like?

Huge hint:  Even though this is absolutely not an AC circuit, you might want
to review the formula for the one-terminal self-capacitance of an isolated
sphere.

=====================

Remark:  The D=2 version of this problem is also worth investigating (in contrast
to water, which is a D=3 system).  This can be done on a tabletop using resistor 
paper.  This is a standard high-school physics activity.  The D=2 mathematics is 
not quite as clean as equation [1], in particular the integral does not converge
if you push it all the way to infinity, but this isn't a major problem in practice.

==================================================

> Can we make relations with measuring earth resistivity problem

Yes.