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> Resistance Between Two Points In A Solid Medium
Particle
Posted: February 25, 2011 11:41 pm
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I'm curious. Every day, we all calculate the resistance of conductors. Usually, it's in the form of either insulators or wires. The impedance of a given length of wire is an easily obtainable quantity. There's one decent path for the charge to conduct along. It's essentially a one-dimensional problem.

What about when the media is solid instead of a wire, bar, or other easily conceivable "path". Let's say you live in a ball of iron, somehow, that is 10 miles in diameter. You are at the very center of this perfect sphere of pure elemental iron. What is the resistance between two points whose line passes through the center and are both equidistant from that center and say 3 feet apart from each other?

This was spawned out of a natural progression of thought such as when considering a wire/conductor that is suitably large as opposed to what attaches to it. Say you have two TO-220 devices where one leg of each is connected to the other by a 1" diameter copper wire. I somehow doubt the edges of the outside of the cylinder are particularly useful in this case. Conduction isn't a simple matter of a basic path.
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CWB
Posted: February 26, 2011 01:44 am
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it depends on the frequency of the applied voltage .
general rule of thumb : as frequency goes up , current is conducted more and more towards and on the surface of a conductor .
google : "skin effect" .

interesting thought about a "ball of iron" .


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johansen
Posted: February 26, 2011 02:39 am
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QUOTE (Particle @ February 25, 2011 06:41 pm)
I'm curious.  Every day, we all calculate the resistance of conductors.  Usually, it's in the form of either insulators or wires.  The impedance of a given length of wire is an easily obtainable quantity.  There's one decent path for the charge to conduct along.  It's essentially a one-dimensional problem.

What about when the media is solid instead of a wire, bar, or other easily conceivable "path".  Let's say you live in a ball of iron, somehow, that is 10 miles in diameter.  You are at the very center of this perfect sphere of pure elemental iron.  What is the resistance between two points whose line passes through the center and are both equidistant from that center and say 3 feet apart from each other?

This was spawned out of a natural progression of thought such as when considering a wire/conductor that is suitably large as opposed to what attaches to it.  Say you have two TO-220 devices where one leg of each is connected to the other by a 1" diameter copper wire.  I somehow doubt the edges of the outside of the cylinder are particularly useful in this case.

Conduction isn't a simple matter of a basic path.

sure it is.
you just have to figure out how much current is flowing where.

in the case of a large diameter ball of iron...

try modeling it with something that's a really poor conductor
like tap water with one drop of h2so4 per gallon of water, using steel for electrodes.

i think you'll find the answer to your question depends on the surface area of what you consider "points"

in the case of the transistor soldered to a copper bar (more common scenario)
you'll find the resistance IRL to be dominated by the solder, and the 5mm long strip of .5mm^2 wire

once the current gets to the bar it will spread out evenly, just as the magnetic flux lines would if you had placed a bar magnet on a block of steel*
*assuming that the permeability of air was zero, and that the lines would only flow through the iron.


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sherlock ohms
Posted: February 26, 2011 11:47 am
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seems to me to be an infinitely complex equation, ..as soon as a 'preferred' current path is chosen, resistance increases, therefore creating a 'more preferred' option.


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tekwiz
Posted: February 26, 2011 06:59 pm
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Village Idiot
Posted: February 26, 2011 10:32 pm
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Maxwell sorted this out in the mid 1800's. It's covered in Vol. 1 of "A Treatise on Electricity and Magnetism" which is quite readable BTW.

Nowadays we use the same calculations to determine how ground fault currents flow from ground rods through the soil. Many years ago, I wrote a program to model voltage gradients of buried ground grids, because we were doing a lot of electrical substation design at work, and needed to determine voltages in the soil (and at the surface) during ground fault conditions.

It's all a bit foggy in my mind at the moment, but IIRC the resistance between two infinitesimal points is going to be infinite, or at least indeterminate. You have to specify a finite electrode size.
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Sch3mat1c
Posted: February 28, 2011 03:32 pm
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In the limit of an infinite, uniform body around a finitely sized set of electrodes, you get the electric field (and therefore current flow, if the material is conductive) equal to the coulomb potential (the inverse square law) around each electrode. The total is the superposition of both, so that one is positive, the other is negative, voltage is zero in the middle and at infinity, etc. The actual field lines are complicated, because they aren't going straight from one electrode to the other, but curve in an elliptical manner (or hyperbolic, for those lines which run out to infinity). Obviously, integrating along such field lines can become a pain, but it's not an impossible problem.

VI is correct: the electric field at an infinnetessimal point is infinite (one good reason why electrons aren't infinnetessimal particles!), so you need to define size. If the diameter is much less than the distance between points, you can approximate each electrode locally with a spherical Gaussian surface. (As the electrodes get closer, the unipotential surface goes from a round sphere towards a stretched, squashed oblongated thing, until it becomes a perfectly flat surface in the middle, where the voltage is zero.)

Tim


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