HandsOn 31  Resistance on a Fractal Network
Recall how one calculates the net resistance of two resistors in series or in parallel. We offer two different ways to think of resistance: one is electrical resistance, the other is resistance to the flow of water through pipes.
In Figure 8.4(a), two resistors are in series in a circuit with a battery. Resistors literally resist the flow of current. When one resistor follows another, the total resistance of the circuit is R_{1} + R_{2}.
Figure 8.4(b) shows a piston applying pressure to drive water through two porous rocks connected in series with pipes between them. If a constant force is applied to the piston, then the rate of flow of water through the porous rocks is determined by their resistances. Since the two rocks are in a row, the resistances of the two porous rocks are added together.
What is important about series resistances is that they add. If you double the length of a resistor, its resistance doubles. To describe how resistors add in series, we could say that the resistance increases with the total length of the resistors. So, for resistors in series, the total resistance is proportional to length, L, i.e., R L.
By contrast, Figures 8.5(a) and 8.5(b), show two resistors and two porous rocks in parallel. The piston applies the same force to each of the porous rocks. Each rock experiences the same water force as it would if it were alone, with no parallel rock alongside it. Each rock permits the passage of as much water as if it were alone. Taken together then, the two rocks pass twice as much water as either rock alone.
Twice as much water for the same force on the piston means that the two rocks in parallel have half the resistance to water flow as one rock alone. What is important is that the total resistance decreases as more resistors are placed in parallel. If two equal resistors are placed in parallel, their combined resistance is half of the resistance of either one alone.
We could also reason that placing resistors in parallel is equivalent to increasing the crosssectional area A through which current can flow. Since the resistance decreases with more resistors in parallel, and hence greater crosssectional area, we can say that R ^{1}/_{A}.
Combining
our result for series and parallel resistors, we conclude that for
ordinary objects the dependence of resistance R of material of crosssection
A and length L is

(8.8) 
What does this all have to do with fractal dimension?
Let's apply the logic of the preceding section to objects with different dimensions, as shown in Figures 8.6(a),(b), and (c). The wire in Figure 8.6(a) is effectively a onedimensional object, because we vary only its length and not its radius.

In particular, if we double the length of such a wire, its resistance
doubles. From this we conclude that for a onedimensional object,
R L = L^{1} (here we emphasize that the exponent is 1).
Now consider the twodimensional square sheet resistor in Figure 8.6(b). We treat it as twodimensional because throughout the following we hold its thickness t constant.

The crosssectional area of the sheet is A = tL. So, if we double
its length L and double its width L, then the resistance
remains constant since L cancels out in Eq. 8):

(8.9) 

Let's consider the change of resistance of a sheet as we increase
its width and length separately. If we double its length in the
direction of current flow, then the resistance of the sheet doubles.
This action is equivalent to treating the sheet as a onedimensional
object. In terms of water flow through the sheet, we have doubled
the distance over which the water must be driven, and hence doubled
the resistance to the flow of water.
On the other hand, if we double the width of the sheet in Figure 8.6(b), this is equivalent to adding an identical resistor in parallel. This cuts the resistance in half. Or, in terms of water flow, we have doubled the quantity of water that can flow through when the same force is applied to the piston, which means the resistance to fluid flow has been cut in half.
This combination of series and parallel resistance changes yields the surprising result that in two dimensions, the resistance of a square sheet (of a given thickness) is not dependent on the edge length L of the sheet, but only on the material (that is, on the value of its resistivity r). Equivalently, for flow through a slab of uniform twodimensional rock, the resistance is not dependent on the width or length of the slab!

Finally, let's consider the resistance of a threedimensional cube
as in Figure 8.6(c). If we double the cube's length in the direction
of fluid or electric flow (making it no longer a cube but something
called "a rectangular parallelepiped''), this doubles its resistance.
If we double its width (equivalent to putting two of the new parallelepipeds
in parallel), we halve its resistance. Finally, if we double the
height of the cube, equivalent to putting four more parallelepipeds
in parallel, we halve the resistance again. The result is a new
cube with resistance


All
of the above results regarding resistance in 1, 2, and 3dimensional
objects (with L describing the size of the object), can be summarized
by writing

(8.10) 

Now we are experts on the how resistance behaves for objects with
integer dimensions. We have attempted to understand the behavior
of such objects using simple arguments about resistors in series
and parallel.
But what about objects which have a more complicated geometry than that described by integer dimensions? What about fractal objects with noninteger dimensions? For example, what is the resistance of the Sierpinski gasket measured between a variety of points on its structure?
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