Two nichrome wires of equal length (179 cm) but different diameters (0.643 mm (22 AWG, 0.0253″) and 0.404 mm (26 AWG, 0.0159″)) are mounted between banana plugs on the black board. Resistance measurements of equal lengths of the two wires, and different lengths of the same wire, show that resistance is inversely proportional to the cross-sectional area of the wire, and proportional to the length of the wire. A white stripe marks the midpoint of the two wires. If you wish, you can use a two-meter stick (not shown) to measure length.
The resistance of a piece of conductive material depends on its dimensions and on an intrinsic property of the material called its resistivity. Here we assume that the material is of uniform composition, and that its resistivity does not change with the voltage applied across it, in other words, that it behaves ohmically. The resistivity is the ratio of the electric field in the conductor to the current per unit cross section, or:
ρ = E/(i/A)
E is in volts per unit length. If we call the potentials at two points on the material, say the ends, Va and Vb, then E = (Va - Vb)/L, where L is the length of the material. If we call Va - Vb, the potential difference between the ends of the material, V, then
ρ = V/(Li/A).
According to Ohm’s Law (see demonstration 64.09 -- Ohm’s Law), V = iR, or V/i = R, where V is the voltage across the resistor (our piece of conductive material), i is the current through it, and R is its resistance, the unit of which is the Ohm. We can then write the equation above as
V/i = ρL/A,
from which we see that R = ρL/A. The units of ρ are volt·meters/ampere, which equal ohm·meters. Resistivity is frequently quoted in ohm·cm.
We see that the resistance of a piece of material is directly proportional to its length, and inversely proportional to its cross-sectional area. Increasing the length of the material adds resistance linearly, and is analogous to connecting resistors in series. Increasing the cross-sectional area of the material decreases the resistance in proportion to the increase in the area. Increasing the area through which current can flow is analogous to connecting resistors in parallel. (See demonstration 64.39 -- Resistors in series and parallel.)
As we see from the readout in the photograph, the resistance of the 179-cm length of 26-gauge nichrome wire between the banana jacks is 15.75 ohms. The resistance of the same length of 22-gauge wire is 6.13 ohms. Since the lengths are equal, this difference must be due to the difference in the cross-sectional area of the two wires. For the 26-gauge wire, this is π(0.0404 cm/2)2, or 1.28× 10-3 cm2, and for the 22-gauge wire, it is π(0.0643 cm/2)2, or 3.25× 10-3 cm2. The cross-sectional area of the 22-gauge wire is thus 3.25× 10-3/1.28 × 10-3 = 2.53 times as great as that of the 26-gauge wire. The resistance of the 26-gauge wire is 15.75/6.13 = 2.57 times as great as that of the 22-gauge wire. We see that the resistances of the two wires are in the inverse ratio of their cross-sectional areas.
If you measure the resistance of either wire between one end and any point along its length, the value you obtain is proportional to the resistance of the wire over its entire length, in the same proportion as the length over which you are measuring relative to the overall length of the wire. For example, the resistance of the 26-gauge wire from one end to its midpoint should measure 15.75 Ω/2, or 7.88 Ω, and of the 22-gauge wire from one end to its midpoint should measure 6.13 Ω/2, or 3.06 Ω.
From the above, we can calculate ρ = RA/L for the wire. From the resistance measurement for the 26-gauge wire, we have ρ = (15.75 Ω × 1.28× 10-3 cm2)/179 cm = 1.13 × 10-4 ohm·cm, and for the 22-gauge wire, ρ = (6.13 Ω × 3.25 × 10-3 cm2)/179 cm = 1.11 × 10-4 ohm·cm. These values are close to published values for the resistivity of nichrome. (See, for example, http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/rstiv.html. Note that this value is given in ohm·m.)
References:
1) Sears, Francis Weston and Zemansky, Mark W. College Physics, Third Edition (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1960), p541-543.