Resistivity
The overall resistance (R) of a conductor depends on the length (l) and cross-sectional area (A) of the specimen as well as the resistivity (ρ) of the material (where ρ is the Greek letter rho).
.The SI unit of resistivity is the ohm metre (Ωm), which may be confirmed by transposing to make resistivity the subject of the expression and then analysing the dimensions of the terms on the right-hand side (a method known as dimensional analysis).
As might be expected, increasing the length of a conductor will increase its resistance, which is analagous to the resistors in series concept. Conversely, increasing the cross-sectional area has the effect of reducing the overall resistance, which is similar to resistors in parallel.
The resistivity depends on the chosen material, and on a number of other factors including the purity of the sample, its temperature, and even on the internal microscopic structure.
Manufacturing of electrical wires involves processes such as rolling of metal ingots to reduce their diameter, and drawing wire through a series of progressively narrower dies to reduce its diameter, while increasing the length. These mechanical forming processes cause the microscopic crystal structure to become elongated along the direction of extension, making the cable brittle. To overcome this problem, the metal is periodically heated to a sufficiently high temperature to allow it to recrystallise, but without melting. This annealing process restores the ductility of the metal, allowing it to bend or stretch without breaking. Multiple strand conductors are preferred where there is a need for flexibility in use, with single strand wire being more rigid, but offering lower resistance – and being cheaper to manufacture.
Copper and aluminium are the most widely used general purpose metallic conductors, with gold being employed on a small scale as a coating for electrical contacts, due to its non-corrosive properties. Typical resistivity values for pure samples are given in the table below:
Material | Resistivity (Ωm) | Data Source |
---|---|---|
Aluminium (pure) | 2.826×10^{-8} | |
Copper (pure, annealed) | 1.724×10^{-8} | |
Gold (pure) | 2.463×10^{-8} |
The reciprocal of resistivity is electrical conductivity (σ), which is measured in siemens per metre (S/m).
Worked Example 1
A wire of length 10 m and cross-sectional area 6 mm^{2} has a resistance of 0.1 Ω. Find the resistivity of the conducting material.
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First, remember to convert the cross-sectional area into m^{2} using the conversion factor 1 mm^{2} = 1 × 10^{−6} m^{2}. Hence, A = 6 × 10^{-6} m^{2}.
ρ = R × A ÷ l
= 0.1 × 6 × 10^{-6} ÷ 10
= 6 × 10^{-8} Ω m
Worked Example 2
Calculate the expected resistance of a circular aluminium cable of diameter 16 mm and length 10 km. Take the resistivity of aluminium to be 2.8 × 10^{-8} Ωm.
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Begin by converting the cable length l to metres and then calculate the cross-sectional area A in m^{2}.
l = 10 km = 10,000 m
The cross-sectional area of a circular cable is given by A = π r^{2} (see Areas of basic shapes for more details). Hence:
A = π × 8^{2} mm^{2}
= 64π × 10^{-6} m^{2}
The resistance R is given by R = ρ × l ÷ A.
= 2.8 × 10^{-8} × 10,000 ÷ (64π × 10^{-6})
= 1.4 Ω
Worked Example 3
If the resistance of a 50 m coil of wire is 4 Ω, find the resistance 80 m of the same cable.
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A simple way to solve this type of problem is to find the resistance per metre and then multiply by the new cable length.
R_{(1 m)} = 4 / 50 = 0.08 Ω/m
R_{(80 m)} = 0.08 × 80 = 6.4 Ω