9.4: Resistivity and Resistance

Resistivity

When a voltage is applied to a conductor, an electrical field (vec{E}) is created, and charges in the conductor feel a force due to the electrical field. The current density (vec{J}) that results depends on the electrical field and the properties of the material. This dependence can be very complex. In some materials, including metals at a given temperature, the current density is approximately proportional to the electrical field. In these cases, the current density can be modeled as

[vec{J} = sigma vec{E},]

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where (sigma) is the electrical conductivity. The electrical conductivity is analogous to thermal conductivity and is a measure of a material’s ability to conduct or transmit electricity. Conductors have a higher electrical conductivity than insulators. Since the electrical conductivity is (sigma = J/E), the units are

[sigma = dfrac{|J|}{|E|} = dfrac{A/m^2}{V/m} = dfrac{A}{V cdot m}.]

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Here, we define a unit named the ohm with the Greek symbol uppercase omega, (Omega). The unit is named after Georg Simon Ohm, whom we will discuss later in this chapter. The (Omega) is used to avoid confusion with the number 0. One ohm equals one volt per amp: (1 , Omega = 1 , V/A). The units of electrical conductivity are therefore ((Omega cdot m)^{-1}).

Conductivity is an intrinsic property of a material. Another intrinsic property of a material is the resistivity, or electrical resistivity. The resistivity of a material is a measure of how strongly a material opposes the flow of electrical current. The symbol for resistivity is the lowercase Greek letter rho, (rho), and resistivity is the reciprocal of electrical conductivity:

[rho = dfrac{1}{sigma}.]

The unit of resistivity in SI units is the ohm-meter ((Omega cdot m). We can define the resistivity in terms of the electrical field and the current density.

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[rho = dfrac{E}{J}.]

The greater the resistivity, the larger the field needed to produce a given current density. The lower the resistivity, the larger the current density produced by a given electrical field. Good conductors have a high conductivity and low resistivity. Good insulators have a low conductivity and a high resistivity. Table (PageIndex{1}) lists resistivity and conductivity values for various materials.

Table (PageIndex{1}): Resistivities and Conductivities of Various Materials at 20 °C[1] Values depend strongly on amounts and types of impurities. Material Conductivity, (sigma) ((Omega cdot m)^{-1}) Resistivity, (rho) ((Omega cdot m)) Temperature Coefficient (alpha) ((^oC)^{-1}) Conductors Silver (6.29 times 10^7) (1.59 times 10^{-8}) 0.0038 Copper (5.95 times 10^7) (1.68 times 10^{-8}) 0.0039 Gold (4.10 times 10^7) (2.44 times 10^{-8}) 0.0034 Aluminum (3.77 times 10^7) (2.65 times 10^{-8}) 0.0039 Tungsten (1.79 times 10^7) (5.60 times 10^{-8}) 0.0045 Iron (1.03 times 10^7) (9.71 times 10^{-8}) 0.0065 Platinum (0.94 times 10^7) (10.60 times 10^{-8}) 0.0039 Steel (0.50 times 10^7) (20.00 times 10^{-8}) Lead (0.45 times 10^7) (22.00 times 10^{-8}) Manganin (Cu, Mn. Ni alloy) (0.21 times 10^7) (48.20 times 10^{-8}) 0.000002 Constantan (Cu, Ni alloy) (0.20 times 10^7) (49.00 times 10^{-8}) 0.00003 Mercury (0.10 times 10^7) (98.00 times 10^{-8}) 0.0009 Nichrome (Ni, Fe, Cr alloy) (0.10 times 10^7) (100.00 times 10^{-8}) 0.0004 Semiconductors [1] Carbon (pure) (2.86 times 10^{4}) (3.50 times 10^{-5}) -0.0005 Carbon ((2.86 – 1.67) times 10^{-6}) ((3.5 – 60) times 10^{-5}) -0.0005 Germanium (pure) (600 times 10^{-3}) -0.048 Germanium ((1 – 600) times 10^{-3}) -0.050 Silicon (pure) 2300 -0.075 Silicon 0.1 – 2300 -0.07 Insulators Amber (2.00 times 10^{-15}) (5 times 10^{14}) Glass (10^{-9} – 19^{-14}) (10^9 – 10^{14}) Lucite (< 10^{-13}) (> 10^{13}) Mica (10^{-11} – 10^{-15}) (10^{11} – 10^{15}) Quartz (fused) (1.33 times 10^{-18}) (75 times 10^{16}) Rubber (hard) (10^{-13} – 10^{-16}) (10^{13} – 10^{16}) Sulfur (10^{-15}) (10^{15}) TeflonTM (< 10^{-13}) (> 10^{13}) Wood (10^{-8} – 10^{-11}) (10^8 – 10^{11})

The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivity. Conductors have the smallest resistivity, and insulators have the largest; semiconductors have intermediate resistivity. Conductors have varying but large, free charge densities, whereas most charges in insulators are bound to atoms and are not free to move. Semiconductors are intermediate, having far fewer free charges than conductors, but having properties that make the number of free charges depend strongly on the type and amount of impurities in the semiconductor. These unique properties of semiconductors are put to use in modern electronics, as we will explore in later chapters.

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