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8.4: Resistance and Resistivity

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Learning Objectives

By the end of this section, you will be able to:

  • Explain the concept of resistivity.
  • Use resistivity to calculate the resistance of specified configurations of material.
  • Use the thermal coefficient of resistivity to calculate the change of resistance with temperature.

Material and Shape Dependence of Resistance

The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in Figure 1 is easy to analyze, and, by so doing, we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder’s electric resistance R is directly proportional to its length L, similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again similar to the flow of fluid through a pipe). In fact, R is inversely proportional to the cylinder’s cross-sectional area A.

A cylindrical conductor of length L and cross section A is shown. The resistivity of the cylindrical section is represented as rho. The resistance of this cross section R is equal to rho L divided by A. The section of length L of cylindrical conductor is shown equivalent to a resistor represented by symbol R.
Figure 8.4.1: A uniform cylinder of length L and cross-sectional area A. Its resistance to the flow of current is similar to the resistance posed by a pipe to fluid flow. The longer the cylinder, the greater its resistance. The larger its cross-sectional area A, the smaller its resistance.

For a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the resistivity ρ of a substance so that the resistance R of an object is directly proportional to ρ. Resistivity ρ is an intrinsic property of a material, independent of its shape or size. The resistance R of a uniform cylinder of length L, of cross-sectional area A, and made of a material with resistivity ρ, is R=ρLA.

The table below gives representative values of ρ. The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. 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 will be explored in later chapters.

Table 8.4.1 gives representative values of ρ. The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. 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 will be explored in later chapters.

Table 8.4.1: Resistivities ρ of Various materials at 20ºC
Material Resistivity ρ ( Ω⋅m )
Conductors
Silver 1.59x108
Copper 1.72x108
Gold 2.44x108
Aluminum 2.65x108
Tungsten 5.6x108
Iron 9.71x108
Platinum 10.6x108
Steel 20x108
Lead 22x108
Manganin (Cu, Mn, Ni alloy) 44x108
Constantan (Cu, Ni alloy) 49x108
Mercury 96x108
Nichrome (Ni, Fe, Cr alloy) 100x108
Semiconductors
Carbon (pure) 3.5x105
Carbon (3.56.0)x105
Germanium (pure) 600x103
Germanium (1600)x103
Silicon (pure) 2300
Silion 0.1−2300
Insulators
Amber 5x1014
Glass 1091014
Lucite >1013
Mica 10111015
Quartz (fused) 75x1016
Rubber (hard) 10131016
Sulfur 1015
Teflon >1013
Wood 1081011

Example 8.4.1:Calculating Resistor Diameter: A Headlight Filament

A car headlight filament is made of tungsten and has a cold resistance of 0.350Ω. If the filament is a cylinder 4.00 cm long (it may be coiled to save space), what is its diameter?

Strategy

We can rearrange the equation R=ρLA to find the cross-sectional area A of the filament from the given information. Then its diameter can be found by assuming it has a circular cross-section.

Solution

The cross-sectional area, found by rearranging the expression for the resistance of a cylinder given in R=ρLA, is

A=ρLR.

Substituting the given values, and taking ρ from Table 8.4.1, yields

A=(5.6×108Ωm)(4.00×102m)0.350Ω=6.40×109m2.

The area of a circle is related to its diameter D by

A=πD24.

Solving for the diameter D, and substituting the value found for A, gives

D=2(Ap)12=2(6.40×109m23.14)12=9.0×105m.

Discussion

The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because ρ is known to only two digits.

Temperature Variation of Resistance

The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See Figure 2.) Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal make more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about 100C or less), resistivity ρ varies with temperature change ΔT as expressed in the following equation ρ=ρ0(1+αΔT),

where ρ0 is the original resistivity and α is the temperature coefficient of resistivity. (See the values of α in the table below.) For larger temperature changes, α may vary or a nonlinear equation may be needed to find ρ. Note that α is positive for metals, meaning their resistivity increases with temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (which is made of copper, manganese and nickel), for example, has α close to zero (to three digits on the scale in the table), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.

A graph for variation of resistance R with temperature T for a mercury sample is shown. The temperature T is plotted along the x axis and is measured in Kelvin, and the resistance R is plotted along the y axis and is measured in ohms. The curve starts at x equals zero and y equals zero, and coincides with the X axis until the value of temperature is four point two Kelvin, known as the critical temperature T sub c. At temperature T sub c, the curve shows a vertical rise, represented by a dotted line, until the resistance is about zero point one one ohms. After this temperature the resistance shows a nearly linear increase with temperature T.
Figure 8.4.2: The resistance of a sample of mercury is zero at very low temperatures—it is a superconductor up to about 4.2 K. Above that critical temperature, its resistance makes a sudden jump and then increases nearly linearly with temperature.
Table 8.4.2: Tempature Coefficients of Resistivity α
Material Coefficient α(1/°C)
Conductors
Silver 3.8x103
Copper 3.9x103
Gold 3.4x103
Aluminum 3.9x103
Tungsten 4.5x103
Iron 5.0x103
Platinum 3.93x103
Lead 3.9x103
Manganin (Cu, Mn, Ni alloy) 0.000x103
Constantan (Cu, Ni, alloy) 0.002x103
Mercury 0.89x103
Nichrome (Ni, Fe, Cr alloy) 0.4x103
Semiconductors
Carbon (pure) 0.5x103
Germanium (pure) 50x103
Silicon (pure) 70x103

Note also that α is negative for the semiconductors listed in the table, meaning that their resistivity decreases with increasing temperature. They become better conductors at higher temperature, because increased thermal agitation increases the number of free charges available to carry current. This property of decreasing ρ with temperature is also related to the type and amount of impurities present in the semiconductors.

The resistance of an object also depends on temperature, since R0 is directly proportional to ρ. For a cylinder we know R=ρL/A, and so, if L and A do not change greatly with temperature, R will have the same temperature dependence as ρ. (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on L and A is about two orders of magnitude less than on ρ.) Thus, R=R0(1+αΔT)

is the temperature dependence of the resistance of an object, where R0 is the original resistance and R is the resistance after a temperature change ΔT. Numerous thermometers are based on the effect of temperature on resistance. (See Figure 3.) One of the most common is the thermistor, a semiconductor crystal with a strong temperature dependence, the resistance of which is measured to obtain its temperature. The device is small, so that it quickly comes into thermal equilibrium with the part of a person it touches.

A photograph showing two digital thermometers used for measuring body temperature.
Figure 8.4.3:These familiar thermometers are based on the automated measurement of a thermistor’s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)

Example 8.4.2:Calculating Resistance: Hot-Filament Resistance:

Although caution must be used in applying ρ=ρ0(1+αΔT) and R=R0(1+αΔT) for temperature changes greater than 100C, for tungsten the equations work reasonably well for very large temperature changes. What, then, is the resistance of the tungsten filament in the previous example if its temperature is increased from room temperature (20C) to a typical operating temperature of 2850C?

Strategy

This is a straightforward application of R=R0(1+αΔT), since the original resistance of the filament was givent o eb R0=0.350Ω, and the temperature change is ΔT=2830C.

Solution

The hot resistance R is obtained by entering known values into the above equation:

R=R0(1+αΔT)=(0.350Ω)[1+(4.5×103/C)]=4.8Ω

Discussion

This value is consistent with the headlight resistance example in 20.3.

PHET EXPLORATIONS: RESISTANCE IN A WIRE

Learn about the physics of resistance in a wire. Change its resistivity, length, and area to see how they affect the wire's resistance. The sizes of the symbols in the equation change along with the diagram of a wire.

PhET_Icon.png
Figure 8.4.4: Resistance in a Wire

Summary

  • The resistance R of a cylinder of length L and cross-sectional area A is R=ρLA, where ρ is the resistivity of the material.
  • Values of ρ in the Table show that materials fall into three groups—conductors, semiconductors, and insulators.
  • Temperature affects resistivity; for relatively small temperature changes ΔT, resistivity is ρ=ρ0(1+αΔT),where\(ρ0 is the original resistivity and α is the temperature coefficient of resistivity.
  • The Table gives values for α, the temperature coefficient of resistivity.
  • The resistance R of an object also varies with temperature: R=R0(1+αΔT), where R0 is the original resistance, and R is the resistance after the temperature change.

Footnotes

1 Values depend strongly on amounts and types of impurities

2 Values at 20°C.

Glossary

resistivity
an intrinsic property of a material, independent of its shape or size, directly proportional to the resistance, denoted by ρ
temperature coefficient of resistivity
an empirical quantity, denoted by α, which describes the change in resistance or resistivity of a material with temperature

Contributors and Attributions

Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).


This page titled 8.4: Resistance and Resistivity is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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