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 RR size 12{R} {} is directly proportional to its length LL size 12{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, RR size 12{R} {} is inversely proportional to the cylinder’s cross-sectional area AA size 12{A} {}.

Figure 1: A uniform cylinder of length LL size 12{L} {} and cross-sectional area AA size 12{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 AA size 12{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 ρρ size 12{ρ} {} of a substance so that the resistance RR size 12{R} {} of an object is directly proportional to ρρ size 12{ρ} {}. Resistivity ρρ size 12{ρ} {} is an intrinsic property of a material, independent of its shape or size. The resistance RR size 12{R} {} of a uniform cylinder of length LL size 12{L} {}, of cross-sectional area AA size 12{A} {}, and made of a material with resistivity ρρ size 12{ρ} {}, is

Table 1 gives representative values of ρρ size 12{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.

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 100ºC100ºC size 12{"100"°C} {} or less), resistivity ρρ size 12{ρ} {} varies with temperature change ΔTΔT size 12{DT} {} as expressed in the following equation

where ρ0ρ0 size 12{ρ rSub { size 8{0} } } {} is the original resistivity and αα size 12{α} {} is the temperature coefficient of resistivity. (See the values of αα size 12{α} {} in Table 2 below.) For larger temperature changes, αα size 12{α} {} may vary or a nonlinear equation may be needed to find ρρ size 12{ρ} {}. Note that αα size 12{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 αα size 12{α} {} close to zero (to three digits on the scale in Table 2), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.

Figure 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.

Note also that αα size 12{α} {} is negative for the semiconductors listed in Table 2, 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 ρρ size 12{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 R0R0 size 12{R rSub { size 8{0} } } {} is directly proportional to ρρ size 12{ρ} {}. For a cylinder we know R=ρL/AR=ρL/A size 12{R=ρL/A} {}, and so, if LL size 12{L} {} and AA size 12{A} {} do not change greatly with temperature, RR size 12{R} {} will have the same temperature dependence as ρρ size 12{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 LL size 12{L} {} and AA size 12{A} {} is about two orders of magnitude less than on ρρ size 12{ρ} {}.) Thus,

is the temperature dependence of the resistance of an object, where R0R0 size 12{R rSub { size 8{0} } } {} is the original resistance and RR size 12{R} {} is the resistance after a temperature change ΔTΔT size 12{DT} {}. 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.

Figure 3: These familiar thermometers are based on the automated measurement of a thermistor’s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)

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.

Figure 4: Resistance in a Wire