Material and Shape Dependence of Resistance

The resistance of an object depends on both its shape and the material it is made from. The uniform cylindrical conductor in
Figure 21.1 is a useful model because it is easy to analyze, and it helps us understand more complicated shapes (wires, electrodes, leads, and conductive traces).
As you might expect, the resistance [latex]R[/latex] of a cylinder is directly proportional to its length [latex]L[/latex]:
the longer the path, the more collisions charge carriers make as they drift.
The cylinder’s resistance is also inversely proportional to its cross-sectional area [latex]A[/latex]:
a thicker wire provides more “room” for charge carriers to move, allowing more current for the same voltage.
This is similar to fluid flow—longer pipes resist flow more, while wider pipes resist flow less.

For a given shape, resistance also depends on the material. Different materials oppose charge flow by different amounts.
We define the resistivity [latex]\rho[/latex] of a material so that the resistance [latex]R[/latex] of an object is directly proportional to [latex]\rho[/latex].
Resistivity is an intrinsic material property—independent of the object’s size or shape.
For a uniform cylinder of length [latex]L[/latex] and cross-sectional area [latex]A[/latex], made from a material with resistivity [latex]\rho[/latex], the resistance is

This equation captures three practical ideas you will use repeatedly:
(1) longer conductors have greater resistance,
(2) thicker conductors have smaller resistance,
and (3) the material matters through [latex]\rho[/latex].
In biomedical contexts, these same dependencies help explain why a thin lead wire heats more easily than a thick one for the same current,
and why electrode materials are carefully selected to control resistance and heating at the skin–electrode interface.

Table 21.1 lists representative resistivities at [latex]20^\circ\text{C}[/latex].
The table groups materials as conductors, semiconductors, and insulators based on the typical size of [latex]\rho[/latex].
Conductors have small resistivity because they have many mobile charge carriers.
Insulators have extremely large resistivity because most charges are bound to atoms and cannot move freely.
Semiconductors fall in between; their charge-carrier density (and therefore resistivity) can change dramatically with temperature and with added impurities (doping)—a key idea behind modern electronics.

Temperature Variation of Resistance

The resistivity of all materials depends on temperature. Some materials become superconductors (zero resistivity) at very low temperatures. More commonly in everyday circuits, the resistivity of metals increases as temperature increases. At higher temperatures, atoms in a metal lattice vibrate more vigorously, and drifting electrons undergo more frequent collisions—this raises resistivity.
For relatively small temperature changes (roughly [latex]100^\circ\text{C}[/latex] or less), resistivity often changes approximately linearly with temperature:

Here [latex]\rho_0[/latex] is the resistivity at the reference temperature (often [latex]20^\circ\text{C}[/latex]), [latex]\Delta T[/latex] is the temperature change, and [latex]\alpha[/latex] is the temperature coefficient of resistivity.
For larger temperature changes, the relationship can become nonlinear because [latex]\alpha[/latex] itself may vary with temperature.
For metals, [latex]\alpha[/latex] is typically positive, so resistivity increases with temperature.
Some alloys (such as manganin) are engineered to have [latex]\alpha[/latex] close to zero, which makes them useful when a nearly constant resistance is needed.

Notice that [latex]\alpha[/latex] is negative for the semiconductors listed in Table 21.2. This means their resistivity decreases as temperature increases: they become better conductors at higher temperature. The reason is that thermal energy can free more charge carriers, increasing the number available to carry current. This behavior is also strongly influenced by impurities (doping), which is why semiconductor properties can be engineered for specific devices.

Because the resistance of a cylinder is [latex]R=\rho L/A[/latex], resistance changes with temperature in the same way that resistivity changes—provided the dimensions [latex]L[/latex] and [latex]A[/latex] do not change much. (Thermal expansion effects are usually much smaller than resistivity effects for modest temperature changes.) Therefore, for many materials and moderate temperature ranges:

Many thermometers use this temperature dependence of resistance. A common example is the thermistor, typically made of a semiconductor with a strong temperature dependence. A thermistor is small, so it quickly comes into thermal equilibrium with whatever it touches—useful for measuring body temperature (see Figure 21.3).