Material and Shape Dependence of Resistance

In the previous section, we learned that resistance determines how much a material opposes the flow of electric current. But what determines the resistance of an object? Two factors are important:

• The geometry of the object (its length and cross-sectional area)
• The material from which it is made

The uniform cylindrical conductor shown in Figure 21.1 provides a simple model for understanding these effects. Although real conductors come in many shapes, this model applies well to wires, ECG leads, stimulation electrodes, printed circuit traces, and many other electrical components.

As you might expect, a longer conductor has a larger resistance because charge carriers must travel farther and experience more collisions with atoms along the way. Likewise, a thicker conductor has a smaller resistance because it provides more space for charge carriers to move simultaneously. This is similar to water flowing through a pipe: long pipes offer more resistance to flow, while wider pipes allow water to flow more easily.

The material itself also plays an important role. Some materials allow charges to move easily, while others strongly oppose their motion. This property is described by the material's resistivity, denoted by [latex]\rho[/latex]. Resistivity is an intrinsic property, meaning it depends only on the material—not on the object's size or shape.

For a uniform cylindrical conductor of length [latex]L[/latex], cross-sectional area [latex]A[/latex], and resistivity [latex]\rho[/latex], the resistance is

This equation summarizes three important relationships:

• Increasing the length increases the resistance.
• Increasing the cross-sectional area decreases the resistance.
• Using a material with a larger resistivity increases the resistance.

These relationships have many practical applications in medicine and biology. For example, ECG leads and power cables are made relatively thick to reduce resistance and heating, while the thin tungsten filament inside an incandescent light bulb is intentionally designed with a high resistance so that it becomes hot enough to emit visible light. Likewise, engineers carefully select electrode materials used in medical devices to control both electrical resistance and heat generation where the electrode contacts the skin.

Representative values of resistivity at [latex]20^\circ\text{C}[/latex] are listed in Table 21.1. Materials are commonly grouped into three categories:

• Conductors, which have very low resistivity because they contain many mobile charge carriers.
• Semiconductors, whose resistivity lies between conductors and insulators and can change dramatically with temperature or with the addition of impurities (doping).
• Insulators, which have extremely high resistivity because their electrons are tightly bound and cannot move freely.

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Temperature Variation of Resistance

The resistivity of every material depends on temperature. In most everyday electrical circuits, increasing the temperature increases the resistance of metals. The reason is microscopic: as the temperature rises, the atoms in the metal vibrate more vigorously about their equilibrium positions. Electrons moving through the material collide more frequently with these vibrating atoms, making it more difficult for current to flow.

For relatively small temperature changes (typically less than about
[latex]100^\circ\text{C}[/latex]), the resistivity of many materials changes approximately linearly with temperature. This relationship is described by

[latex]\rho=\rho_0\left(1+\alpha\Delta T\right)[/latex]

where

• [latex]\rho[/latex] is the resistivity at the new temperature,
• [latex]\rho_0[/latex] is the resistivity at a reference temperature (usually [latex]20^\circ\text{C}[/latex]),
• [latex]\Delta T[/latex] is the change in temperature, and
• [latex]\alpha[/latex] is the temperature coefficient of resistivity.

The coefficient [latex]\alpha[/latex] tells us how sensitive a material's resistivity is to temperature. A large positive value means the resistivity increases rapidly as temperature rises. For larger temperature changes, however, this simple linear relationship is only an approximation because [latex]\alpha[/latex] itself may vary with temperature.

Most metals have positive values of [latex]\alpha[/latex], meaning they become poorer conductors as they get hotter. Some alloys, such as manganin and constantan, have values of [latex]\alpha[/latex] that are extremely close to zero. Their resistance changes very little with temperature, making them useful in precision electrical instruments where a stable resistance is essential.

Table 21.2. Temperature Coefficients of Resistivity [latex]\alpha[/latex]

Material	Coefficient [latex]\alpha[/latex] (1/°C)
Conductors
Silver	[latex]3.8\times10^{-3}[/latex]
Copper	[latex]3.9\times10^{-3}[/latex]
Gold	[latex]3.4\times10^{-3}[/latex]
Aluminum	[latex]3.9\times10^{-3}[/latex]
Tungsten	[latex]4.5\times10^{-3}[/latex]
Iron	[latex]5.0\times10^{-3}[/latex]
Platinum	[latex]3.93\times10^{-3}[/latex]
Lead	[latex]3.9\times10^{-3}[/latex]
Manganin (Cu, Mn, Ni alloy)	[latex]0.000\times10^{-3}[/latex]
Constantan (Cu, Ni alloy)	[latex]0.002\times10^{-3}[/latex]
Mercury	[latex]0.89\times10^{-3}[/latex]
Nichrome (Ni, Fe, Cr alloy)	[latex]0.4\times10^{-3}[/latex]
Semiconductors
Carbon (pure)	[latex]-0.5\times10^{-3}[/latex]
Germanium (pure)	[latex]-50\times10^{-3}[/latex]
Silicon (pure)	[latex]-70\times10^{-3}[/latex]

Notice that the semiconductors listed in Table 21.2 have negative values of [latex]\alpha[/latex]. Their resistivity decreases as temperature increases, meaning they become better conductors when heated. Unlike metals, where increasing temperature mainly increases collisions, semiconductors respond to heating by freeing additional charge carriers. The increase in available charge carriers more than compensates for the additional collisions, so the overall resistance decreases.

Since the resistance of a cylindrical conductor is

[latex]R=\frac{\rho L}{A},[/latex]

and the dimensions of the conductor usually change only slightly with temperature, resistance follows the same temperature dependence as resistivity:

[latex]R=R_0\left(1+\alpha\Delta T\right)[/latex]

where [latex]R_0[/latex] is the resistance at the reference temperature. For most engineering applications, changes in length and cross-sectional area due to thermal expansion are much smaller than the change caused by the resistivity itself, so they are usually neglected.

The temperature dependence of resistance is extremely useful in sensing applications. One of the most common examples is the thermistor, a resistor whose resistance changes dramatically with temperature. Because thermistors are small, they rapidly come into thermal equilibrium with their surroundings, making them ideal temperature sensors.

In healthcare, thermistors are widely used in digital thermometers, incubators, respiratory monitoring equipment, and many medical devices that require accurate temperature measurements. By measuring the resistance of the thermistor, the electronics can determine the patient's temperature with excellent precision.

Interactive Exploration: Resistance in a Wire

The resistance of a conductor depends on both the material it is made from and its geometry. In this simulation, you will investigate how the resistance of a wire changes as you vary its length, cross-sectional area, and material. By experimenting with these variables, you will develop an intuitive understanding of the resistance equation introduced in this section.

Change one property of the wire at a time while observing the calculated resistance. As you explore, relate your observations to the equation
[latex]R=\dfrac{\rho L}{A}[/latex]
and consider why different materials and wire dimensions are used for different electrical applications.