# 20.3 Resistance and resistivity  (Page 2/6)

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Resistivities $\rho$ Of various materials at $\text{20º}\text{C}$
Material Resistivity $\rho$ ( $\Omega \cdot \text{m}$ )
Conductors
Silver $1\text{.}\text{59}×{\text{10}}^{-8}$
Copper $1\text{.}\text{72}×{\text{10}}^{-8}$
Gold $2\text{.}\text{44}×{\text{10}}^{-8}$
Aluminum $2\text{.}\text{65}×{\text{10}}^{-8}$
Tungsten $5\text{.}6×{\text{10}}^{-8}$
Iron $9\text{.}\text{71}×{\text{10}}^{-8}$
Platinum $\text{10}\text{.}6×{\text{10}}^{-8}$
Steel $\text{20}×{\text{10}}^{-8}$
Lead $\text{22}×{\text{10}}^{-8}$
Manganin (Cu, Mn, Ni alloy) $\text{44}×{\text{10}}^{-8}$
Constantan (Cu, Ni alloy) $\text{49}×{\text{10}}^{-8}$
Mercury $\text{96}×{\text{10}}^{-8}$
Nichrome (Ni, Fe, Cr alloy) $\text{100}×{\text{10}}^{-8}$
Semiconductors Values depend strongly on amounts and types of impurities
Carbon (pure) $\text{3.5}×{\text{10}}^{5}$
Carbon $\left(3.5-\text{60}\right)×{\text{10}}^{5}$
Germanium (pure) $\text{600}×{\text{10}}^{-3}$
Germanium $\left(1-\text{600}\right)×{\text{10}}^{-3}$
Silicon (pure) $\text{2300}$
Silicon $\text{0.1–2300}$
Insulators
Amber $5×{\text{10}}^{\text{14}}$
Glass ${\text{10}}^{9}-{\text{10}}^{\text{14}}$
Lucite ${\text{>10}}^{\text{13}}$
Mica ${\text{10}}^{\text{11}}-{\text{10}}^{\text{15}}$
Quartz (fused) $\text{75}×{\text{10}}^{\text{16}}$
Rubber (hard) ${\text{10}}^{\text{13}}-{\text{10}}^{\text{16}}$
Sulfur ${\text{10}}^{\text{15}}$
Teflon ${\text{>10}}^{\text{13}}$
Wood ${10}^{8}-{10}^{14}$

## Calculating resistor diameter: a headlight filament

A car headlight filament is made of tungsten and has a cold resistance of $0\text{.}\text{350}\phantom{\rule{0.25em}{0ex}}\Omega$ . 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 $R=\frac{\mathrm{\rho L}}{A}$ 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 $R=\frac{\mathrm{\rho L}}{A}$ , is

$A=\frac{\mathrm{\rho L}}{R}\text{.}$

Substituting the given values, and taking $\rho$ from [link] , yields

$\begin{array}{lll}A& =& \frac{\left(5.6×{\text{10}}^{–8}\phantom{\rule{0.25em}{0ex}}\Omega \cdot \text{m}\right)\left(4.00×{\text{10}}^{–2}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}{\text{0.350}\phantom{\rule{0.25em}{0ex}}\Omega }\\ & =& \text{6.40}×{\text{10}}^{–9}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}\text{.}\end{array}$

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

$A=\frac{{\mathrm{\pi D}}^{2}}{4}\text{.}$

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

$\begin{array}{lll}D& =& \text{2}{\left(\frac{A}{p}\right)}^{\frac{1}{2}}=\text{2}{\left(\frac{6.40×{\text{10}}^{–9}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}}{3.14}\right)}^{\frac{1}{2}}\\ & =& 9.0×{\text{10}}^{–5}\phantom{\rule{0.25em}{0ex}}\text{m}\text{.}\end{array}$

Discussion

The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because $\rho$ 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 [link] .) 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 $\text{100º}\text{C}$ or less), resistivity $\rho$ varies with temperature change $\Delta T$ as expressed in the following equation

$\rho ={\rho }_{0}\left(\text{1}+\alpha \Delta T\right)\text{,}$

where ${\rho }_{0}$ is the original resistivity and $\alpha$ is the temperature coefficient of resistivity    . (See the values of $\alpha$ in [link] below.) For larger temperature changes, $\alpha$ may vary or a nonlinear equation may be needed to find $\rho$ . Note that $\alpha$ 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 $\alpha$ close to zero (to three digits on the scale in [link] ), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.

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