(A) Calculate the resistance per unit length of a 22-gauge Nichrome wire, which has a radius of 0.321 mm.

(B) If a potential difference of 10 V is maintained across a 1.0-m length of the Nichrome wire, what is the current in the wire?

Question

(A) Calculate the resistance per unit length of a 22-gauge Nichrome wire, which has a radius of 0.321 mm.

(B) If a potential difference of 10 V is maintained across a 1.0-m length of the Nichrome wire, what is the current in the wire?

Accepted Answer

The cross-sectional area of this wire is

[latex] A=\pi r^{2}=\pi\left(0.321 imes 10^{-3} m ight)^{2}=3.24 imes 10^{-7} m ^{2} [/latex].

[latex] ext { The resistivity of Nichrome is } 1.5 imes 10^{-6} \Omega \cdot m ext { (see } [/latex] Table 27.1). Thus, we can use Equation 27.11 to find the resistance per unit length:

[latex] R= ho \frac{\ell}{A} [/latex] (27.11).

[latex] \frac{R}{\ell}=\frac{ ho}{A}=\frac{1.5 imes 10^{-6} \Omega \cdot m }{3.24 imes 10^{-7} m ^{2}}=4.6 \Omega / m [/latex].

(B) If a potential difference of 10 V is maintained across a 1.0-m length of the Nichrome wire, what is the current in the wire?

Because a 1.0-m length of this wire has a resistance of 4.6 Ω, Equation 27.8 gives

[latex] R \equiv \frac{\Delta V}{I} [/latex] (27.8).

[latex] I=\frac{\Delta V}{R}=\frac{10 V }{4.6 \Omega}=2.2 A [/latex].

Note from Table 27.1 that the resistivity of Nichrome wire is about 100 times that of copper. A copper wire of the same radius would have a resistance per unit length of only
0.052 Ω/m. A 1.0-m length of copper wire of the same radius would carry the same current (2.2 A) with an applied potential difference of only 0.11 V.
Because of its high resistivity and its resistance to oxidation, Nichrome is often used for heating elements in toasters, irons, and electric heaters.

Table 27.1
Resistivities and Temperature Coefficients of Resistivity for Various Materials
Temperature Coefficient [latex]\alpha\left[\left({ }^{\circ} C ight)^{-1} ight][/latex]	Resistivity (Ω(m)	Material
[latex]3.8 imes 10^{-3}[/latex]	[latex]1.59 imes 10^{-8}[/latex]	Silver
[latex]3.9 imes 10^{-3}[/latex]	[latex]1.7 imes 10^{-8}[/latex]	Copper
[latex]3.4 imes 10^{-3}[/latex]	[latex]2.44 imes 10^{-8}[/latex]	Gold
[latex]3.9 imes 10^{-3}[/latex]	[latex]2.82 imes 10^{-8}[/latex]	Aluminum
[latex]4.5 imes 10^{-3}[/latex]	[latex]5.6 imes 10^{-8}[/latex]	Tungsten
[latex]5.0 imes 10^{-3}[/latex]	[latex]10 imes 10^{-8}[/latex]	Iron
[latex]3.92 imes 10^{-3}[/latex]	[latex]11 imes 10^{-8}[/latex]	Platinum
[latex]3.9 imes 10^{-3}[/latex]	[latex]22 imes 10^{-8}[/latex]	Lead
[latex]0.4 imes 10^{-3}[/latex]	[latex]1.50 imes 10^{-6}[/latex]	Nichrome
[latex]-0.5 imes 10^{-3}[/latex]	[latex]3.5 imes 10^{-5}[/latex]	Carbon
[latex]-48 imes 10^{-3}[/latex]	0.46	Germanium
[latex]-75 imes 10^{-3}[/latex]	640	Silicon
	[latex]10^{10} ext { to } 10^{14}[/latex]	Glass
	[latex]\sim 10^{13}[/latex]	Hard rubber
	[latex] 10^{15}[/latex]	Sulfur
	[latex]75 imes 10^{16}[/latex]	Quartz (fused)

Question 27.3: (A) Calculate the resistance per unit length of a 22-gauge N...

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