/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 A wire \(l=8 \mathrm{~m}\) long ... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 31

A wire \(l=8 \mathrm{~m}\) long of uniform cross-sectional area \(A=8 \mathrm{~mm}^{2}\) has a conductance of \(G=2.45 \Omega^{-1}\). The resistivity of material of the wire will be (A) \(2.1 \times 10^{-7} \Omega \mathrm{m}\) (B) \(3.1 \times 10^{-7} \Omega \mathrm{m}\) (C) \(4.1 \times 10^{-7} \Omega \mathrm{m}\) (D) \(5.1 \times 10^{-7} \Omega \mathrm{m}\)

Short Answer

Expert verified
The resistivity of the material of the wire is approximately \(4.1 \times 10^{-7}~惟m\). The correct answer is (C) \(4.1 \times 10^{-7} \Omega \mathrm{m}\).

Step by step solution

01

Determine the Resistance

First, we need to find the resistance of the wire. We know: \(G = 2.45~惟^{-1}\) Because conductance is the reciprocal of resistance, we can find the resistance by inverting the given conductance value: \(R = \frac{1}{G}\) Calculate the resistance: \(R = \frac{1}{2.45} \approx 0.4082~惟\)
02

Use the Resistance formula to find Resistivity

Now that we have the resistance value, we can use the formula \(R = \frac{\rho L}{A}\) to solve for the resistivity. Rearrange this formula to isolate the resistivity term, \(\rho\): \(\rho = \frac{RA}{L}\) Plug in the given values for resistance, length and cross-sectional area, noting that we need to convert the cross-sectional area from square millimeters (\(mm^2\)) to square meters (\(m^2\)): \(R = 0.4082~惟\) \(L = 8~m\) \(A = 8~mm^2 = 8 \times 10^{-6}~m^2\) Now, calculate the resistivity (\(\rho\)): \(\rho = \frac{(0.4082~惟)(8 \times 10^{-6}~m^2)}{8~m} \approx 4.1 \times 10^{-7}~惟m\) The resistivity of the material of the wire is approximately \(4.1 \times 10^{-7}~惟m\). The correct answer is (C) \(4.1 \times 10^{-7} \Omega \mathrm{m}\).

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