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Chapter 24: Problem 20

A 1.0-cm-diameter rod carries a 50 - A current when the electric field in the rod is \(1.4 \mathrm{V} / \mathrm{m} .\) What's the resistivity of the rod material?

Short Answer

Expert verified
The resistivity of the rod's material is approximately \(0.0000022 \, \Omega.m\)

Step by step solution

01

Calculation of the Cross-Sectional Area of the Rod

Before we can calculate the current density, we will first determine the cross-sectional area of the rod. The rod is circular in shape, so its cross-sectional area \(A\) can be given by \(A = \pi r^2\), where \(r\) is radius of the rod. We're given the diameter, so we divide that by 2 to get the radius. Here, \(r = 1.0 \, cm / 2 = 0.5 \, cm = 0.005 \, m\). Substituting \(r\) into the cross-sectional area formula, we would get \(A = \pi (0.005\, m)^2 \approx 0.0000785 \, m^2\).
02

Calculation of the Current Density

Current density (\(J\)) is the current (\(I\)) divided by the cross-sectional area (\(A\)) through which the current is passing. Mathematically, it can be expressed as \(J = I/A\). We're given \(I = 50 \, A\), and we have calculated \(A\), so substituting these values, we get \(J = 50 \, A / 0.0000785 \, m^2 \approx 637324 \, A/m^2\).
03

Calculation of the Resistivity

Now we'll substitute our current density \(J\) and given electric field \(E = 1.4 \, V/m\) into the Ohm's Law equation \(J = \sigma E\), rearranging it to solve for the conductivity \(\sigma\), which gives us \(\sigma = J / E\). Substituting our values, we get \(\sigma = 637324 \, A/m^2 / 1.4 \, V/m = 454516 \, m/\Omega\). Resistivity \(\rho\) is the reciprocal of conductivity, hence \(\rho = 1/\sigma = 1 / 454516 \, m/\Omega = 0.0000022 \, \Omega.m\).

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Most popular questions from this chapter

An implanted pacemaker supplies the heart with 72 pulses per minute, each pulse providing \(6.0 \mathrm{V}\) for \(0.65 \mathrm{ms}\). The resistance of the heart muscle between the pacemaker's electrodes is \(550 \Omega\). Find (a) the current that flows during a pulse, (b) the energy delivered in one pulse, and (c) the average power supplied by the pacemaker.

A gold film in an integrated circuit measures \(2.5 \mu \mathrm{m}\) thick by \(0.18 \mathrm{mm}\) wide. It carries a current density of \(0.68 \mathrm{MA} / \mathrm{m}^{2}\) What's the total current?

You touch a defective appliance while standing on the ground, and you feel the tingle of a 2.5-mA current. What's your resistance, assuming you're touching the "hot" side of the \(120-\mathrm{V}\) household wiring?

A person and a cow are standing in a field when lightning strikes the ground nearby. Why is the cow more likely to be electrocuted?

The National Electrical Code specifies a maximum current of \(10 \mathrm{A}\) in 16 -gauge \((1.29-\mathrm{mm} \text { -diameter })\) copper wire. What's the corresponding current density?
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