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Chapter 24: Problem 43

The maximum safe current in 12-gauge (2.1-mm-diameter) copper wire is 20 A. Find (a) the current density and (b) the electric field under these conditions.

Short Answer

Expert verified
a) The current density in the wire is approximately 1.43 x 10^6 A/m^2. b) The electric field in the wire under these conditions is approximately 24 V/m.

Step by step solution

01

Find cross-sectional Area of the wire

Firstly, it's needed to calculate the cross-sectional area of the wire. Copper wire is circular in cross section, so the area \(A\) can be calculated using the formula \(A=πr^2\), where \(r\) is the radius of the wire, which is half the diameter given (2.1mm/2). Don't forget to convert mm to meters to keep units consistent.
02

Calculate current density

With the cross-sectional area and the current, the current density \(J\) can be calculated using the formula \(J=I/A\), where \(I\) is the current and \(A\) is the cross-sectional Area.
03

Calculate electric field

With the current density, the electric field \(E\) can be calculated using the formula \(E=ρJ\), where \(ρ\) is the resistivity of copper, and \(J\) is the current density. The resistivity of copper is a known constant (1.68 x 10^-8 Ohm.m).

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Most popular questions from this chapter

You're writing the instruction manual for a power saw, and you have to specify the maximum permissible length for an extension cord made from 18 -gauge copper wire (diameter \(1.0 \mathrm{mm}\) ). The saw draws \(7.0 \mathrm{A}\) and needs a minimum of \(115 \mathrm{V}\) across its motor when the outlet supplies \(120 \mathrm{V}\). What do you specify for the maximum length extension cord, given that they come in 25 -foot increments?

A brownout occurs when an electric utility can't supply enough power to meet demand. Rather than cut off some customers completely, the utility reduces the voltage across its system. Brownouts are most likely on hot summer days, when heavy air-conditioning loads drive up demand for electricity. In a particular brownout, the utility reduces the voltage by \(10 \%.\) Metallic conductors like lightbulb filaments and electric stove burners have resistance that increases with increasing temperature. During the brownout, the current in such devices a. decreases by \(10 \%.\) b. decreases by more than \(10 \%.\) c. decreases by less than \(10 \%.\) d. You can't tell without knowing more about how the resistance varies.

A brownout occurs when an electric utility can't supply enough power to meet demand. Rather than cut off some customers completely, the utility reduces the voltage across its system. Brownouts are most likely on hot summer days, when heavy air-conditioning loads drive up demand for electricity. In a particular brownout, the utility reduces the voltage by \(10 \%.\) During the brownout, the current in conductors whose resistance is nearly independent of temperature a. decreases by approximately \(10 \%.\) b. decreases by approximately \(20 \%.\) c. decreases by approximately \(5 \%.\) d. You can't tell without knowing the resistance.

A 4.5 -W flashlight bulb draws 750 mA. (a) At what voltage does it operate? (b) What's its resistance?

How must the diameters of copper and aluminum wire be related if they're to have the same resistance per unit length?
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