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Chapter 26: Problem 32

32\. The U.S. Electric Code The (United States) National Electric Code, which sets maximum safe currents for insulated copper wires of various diameters, is given (in part) in the table. Plot the safe current density as a function of diameter. Which wire gauge has the maximum safe current density? ("Gauge" is a way of identifying wire diameters, and \(1 \mathrm{mil}=10^{-3}\) in. $$ \begin{array}{lrrrrrrrr} \hline \text { Gauge } & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 \\ \text { Diameter, mils } & 204 & 162 & 129 & 102 & 81 & 64 & 51 & 40 \\ \text { Safe current, A } & 70 & 50 & 35 & 25 & 20 & 15 & 6 & 3 \\ \hline \end{array} $$

Short Answer

Expert verified
Wire gauge 18 has the maximum safe current density.

Step by step solution

01

Understanding the Given Data

Extract the diameters and safe currents from the table. The diameters are given in mils (1 mil = 10^{-3} inches) and the safe currents are in amperes (A).
02

Convert Diameter to Inches

Convert the diameters from mils to inches using the relation: \ 1 \text{ mil} = 10^{-3} \text{ inches}.\ For example, \( 204 \text{ mils} = 204 \times 10^{-3} \text{ inches} \).
03

Calculate the Cross-sectional Area

Assuming the wire is circular, compute the cross-sectional area using the formula for the area of a circle \( A = \frac{\text{蟺}d^2}{4} \), where \( d \) is the diameter in inches.
04

Calculate the Current Density

Calculate the current density using \( J = \frac{I}{A} \) (current over area). Substitute \( I \) (current in amperes) and \( A \) (cross-sectional area in square inches) for each wire gauge.
05

Create a Plot

Use a graphing tool to plot the safe current density as a function of the diameter.
06

Identify Maximum Current Density

Examine the plot to determine which wire gauge has the highest safe current density.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

National Electric Code
The National Electric Code (NEC) is a standard for ensuring electrical safety. It's a set of guidelines and regulations that detail safe electrical practices.
One vital aspect of the NEC is the specification of maximum safe currents for various types of wires.
This ensures that wires don't overheat and cause hazards.
Following the NEC helps prevent electrical fires and other dangers.
This code applies to buildings and other structures, ensuring they have safe and reliable electrical systems.
Safe Current
Safe current refers to the maximum amount of electrical current a wire can carry without overheating or causing damage.
When electricity flows through a wire, it generates heat. If the current exceeds the wire's safe limit, the wire can overheat.
Overheated wires may lead to insulation melting and potential fires.
Safe current ratings are determined based on factors such as wire material (often copper or aluminum) and insulation type.
It's crucial to choose the correct wire gauge for the intended electrical load to ensure safety.
Wire Gauge
Wire gauge is a measure of the wire diameter and is an important factor in electrical applications.
The American Wire Gauge (AWG) system is commonly used to denote wire sizes.
Smaller gauge numbers correspond to larger wire diameters.
For example, a 4-gauge wire is thicker than a 10-gauge wire.
Thicker wires can carry more current, making them suitable for high-power applications.
Conversely, thinner wires are used for lower current levels.
Choosing the right wire gauge is essential for maintaining safe current levels and ensuring efficient operation.
Cross-sectional Area
The cross-sectional area of a wire is the area of its circular face when sliced perpendicular to its length.
It can be calculated using the formula for the area of a circle: \ A = \frac{\pi d^2}{4}
where \(d\) is the diameter.
The cross-sectional area is important because it affects the wire's ability to carry current.
Larger cross-sectional areas provide more pathways for electrons, allowing higher current flow.
This is why wires with larger diameters can handle more current safely.
In electrical engineering, calculating the cross-sectional area helps in designing circuits to ensure they operate within safe limits.
Current Density Calculation
Current density is defined as the amount of electrical current flowing per unit area of the wire's cross-section.
It can be expressed with the formula: \ J = \frac{I}{A} , where \(J\) is current density, \(I\) is current, and \(A\) is the cross-sectional area.
High current density means more current is flowing through a smaller area, which can lead to overheating.
Calculating current density ensures that wires are used within their safe operating limits.
This is crucial in electrical engineering to prevent failures and maintain safety standards.
Proper calculation and adherence to guidelines like the NEC help in choosing the right wire for different applications.

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Most popular questions from this chapter

34\. Near Earth Near the Earth, the density of protons in the solar wind (a stream of particles from the Sun) is \(8.70 \mathrm{~cm}^{-3}\), and their speed is \(470 \mathrm{~km} / \mathrm{s}\). (a) Find the current density of these protons. (b) If the Earth's magnetic field did not deflect them, the protons would strike the planet. What total current would the Earth then receive?

\begin{array}{l} \text { 49. Increases Over Time The charge passing through a conductor }\\\ \text { increases over time as } q(t)=\left(1.5 \mathrm{C} / \mathrm{s}^{3}\right) t^{3}-\left(4.5 \mathrm{C} / \mathrm{s}^{2}\right) t^{2}+(2 \mathrm{C} / \mathrm{s}) t, \end{array} where \(t\) is in seconds. (a) What equation describes the current in the circuit as a function of time? (b) What is the current in the conductor at \(t=0.0 \mathrm{~s}\) and at \(t=1.0 \mathrm{~s}\) ?

50\. 1994 Honda Accord Consider a 1994 Honda Accord with a battery that is rated at 52 ampere-hours. This battery is supposed to be able to deliver 1 ampere of current to electrical devices in a car for at least 52 hours or 2 amperes for 26 hours, and so on. Suppose you leave the car lights turned on when you park the car and the car lights draw 20 amperes of current. How long will it be before your battery is dead?

25\. Nichrome Heater A Nichrome heater dissipates \(500 \mathrm{~W}\) when the applied potential difference is \(110 \mathrm{~V}\) and the wire temperature is \(800^{\circ} \mathrm{C}\). What would be the dissipation rate if the wire temperature were held at \(200^{\circ} \mathrm{C}\) by immersing the wire in a bath of cooling oil? The applied potential difference remains the same, and \(\alpha\) for Nichrome at \(800^{\circ} \mathrm{C}\) is \(4.0 \times 10^{-4} / \mathrm{K}\).

31\. A Beam A beam contains \(2.0 \times 10^{8}\) doubly charged positive ions per cubic centimeter, all of which are moving north with a speed of \(1.0 \times 10^{5} \mathrm{~m} / \mathrm{s}\). (a) What are the magnitude and direction of the current density \(\vec{J}\) ? (b) Can you calculate the total current \(i\) in this ion beam? If not what additional information is needed?
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