Ch 27: Current and Resistance

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 27: Current and Resistance

Problem 64d

Chapter 27, Problem 64d

An aluminum wire consists of the three segments shown in FIGURE P27.64. The current in the top segment is 10 A. For each of these three segments, find the drift velocity v_d. Place your results in a table for easy viewing.

Verified step by step guidance

Understand the problem: The drift velocity (v_d) of electrons in a conductor is given by the formula:

v

_d = InqA, where I is the current, n is the number density of conduction electrons, q is the charge of an electron, and A is the cross-sectional area of the wire. We need to calculate v_d for each segment of the aluminum wire using this formula.

Step 1: Identify the given values. The current (I) in the top segment is 10 A. For aluminum, the number density of conduction electrons (n) is approximately

2.5 × 10

²⁸ m^-3, and the charge of an electron (q) is

1.6 × 10

^-19 C. The cross-sectional area (A) for each segment will depend on the dimensions provided in the figure (e.g., diameter or radius).

Step 2: Calculate the cross-sectional area (A) for each segment. If the diameter (d) or radius (r) of the wire is given, use the formula for the area of a circle:

A = π r

². Ensure that the radius is in meters for consistency in SI units.

Step 3: Substitute the known values into the drift velocity formula for each segment. For example, for the top segment:

v

_d = 10(2.5 × 10²⁸)(1.6 × 10^-19)A. Repeat this calculation for the other two segments, using their respective cross-sectional areas.

Step 4: Organize the results into a table. For each segment, list the current (I), cross-sectional area (A), and calculated drift velocity (v_d). Ensure that the units are consistent and clearly labeled in the table.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Drift Velocity

Drift velocity is the average velocity that charged particles, such as electrons, attain due to an electric field. It is a crucial concept in understanding how current flows through a conductor. The drift velocity can be calculated using the formula v_d = I/(nAe), where I is the current, n is the charge carrier density, A is the cross-sectional area, and e is the charge of an electron.

Current

Current is the rate of flow of electric charge in a circuit, measured in amperes (A). In this context, the current in the aluminum wire segments affects the drift velocity of the electrons. Understanding how current relates to drift velocity is essential for calculating the drift velocity in each segment of the wire.

Conductivity and Resistivity

Conductivity is a measure of a material's ability to conduct electric current, while resistivity is its opposition to current flow. These properties are influenced by the material's composition and temperature. In the context of the aluminum wire, knowing the conductivity allows for the calculation of drift velocity, as it directly affects the relationship between current and electric field in the wire.

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Related Practice

Textbook Question

You've decided to protect your house by placing a 5.0-m-tall iron lightning rod next to the house. The top is sharpened to a point and the bottom is in good contact with the ground. From your research, you've learned that lightning bolts can carry up to 50 kA of current and last up to 50 μs. You don't want the potential difference between the top and bottom of the lightning rod to exceed 100 V. What minimum diameter must the rod have?

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Textbook Question

A 2.0-mm-diameter wire formed from a composite material has a resistivity that decreases with distance along the wire as ρ=ρ₀e^−αx, where ρ₀=4.0×10⁻⁵ Ω m, x (in m) is measured from one end of the wire, and the constant α=4.0 m⁻¹. What is the resistance of a 50-cm-long length of this wire?

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Textbook Question

You've decided to protect your house by placing a 5.0-m-tall iron lightning rod next to the house. The top is sharpened to a point and the bottom is in good contact with the ground. From your research, you've learned that lightning bolts can carry up to 50 kA of current and last up to 50 μs. How much charge is delivered by a lightning bolt with these parameters?

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Textbook Question

An aluminum wire consists of the three segments shown in FIGURE P27.64. The current in the top segment is 10 A. For each of these three segments, find the current density J. Place your results in a table for easy viewing.

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Textbook Question

What diameter should the nichrome wire in FIGURE P27.62 be in order for the electric field strength to be the same in both wires?

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Textbook Question

The two wires in FIGURE P27.63 are made of the same material. What is the electron drift speed in the 2.0-mm-diameter segment of the wire?

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An aluminum wire consists of the three segments shown in FIGURE P27.64. The current in the top segment is 10 A. For each of these three segments, find the drift velocity vd. Place your results in a table for easy viewing.

Verified video answer for a similar problem:

Key Concepts

Drift Velocity

Current

Conductivity and Resistivity

An aluminum wire consists of the three segments shown in FIGURE P27.64. The current in the top segment is 10 A. For each of these three segments, find the drift velocity v_d. Place your results in a table for easy viewing.