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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
All textbooksKnight Calc 5th EditionCh 27: Current and ResistanceProblem 64b
Chapter 27, Problem 64b

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.
Illustration of an aluminum wire with three segments, showing current direction and diameters of 2.0 mm and 1.0 mm.

Verified step by step guidance
1
Determine the formula for current density. The current density \( J \) is defined as the current \( I \) divided by the cross-sectional area \( A \) of the wire. Mathematically, \( J = \frac{I}{A} \).
Identify the current \( I \) in the wire. From the problem, the current in the top segment is given as \( I = 10 \; \text{A} \). This current is the same for all three segments since the wire is continuous and the current is conserved.
Calculate the cross-sectional area \( A \) for each segment. The cross-sectional area is related to the diameter \( d \) of the wire by the formula \( A = \pi \left( \frac{d}{2} \right)^2 \). Use the given diameters for each segment to compute \( A \).
Substitute the values of \( I \) and \( A \) into the formula \( J = \frac{I}{A} \) for each segment. Perform this calculation for all three segments to find the current density \( J \) for each one.
Organize the results into a table. The table should include the segment number, the diameter \( d \), the cross-sectional area \( A \), and the current density \( J \) for each segment.

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Key Concepts

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

Current Density

Current density (J) is defined as the amount of electric current flowing per unit area of a cross-section of a conductor. It is expressed in amperes per square meter (A/m²). The formula to calculate current density is J = I/A, where I is the current in amperes and A is the cross-sectional area in square meters. Understanding current density is crucial for analyzing how current distributes across different segments of a wire.
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Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. It is mathematically expressed as V = IR. This law is fundamental in understanding how voltage, current, and resistance interact in electrical circuits, which is essential for calculating current density in different wire segments.
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Cross-Sectional Area

The cross-sectional area of a wire is the area of its cut surface perpendicular to its length. It is a critical factor in determining the current density, as a larger cross-sectional area allows more current to flow through it without increasing the current density. The area can be calculated using geometric formulas depending on the wire's shape, such as A = πr² for a circular wire. This concept is vital for understanding how the physical dimensions of a wire affect its electrical properties.
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Related Practice
Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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