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1. Intro to Physics Units1h 29m
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24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
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30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

27. Resistors & DC Circuits

Intro to Current

Physics

27. Resistors & DC Circuits

Intro to Current

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Problem 14

Textbook Question

The current in a 2.0 mm x 2.0 mm square aluminum wire is 2.5 A. What are (a) the current density and (b) the electron drift speed?

Verified step by step guidance

To calculate the current density \( J \), use the formula \( J = \frac{I}{A} \), where \( I \) is the current and \( A \) is the cross-sectional area of the wire. First, calculate the area of the square cross-section: \( A = \text{side}^2 = (2.0 \times 10^{-3} \ \text{m})^2 \).

Substitute the given current \( I = 2.5 \ \text{A} \) and the calculated area \( A \) into the formula \( J = \frac{I}{A} \) to find the current density.

To calculate the electron drift speed \( v_d \), use the formula \( v_d = \frac{I}{n e A} \), where \( n \) is the number density of electrons, \( e \) is the elementary charge \( (1.6 \times 10^{-19} \ \text{C}) \), and \( A \) is the cross-sectional area of the wire.

For aluminum, the number density of conduction electrons \( n \) is approximately \( 6.02 \times 10^{28} \ \text{electrons/m}^3 \). Substitute \( I = 2.5 \ \text{A} \), \( n = 6.02 \times 10^{28} \ \text{electrons/m}^3 \), \( e = 1.6 \times 10^{-19} \ \text{C} \), and the previously calculated \( A \) into the formula \( v_d = \frac{I}{n e A} \).

Simplify the expression to find the electron drift speed \( v_d \). Note that the drift speed is typically very small, as the number density of electrons is very large.

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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 is defined as the amount of electric current flowing per unit area of a conductor. It is calculated using the formula J = I/A, where J is the current density, I is the current, and A is the cross-sectional area. In this case, the area of the square aluminum wire can be determined by squaring its side length, allowing for the calculation of current density in units of A/m².

Electron Drift Speed

Electron drift speed refers to the average velocity that a charge carrier, such as an electron, attains due to an electric field. It is calculated using the formula v_d = J/(nq), where v_d is the drift speed, J is the current density, n is the number density of charge carriers, and q is the charge of an electron. Understanding this concept is crucial for determining how quickly electrons move through the conductor under the influence of the applied current.

Aluminum Properties

Aluminum is a conductive material with specific properties that affect its electrical behavior, including resistivity and the number density of free electrons. The resistivity of aluminum is relatively low, making it an efficient conductor. Knowing the properties of aluminum helps in calculating both current density and electron drift speed, as these calculations depend on the material's characteristics and the number of charge carriers available.

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Multiple Choice

A wire is connected to the positive and negative plates of a capacitor. Electrons in the wire feel an attraction toward the positive plate of a capacitor, and a repulsion from the negative plate. What is true about the motion of these electrons?

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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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You need to design a 1.0 A fuse that 'blows' if the current exceeds 1.0 A. The fuse material in your stockroom melts at a current density of 500 A/cm². What diameter wire of this material will do the job?

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