### Current Density in a Cylindrical WireIn this exercise, we analyze the current within a cylindrical wire by examining the relationship between the current \( i \) and the square of the radius \( r^2 \). **Problem Statement:**A certain cylindrical wire carries current. We draw a circle of radius \( r \) around its central axis in Figure \((a)\) to determine the current \( i \) within the circle. Figure \((b)\) shows current \( i \) as a function of \( r^2 \). The vertical scale is set by \( i_s = 5.3 \, \text{mA} \), and the horizontal scale is set by \( r_s^2 = 4.7 \, \text{mm}^2 \).**Questions:**(a) Is the current density uniform?(b) If so, what is its magnitude?**Explanation of Graphs:**- **Figure (a):** This is a visual representation of the cross-section of the cylindrical wire, depicting a circle of radius \( r \) around its central axis. The current \( i \) is measured within this circle. - **Figure (b):** This graph illustrates the relationship between the current \( i \), in milliamps (mA), and the square of the radius \( r^2 \), in square millimeters (\( \text{mm}^2 \)). - The horizontal axis represents \( r^2 \) ranging from 0 to \( 4.7 \, \text{mm}^2 \). - The vertical axis shows the current \( i \) ranging up to \( 5.3 \, \text{mA} \). - The data points form a straight line, indicating a linear relationship between current \( i \) and \( r^2 \).**Analysis:**(a) To determine if the current density is uniform, observe if the current \( i \) increases linearly with \( r^2 \). In a uniform current density scenario, \( i \) should be proportional to \( r^2 \).(b) If indeed the current density is uniform, the magnitude can be calculated using the slope of the linear relationship observed in Figure (b).**Interactive Components:**- **(a) Dropdown Menu:** Select "Yes" or "No" to indicate whether the current density is uniform.-

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A certain cylindrical wire carries current. We draw a circle of radius r around its central axis in Figure (a) to determine the current i within the circle. Figure (b) shows current i as a function of r². The vertical scale is set by is = 5.3 mA, and the horizontal scale is set by r² = 4.7 mm². (a) is the current density uniform? (b) If so, what is its magnitude? (a) MI (b) Number i Units i (mA) (()) Ⓡ -² (mm²) (b)

A certain cylindrical wire carries current. We draw a circle of radius r around its central axis in Figure (a) to determine the current i within the circle. Figure (b) shows current i as a function of r². The vertical scale is set by is = 5.3 mA, and the horizontal scale is set by r² = 4.7 mm². (a) is the current density uniform? (b) If so, what is its magnitude? (a) MI (b) Number i Units i (mA) (()) Ⓡ -² (mm²) (b)

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Resistivity

Resistivity is a characteristic property of the conductor’s material that measures the extent to which the material opposes the flow of current. It is denoted by the Greek letter “rho”. Electrical resistivity, also known as specific resistance, depends on the nature of the material of the conductor and is unaffected by its dimensions.

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### Current Density in a Cylindrical Wire

In this exercise, we analyze the current within a cylindrical wire by examining the relationship between the current \( i \) and the square of the radius \( r^2 \).

**Problem Statement:**

A certain cylindrical wire carries current. We draw a circle of radius \( r \) around its central axis in Figure \((a)\) to determine the current \( i \) within the circle. Figure \((b)\) shows current \( i \) as a function of \( r^2 \). The vertical scale is set by \( i_s = 5.3 \, \text{mA} \), and the horizontal scale is set by \( r_s^2 = 4.7 \, \text{mm}^2 \).

**Questions:**

(a) Is the current density uniform?
(b) If so, what is its magnitude?

**Explanation of Graphs:**

- **Figure (a):** This is a visual representation of the cross-section of the cylindrical wire, depicting a circle of radius \( r \) around its central axis. The current \( i \) is measured within this circle.

- **Figure (b):** This graph illustrates the relationship between the current \( i \), in milliamps (mA), and the square of the radius \( r^2 \), in square millimeters (\( \text{mm}^2 \)).
- The horizontal axis represents \( r^2 \) ranging from 0 to \( 4.7 \, \text{mm}^2 \).
- The vertical axis shows the current \( i \) ranging up to \( 5.3 \, \text{mA} \).
- The data points form a straight line, indicating a linear relationship between current \( i \) and \( r^2 \).

**Analysis:**

(a) To determine if the current density is uniform, observe if the current \( i \) increases linearly with \( r^2 \). In a uniform current density scenario, \( i \) should be proportional to \( r^2 \).

(b) If indeed the current density is uniform, the magnitude can be calculated using the slope of the linear relationship observed in Figure (b).

**Interactive Components:**

- **(a) Dropdown Menu:** Select "Yes" or "No" to indicate whether the current density is uniform.
-

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