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)

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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.
-
Transcribed Image Text:### 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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