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The current density in a wire of radius ...

The current density in a wire of radius a varies with radial distance r as `J=(J_0r^2)/a`, where `J_0` is a constant. Choose the incorrect statement.

A

Total current passing through the cross-section of the wire is`I=(piJ_0a^3)/2`

B

Total current passing through the cross-section of the wire is`I=(3piJ_0a^3)/2`

C

The field at a distance r gt a is `B=(mu_0J_0a^3)/(4r)`

D

The field at a distance r gt a is `B=(mu_0J_0a^3)/(4a)`

Text Solution

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The correct Answer is:
To solve the problem regarding the current density \( J = \frac{J_0 r^2}{a} \) in a wire of radius \( a \), we need to analyze the given information and determine which statement is incorrect. ### Step-by-Step Solution: 1. **Understanding Current Density**: The current density \( J \) is defined as the current per unit area. In this case, it varies with the radial distance \( r \) from the center of the wire. The formula given is \( J = \frac{J_0 r^2}{a} \). 2. **Determine Total Current**: To find the total current \( I \) flowing through the wire, we need to integrate the current density over the cross-sectional area of the wire. The differential area element in polar coordinates is \( dA = 2\pi r \, dr \). \[ I = \int_0^a J \, dA = \int_0^a \frac{J_0 r^2}{a} \cdot 2\pi r \, dr \] 3. **Set Up the Integral**: Substitute \( J \) into the integral: \[ I = \int_0^a \frac{J_0 r^2}{a} \cdot 2\pi r \, dr = \frac{2\pi J_0}{a} \int_0^a r^3 \, dr \] 4. **Calculate the Integral**: The integral \( \int_0^a r^3 \, dr \) can be calculated as follows: \[ \int_0^a r^3 \, dr = \left[ \frac{r^4}{4} \right]_0^a = \frac{a^4}{4} \] 5. **Substituting Back**: Now substitute the result of the integral back into the expression for \( I \): \[ I = \frac{2\pi J_0}{a} \cdot \frac{a^4}{4} = \frac{\pi J_0 a^3}{2} \] 6. **Analyzing Statements**: Now, we need to analyze the statements provided in the question. We can check for the following potential statements: - The total current is proportional to \( a^3 \). - The current density is maximum at the center of the wire. - The current density is uniform across the cross-section. - The current density increases with increasing radius. Based on our calculations, we can identify the incorrect statement. ### Conclusion: The incorrect statement is that "The current density is uniform across the cross-section." This is incorrect because the current density varies with \( r^2 \), meaning it is not uniform.
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