Question

Current density varies with radial distance \( r \) as \( J = a r^2 \), in a cylindrical wire of radius \( R \). The current passing through the wire between radial distance \( R/3 \) and \( R/2 \) is

• A. \( \frac{65 \pi a R^4}{2592} \)
• B. \( \frac{25 \pi a R^4}{72} \)
• C. \( \frac{65 \pi a R^3}{2938} \)
• D. \( \frac{81 \pi a R^4}{144} \)

Hint:

Current passing through a cylindrical conductor can be found by integrating the current density over the area of interest.

Answer 1

The Correct Option is A

Step 1: Current Density Formula.
The current passing through a radial strip in the wire is given by: \[ I = \int_{R/3}^{R/2} J(r) \, 2\pi r \, dr \] where \( J(r) = a r^2 \) and \( r \) is the radial distance.
Step 2: Integration.
Integrating the current density over the given radial limits, we obtain the result: \[ I = \frac{65 \pi a R^4}{2592}. \] Step 3: Conclusion.
The correct answer is (A), \( \frac{65 \pi a R^4}{2592} \).