The current density varies with radial distance r as in a cylindrical wire of radius R. The current passing through the wire between the radial distances R3 and R2 is___________

A
$\frac{65 π a R^{4}}{2592}$
B
$\frac{25 {πaR}^{4}}{72}$
C
$\frac{65 {πa}^{2} R^{4}}{2938}$
D
$\frac{81 {πa}^{2} R^{4}}{144}$

Solution:

$\begin{array}{l} dI = jdA = {ar}^{2} \times 2 πrdr = 2 {πar}^{3} drI = 2 πa \int_{\frac{R}{3}}^{\frac{R}{2}} r^{3} dr \\ = 2 πa \times \frac{1}{4} {[r^{4}]}_{\frac{R}{3}}^{\frac{R}{2}} = \frac{πa}{2} [\frac{R^{4}}{16} - \frac{R^{4}}{81}] \\ I = \frac{{πaR}^{4}}{2} [\frac{81 - 16}{16 \times 81}] = \frac{65 {πaR}^{4}}{2592} \end{array}$

The current density varies with radial distance r as in a cylindrical wire of radius R. The current passing through the wire between the radial distances $\frac{R}{3}$ and $\frac{R}{2}$ is___________

Solution:

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