Question

The 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. $\frac{81\pi a{R}^4}{144}$

Answer 1

Answer: (A)

$\frac{65\pi a{R}^4}{2592}$

Given; $J=a{r}^2$
$i={\int }_1^{2}J\times 2\pi rdr={\int }_{R/3}^{R/2}a{r}^2\times 2\pi rdr$
$=2\pi a{\int }_{R/3}^{R/2}{r}^{3}dr=2\pi a{\left|\frac{r^4}{4}\right|}_{R/3}^{R/2}$
$=\frac{\pi a}{2}[{\left(\frac{R}{2}\right)}^4-{\left(\frac{R}{3}\right)}^4]$
$=\frac{\pi a{R}^4}{2}\times \frac{65}{81\times 16}=\frac{65\pi a{R}^4}{2592}$