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. $\frac{65\pi a^2{R}^3}{2938}$
• D. $\frac{81\pi a^2{R}^4}{144}$

Answer 1

Answer: (A)

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

Current density $J=a{r}^2$
Current $I=\int JdA$
$I={\int }_{R/3}^{R/2}a{r}^2\left(2\pi rdr\right)$, [where dA$=2\pi rdr$]
$=2\pi a{\left[\frac{r^4}{4}\right]}_{R/3}^{R/2}=\frac{65\pi a{R}^4}{2592}$