Question

The current density varies with radial distance $r$ as $J= ar^2$, 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. $\frac{81\,\pi\,a^{2}\,R^{4}}{144}$

Answer 1

Answer: (A)

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