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. $\frac{65 \pi aR ^{4}}{2592}$
• 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)

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