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___________

# 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  $\mathrm{R}}{3}$ and  $\mathrm{R}}{2}$ is___________

1. A

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

2. B

$\frac{25{\mathrm{\pi aR}}^{4}}{72}$

3. C

$\frac{65{\mathrm{\pi a}}^{2}{\mathrm{R}}^{4}}{2938}$

4. D

$\frac{81{\mathrm{\pi a}}^{2}{\mathrm{R}}^{4}}{144}$

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

### Solution:

$\begin{array}{l}\mathrm{dI}=\mathrm{jdA}={\mathrm{ar}}^{2}×2\mathrm{\pi rdr}=2{\mathrm{\pi ar}}^{3}\mathrm{drI}=2\mathrm{\pi a}\underset{\mathrm{R}}{3}}{\overset{\mathrm{R}}{2}}{\int }}{\mathrm{r}}^{3}\mathrm{dr}\\ =2\mathrm{\pi a}×\frac{1}{4}{\left[{\mathrm{r}}^{4}\right]}_{\mathrm{R}}{3}}^{\mathrm{R}}{2}}=\frac{\mathrm{\pi a}}{2}\left[\frac{{\mathrm{R}}^{4}}{16}-\frac{{\mathrm{R}}^{4}}{81}\right]\\ \mathrm{I}=\frac{{\mathrm{\pi aR}}^{4}}{2}\left[\frac{81-16}{16×81}\right]=\frac{65{\mathrm{\pi aR}}^{4}}{2592}\end{array}$  Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)