Current density in a cylindrical wire of radius R is given as $\mathrm{J}=\left\{\begin{array}{c}{J}_0\left(\frac{x}{R}-1\right)\text{ for }0\le x<\frac{R}{2}\\ J_0\frac{x}{R}\text{ for }\frac{R}{2}\le x\le R\end{array}\right.$ If the current flowing in the wire is $\frac{\mathrm{k}}{12}\pi {\mathrm{J}}_0{\mathrm{R}}^2,$ find the value of k. 

${\mathrm{dI}}_1={\mathrm{J}}_1\mathrm{dA}={\mathrm{J}}_0\left(\frac{\mathrm{x}}{\mathrm{R}}-1\right)2\pi \mathrm{xdx}\text{ for }0\le \mathrm{x}<\frac{\mathrm{R}}{2}$

${\mathrm{I}}_1=2\pi {\mathrm{J}}_0{\int }_0^{\mathrm{R}/2} \left(\frac{x}{\mathrm{R}}-1\right)x\mathrm{d}x$

$=2\pi {\mathrm{J}}_0{\left[\frac{x^3}{3\mathrm{R}}-\frac{x^2}{2}\right]}_0^{\mathrm{R}/2}$

$=\frac{-\pi {\mathrm{J}}_0{\mathrm{R}}^2}{6}$

$\mathrm{Similarly} {\mathrm{I}}_2=\int {\mathrm{J}}_2\mathrm{dA}$

$={\int }_{\mathrm{R}/2}^{\mathrm{R}} {\mathrm{J}}_0\frac{x}{\mathrm{R}}2\pi x\mathrm{d}x\text{ for }\frac{\mathrm{R}}{2}\le x\le \mathrm{R}$

$=\frac{2\pi {\mathrm{J}}_0}{\mathrm{R}}={\left[\frac{x^3}{3}\right]}_{\mathrm{R}/2}^{\mathrm{R}}$

$=\frac{7}{12}\pi {\mathrm{J}}_0{\mathrm{R}}^2$

${\mathrm{I}}_{\text{net }}={\mathrm{I}}_1+{\mathrm{I}}_2$

$=\pi {\mathrm{J}}_0{\mathrm{R}}^2\left[\frac{7}{12}-\frac{1}{6}\right]=\frac{5}{12}\pi {\mathrm{J}}_0{\mathrm{R}}^2$