Question

Energy shared in cylindrical region of length l extending from radius $R_1$ to $R_2$ and coaxial with long wire carrying current i is

• A. $\frac{{\mu }_0}{\pi }{i}^{2}l\left(\frac{R_2-R_1}{R_1}\right)$
• B. $\frac{{\mu }_0}{\pi }{i}^{2}l\frac{R_2}{R_1}$
• C. $\frac{{\mu }_0}{4\pi }{i}^{2}lln\left(\frac{R_2}{R_1}\right)$
• D. $\frac{{\mu }_0}{2\pi }{i}^{2}lln\left(\frac{R_2}{R_1}\right)$

Answer 1

The correct option is C $\frac{{\mu }_0}{4\pi }{i}^{2}lln\left(\frac{R_2}{R_1}\right)$


Energy shared in small volume is dv $dv=\frac{B^2}{2{\mu }_0}\times 2\pi rdrl$
here $B=\frac{{\mu }_0}{2\pi }\frac{i}{r}\Rightarrow dv=\frac{{\mu }_0^2{i}^2}{4{\pi }^2{r}^2}.\frac{2\pi rdrl}{2{\mu }_0}$
$v=\int dv=\frac{{\mu }_0}{4\pi }{i}^{2}l{\int }_{R_1}^{R^2}\frac{dr}{r}=\frac{{\mu }_0}{4\pi }{i}^{2}lln\left(\frac{R_2}{R_1}\right)$