Question

Two long straight parallel current carrying conductors carry current in opposite direction. Each is of length l and current i and are placed at a distance r₀ from each other. The total work done by an external agent in slowly reducing their distance of separation to $\frac{r_0}{2}$ is

• A. $\frac{H_0{i}^{2}l}{2\pi }ln2$
• B. $\frac{{\mu }_0{i}^2}{2\pi l}ln2$
• C. $\frac{{\mu }_0{i}^{2}l}{2\pi r_0}$
• D. $\frac{{\mu }_0{i}^{2}l}{\pi r_0}$

Answer 1

The correct option is A $\frac{H_0{i}^{2}l}{2\pi }ln2$

F =Bil
$F=il\left[\frac{{\mu }_{0}i}{2\pi r}\right]F=\frac{{\mu }_0{i}^{2}l}{2\pi r}dw=F.d\overrightarrow{r}=Fdrcos{180}^{\circ }dw=-\frac{{\mu }_0{i}^{2}l}{2\pi r}dr$
Total workdone
$W={\int }_{r_0}^{\frac{r_0}{2}}-\frac{{\mu }_0{i}^{2}l}{2\pi r}dr=-\frac{{\mu }_0{i}^{2}l}{2\pi }[l_{n}r{]}_{r_0}^{r_0/2}=\frac{{\mu }_0{i}^{2}l}{2\pi }[ln{r}_0-ln\frac{r_0}{2}]=\frac{{\mu }_0{i}^{2}l}{2\pi }[ln2]$