Question

A conductor AB of length $L$ carrying a current $I_2$, is placed perpendicular to a long straight conductor $XY$ carrying current $I_1$, as shown in figure. The force on $AB$ in magnitude is

• A. $\frac{3{\mu }_0{I}_1{I}_2}{2\pi }$
• B. $\frac{{\mu }_0{I}_1{I}_2}{2\pi }\ln 2$
• C. $\frac{{\mu }_0{I}_1{I}_2}{2\pi }\ln 3$
• D. $\frac{2{\mu }_0{I}_1{I}_2}{3\pi }$

Answer 1

Answer: (C)

Consider an element of length $dx$ on $AB$ at a distance $x$ from $XY$. Force on the element is

$dF=\frac{{\mu }_0}{4\pi }\frac{2I_1}{x}\times I_2\times dx=\frac{{\mu }_0{I}_1{I}_2}{2\pi }\frac{dx}{x}$
Total force on $AB$ is
$F=\frac{{\mu }_0}{2\pi }{I}_1{I}_2\underset{L/2}{\overset{(3L/2)}{\int }}\frac{dx}{x}$
$=\frac{{\mu }_0{I}_1{I}_2}{2\pi }\ln 3$