Question

A conductor AB of length $L$ carrying a current $I_{2}$, is placed perpendicular to a long straight conductor $X Y$ carrying current $I_{1}$, as shown in figure. The force on $A B$ 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

$dF=\frac{\mu_{0}}{4 \pi} \frac{2 I_{1}}{x} \times I_{2} \times d x=\frac{\mu_{0} I_{1} I_{2}}{2 \pi} \frac{dx}{x}$
Total force on $AB$ is
$F=\frac{\mu_{0}}{2 \pi} I_{1} I_{2} \int\limits_{L / 2}^{(3 L / 2)} \frac{d x}{x}$
$=\frac{\mu_{0} I_{1} I_{2}}{2 \pi} \ln 3$