Question

An infinite wire placed along z-axis has current $i_1$ in positive z-axis. Another wire PQ placed in x-y plane along y-axis carry current $i_2$ along y-axis wire PQ is at distance a from wire carrying current $i_1$. Net force acting on wire PQ is

• A. $\frac{{\mu }_0}{2\pi }{i}_1{i}_{2}2\mu F\left(\frac{2}{\sqrt{3}}\right)$
• B. $\frac{{\mu }_0}{2\pi }{i}_1{i}_{2}ln\left(2\right)$
• C. $0$
• D. $\frac{{\mu }_0}{2\pi }{i}_1{i}_2$

Answer 1

The correct option is C $0$

$dF=i_{2}dyBsin\theta$
$B=\frac{{\mu }_0}{2\pi }\frac{i_1}{r}$ where $cos\theta =\frac{a}{r}$
$tan\theta =\frac{y}{a}\Rightarrow dy=ase{c}^2\theta d\theta$
$dF=i_{2}ase{c}^2\theta \left(d\theta \right)\frac{{\mu }_0}{2\pi }\frac{i_1}{r}sin\theta =\frac{{\mu }_0}{2\pi }{i}_1{i}_{2}tan\theta d\theta$
for segment OP
$\overrightarrow{F}=\int \overrightarrow{d}F=\frac{{\mu }_0}{2\pi }{i}_1{i}_2{\int }_0^{{30}^{\circ }}tan\theta d\theta \left(\hat{k}\right)\text{for segment OQ}\overrightarrow{F}=\int \overrightarrow{d}F=\frac{{\mu }_0}{2\pi }{i}_1{i}_2{\int }_0^{{30}^{\circ }}tan\theta d\theta (-\hat{k})$
for segment QP
$\therefore {\overrightarrow{F}}_{net}=0$