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Two long parallel conductors $S_{1}$ and $S_{2}$ are separated by a distance $10 \,cm$ and carrying currents of $4\, A$ and $2 \,A$ respectively. The conductors are placed along $x$-axis in $X - Y$ plane. There is a point $P$ located between the conductors (as shown in figure).
A charge particle of $3 \pi$ coulomb is passing through the point $P$ with velocity
$\overrightarrow{ v }=(2 \hat{ i }+3 \hat{ j }) \,m / s$; where $\hat{i}$ and $\hat{j} \quad$ represents unit vector along $x$ and $y$ axis respectively.
The force acting on the charge particle is $4 \pi \times 10^{-5}(-x \hat{i}+2 \hat{j}) \,N$. The value of $x$ is
$2$
$1$
$3$
$-3$
Solution
$B _{ net }= B _{1}- B _{2}=\frac{\mu_{0} \times 4}{2 \pi[.04]}-\frac{\mu_{0} \times 2}{2 \pi[.06]}$
$\overrightarrow{ B }_{\text {met }}=\frac{\mu_{0}}{2 \pi}\left[\frac{200}{3}\right](-\hat{ k })$
$\overrightarrow{ F }= q [\overrightarrow{ v } \times \overrightarrow{ B }]$
$=[3 \pi]\left[(2 \hat{ i }+3 \hat{ j }) \times\left(\frac{\mu_{0}}{2 \pi}\right)\left(\frac{200}{3}\right)-\hat{ k }\right]$
$=3 \pi \times \frac{\mu_{0}}{2 \pi}\left(\frac{200}{3}\right)[2 \times \hat{j}-3(\hat{i})]$
$=\left(4 \pi \times 10^{-7}\right)(100)(-3 \hat{i}+2 \hat{j})$
$=4 \pi \times 10^{-5} \times[-3 \hat{i}+2 \hat{j}]$
