Question

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 $\hat{ i } \& \hat{ j }$ represents unit vector along $x \& y$ 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 :

• A. 2
• B. 1
• C. 3
• D. -3

Answer 1

Answer: (C)

$B _{\text {net }}= B _{1}- B _{2}=\frac{\mu_{0} \times 4}{2 \pi[.04]}-\frac{\mu_{0} \times 2}{2 \pi[.06]}$
$\vec{ B }_{\text {net }}=\frac{\mu_{0}}{2 \pi}\left[\frac{200}{3}\right](-\hat{ k })$
$\vec{ F }= q [\vec{ v } \times \vec{ B }]$
$=[3 \pi]\left[(2 \hat{ i }+3 \hat{ j }) \times\left(\frac{\mu_{0}}{2 \pi}\right)\left(\frac{200}{3}\right)-\hat{ k }\right]$
$=3 \pi \times \frac{\mu_{0}}{2 \pi}\left(\frac{200}{3}\right)[2 \times \hat{ j }-3(\hat{ i })]$
$=\left(4 \pi \times 10^{-7}\right)(100)(-3 \hat{ i }+2 \hat{ j })$
$=4 \pi \times 10^{-5} \times[-3 \hat{ i }+2 \hat{ j }]$