Question

Two long parallel conductors $S_1$ and $S_2$ are separated by a distance $10cm$ and carrying currents of $4A$ and $2A$ 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)

$B_{\text{net }}=B_1-B_2=\frac{{\mu }_0\times 4}{2\pi [.04]}-\frac{{\mu }_0\times 2}{2\pi [.06]}$
${\vec{B}}_{\text{net }}=\frac{{\mu }_0}{2\pi }\left[\frac{200}{3}\right](-\hat{k})$
$\vec{F}=q[\vec{v}\times \vec{B}]$
$=[3\pi ]\left[(2\hat{i}+3\hat{j})\times \left(\frac{{\mu }_0}{2\pi }\right)\left(\frac{200}{3}\right)-\hat{k}\right]$
$=3\pi \times \frac{{\mu }_0}{2\pi }\left(\frac{200}{3}\right)[2\times \hat{j}-3(\hat{i})]$
$=\left(4\pi \times 10^{-7}\right)(100)(-3\hat{i}+2\hat{j})$
$=4\pi \times 10^{-5}\times [-3\hat{i}+2\hat{j}]$