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 $3 π$ coulomb is passing through the point P with velocity
$\vec{v} = (2 \hat{i} + 3 \hat{j}) m / s$ ; where $\hat{i} & \hat{j}$ represents unit vector along x & y axis respectively.
The force acting on the charge particle is $4 π \times 10^{- 5} (- x \hat{i} + 2 \hat{j}) N$ . The value of x is:

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answer is C.

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Detailed Solution

$B_{net} = B_{1} - B_{2} = \frac{μ_{0} \times 4}{2 π [. 04]} - \frac{μ_{0} \times 2}{2 π [. 06]} {\vec{B}}_{net} = \frac{μ_{0}}{2 π} [\frac{200}{3}] (- \hat{k}) \vec{F} = q [\vec{v} \times \vec{B}] = [3 π] [(2 \hat{i} + 3 \hat{j}) \times (\frac{μ_{0}}{2 π}) (\frac{200}{3}) - \hat{k}]$

$= 3 π \times \frac{μ_{0}}{2 π} (\frac{200}{3}) [2 \times \hat{j} - 3 (\hat{i})] = (4 π \times 10^{- 7}) (100) (- 3 \hat{i} + 2 \hat{j}) = 4 π \times 10^{- 5} \times [- 3 \hat{i} + 2 \hat{j}]$