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

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Correct Answer

(C) \(3\)

Detailed explanation

\(B _{ net }= B _{1}- B _{2}=\frac{\mu_{0} \times 4}{2 \pi[.04]}-\frac{\mu_{0} \times 2}{2 \pi[.06]}\)

\(\overrightarrow{ B }_{\text {met }}=\frac{\mu_{0}}{2 \pi}\left[\frac{200}{3}\right](-\hat{ k })\)

\(\overrightarrow{ F }= q [\overrightarrow{ v } \times \overrightarrow{ B }]\)

\(=[3 \pi]\left[(2 \hat{ i }+3 \hat{ j }) \times\left(\frac{\mu_{0}}{2 \pi}\right)\left(\frac{200}{3}\right)-\hat{ k }\right]\)

\(=3 \pi \times \frac{\mu_{0}}{2 \pi}\left(\frac{200}{3}\right)[2 \times \hat{j}-3(\hat{i})]\)

\(=\left(4 \pi \times 10^{-7}\right)(100)(-3 \hat{i}+2 \hat{j})\)

\(=4 \pi \times 10^{-5} \times[-3 \hat{i}+2 \hat{j}]\)

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