• # question_answer Two long parallel wires P and Q are both perpendicular to    the plane of the paper with distance of 5 m between them. If            P and Q carry current of $2.5$ amp and 5 amp respectively in the same direction, then the magnetic field at a point half-way between the wires is        A)  $\frac{3{{\mu }_{0}}}{2\pi }$                 B)  $\frac{{{\mu }_{0}}}{\pi }$C)  $\frac{\sqrt{3}{{\mu }_{0}}}{2\pi }$                     D)  $\frac{{{\mu }_{0}}}{2\pi }$

Solution :

When current flow in both wire in same direction then magnetic field at halfway due to P wire. ${{\overrightarrow{B}}_{P}}=\frac{{{\mu }_{0}}{{I}_{1}}}{2\pi \frac{5}{2}}=\frac{{{\mu }_{0}}{{I}_{1}}}{\pi .5}=\frac{{{\mu }_{0}}}{\pi }$ (Where ${{I}_{1}}=5\,A$Amp) The direction of ${{B}_{P}}$ is downward $\odot$                 Magnetic field at halfway due to Q wire ${{\overrightarrow{B}}_{Q}}=\frac{{{\mu }_{0}}{{I}_{2}}}{2\pi \frac{5}{2}}=\frac{{{\mu }_{0}}}{2\pi }$ [upward $\otimes$] [Where ${{I}_{2}}=2.5$ Amp.] Net magnetic field at halfway $\overrightarrow{B}={{\overrightarrow{B}}_{P}}+{{\overrightarrow{B}}_{Q}}=\frac{{{\mu }_{0}}}{\pi }+\frac{{{\mu }_{0}}}{2\pi }=\frac{3{{\mu }_{0}}}{2\pi }$ (downward$\odot$) Hence net magnetic field at midpoint $=\frac{3{{\mu }_{0}}}{2\pi }$

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