Question
Two infinitely long straight parallel wires perpendicular to the plane of the paper are 5 m apart. One of the wires, P carries current I out of the plane of the paper and the other, Q carries the current I into the plane of paper. The magnetic field B at the origin O of the coordinate system with x and y axes as perpendicular and parallel to PQ, respectively, is [Given OP = 4 m and OQ = 3 m]
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Solution
[Key Concept: The field of a long straight wire is μ0I/(2πr), perpendicular to the radius; fields of several wires add as vectors.]
$$PM = \frac{16}{5}\text{ }m\& QM = \frac{9}{5}\text{ }m$$
$$\begin{matrix} & {\overrightarrow{B}}_{P} = \frac{\mu_{0}I}{8\pi}\left\lbrack \frac{4}{5}\widehat{i} + \frac{3}{5}\widehat{j} \right\rbrack \\ & {\overrightarrow{B}}_{Q} = \frac{\mu_{0}I}{6\pi}\left\lbrack \frac{3}{5}\widehat{i} - \frac{4}{5}\widehat{j} \right\rbrack \\ & \ \therefore{\overrightarrow{B}}_{\text{net~}} = \frac{\mu_{0}I}{\pi}\left\lbrack \frac{1}{10}\widehat{i} + \frac{3}{40}\widehat{j} + \frac{1}{10}\widehat{i} - \frac{2}{15}\widehat{j} \right\rbrack \\ & \ = \frac{\mu_{0}I}{\pi}\left\lbrack \frac{1}{5}\widehat{i} - \frac{7}{120}\widehat{j} \right\rbrack = \frac{\mu_{0}I}{5\pi}\left\lbrack \widehat{i} - \frac{7}{24}\widehat{j} \right\rbrack \end{matrix}$$
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