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Two Parallel Currents

Two long mutu...

Question

Two long mutually perpendicular conductors carrying currents $I_{1}$ and $I_{2}$ lie in one plane. Find the locus of points at which the magnetic induction is zero.

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Solution

In first quadrant magnetic field due to $I_{1}$ is outwards and due to $I_{2}$ is inwards.So net magnetic field may be zero.
Similarly in third quadrant magnetic field due to $I_{1}$ is inwards and due to $I_{2}$ magnetic field is outwards.Hence only in first and third quadrants magnetic field may be zero.
Let magnetic field is zero at point $P (x y)$ then :
$B_{I_{1}} = B_{I_{2}}$
$∴ \frac{μ o}{2 π} \frac{I_{1}}{y} = \frac{μ o}{2 π} \frac{I_{2}}{x}$
$∴ y = (\frac{I_{1}}{I_{2}}) x$

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