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An infinitely long current carrying wire is placed along X-axis and magnetic induction produced by this wire at point p, which lies in XY plane is 1 T. When the same wire is placed along Y-axis, magnetic induction at the same point is found to be $\sqrt{3} T$ . What will be the magnetic induction at the same point p when the same current carrying wire is placed along Z-axis?

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$\frac{\sqrt{3}}{2} T$

2 T

0.5 T

$2 \sqrt{3} T$

answer is D.

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Detailed Solution

$r = \sqrt{x^{2} + y^{2}}$

$B_{1} = \frac{μ_{0} i}{2 πy} = \frac{k}{y} \Rightarrow y = \frac{k}{B_{1}}$

$B_{2} = \frac{μ_{0} i}{2 πx} = \frac{k}{x} \Rightarrow x = \frac{k}{B_{2}}$

$∴ B_{3} = \frac{μ_{0} i}{2 πr} = \frac{k}{\sqrt{{(\frac{k}{B_{2}})}^{2} + {(\frac{k}{B_{1}})}^{2}}} = \frac{B_{1} B_{2}}{\sqrt{{B_{1}}^{2} + {B_{2}}^{2}}} = \frac{1 \times \sqrt{3}}{\sqrt{1 + 3}} T$

$∴ B_{3} = \frac{\sqrt{3}}{2} T$

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