A) \[\frac{{{\mu }_{0}}i}{4\pi R}(\hat{i}-\hat{k})\]
B) \[\frac{{{\mu }_{0}}i}{2\pi R}(\hat{j}-\hat{k})\]
C) \[\frac{{{\mu }_{0}}i}{4\pi R}\hat{j}\]
D) \[\frac{{{\mu }_{0}}i}{4\pi R}(\hat{i}+\hat{k})\]
Correct Answer: A
Solution :
The magnitude of magnetic field at \[P\left( \frac{R}{2},y,\frac{R}{2} \right)\] is |
\[B=\frac{{{\mu }_{0}}Jr}{2}=\frac{{{\mu }_{0}}i}{2\pi {{R}^{2}}}\times \frac{R}{\sqrt{2}}=\frac{{{\mu }_{0}}i}{2\sqrt{2}\pi R}\] |
(independent on y-coordinate) |
Unit vector in direction of magnetic field is |
\[\hat{B}=\frac{\hat{i}-\hat{k}}{\sqrt{2}}\] ( shown by dotted lines) |
\[\therefore \,\,\,\vec{B}=B\hat{B}=\frac{{{\mu }_{0}}i}{4\pi R}(\hat{i}-\hat{k})\] |
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