question_answer

Axis of a solid cylinder of infinite length and radius R lies along y-axis, it carries a uniformly distributed current i along  \[+y\] direction. Magnetic field at a point \[\left( \frac{R}{2},y,\frac{R}{2} \right)\] is

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)

\[\therefore \,\,\,\vec{B}=B\hat{B}=\frac{{{\mu }_{0}}i}{4\pi R}(\hat{i}-\hat{k})\]