question_answer

A non-planar loop of conducting wire carrying a current I is placed as shown in the figure. Each of the straight sections of the loop of length 2a. The magnetic field due to this loop at the point P(a, 0, a) points in the direction

A) \[\frac{1}{\sqrt{2}}\,\,(-\,\hat{j}+\hat{k})\]         

B) \[\frac{1}{\sqrt{3}}\,\,(-\,\hat{j}+\hat{k}+\hat{i})\]

C) \[\frac{1}{\sqrt{3}}\,\,(\hat{i}+\hat{j}+\hat{k})\]

D) \[\frac{1}{\sqrt{2}}\,\,(\hat{i}+\hat{k})\]

Correct Answer: D

Solution :

Magnetic field of any coil is in direction of magnetic moment of coil. This coil is in two plane xy and yz, so here we will split this coil in two separate coil in xy and yz plane

Magnetic moment of coil 1 is in \[\hat{k}\]

Magnetic moment of coil 2 is in \[\hat{i}\]

So \[\vec{M}\] is in \[\hat{i}\] and \[\hat{k}\] so B must be in \[\hat{i}\] and \[\hat{k}\]

Only one option has B in \[\hat{i}\] and \[\hat{k}\]

i.e. option [D] where\[B=\frac{1}{\sqrt{2}}\,\,(\hat{i}+\hat{k})\]