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

Options

A. ` (1)/(sqrt (2)) (- hat(j) + hat(k))`
B. ` (1)/(sqrt (3 )) (- hat(j) + hat(k) + hat (i))`
C. ` (1)/(sqrt (3 )) (hat(i) + hat(j) + hat(k))`
D. ` (1)/(sqrt (2)) (hat(i) + hat(k))`

Answer

D

Explanation

Note : If we take individual length for the purpose of calculating the magnetic field in a ` 3 - dimensional` figure then it will be difficult .
Here a smart choice is divide the loop into two loops . One loop is `ADEFA` in ` y - z `plane and the other loop will be ` ABCDA` in the ` x- y` plane .
We actually do not have any current in the segment ` AD` By choosing the loops we find that in one loop we have to take current from ` A to D` and in the other one from `D to A` . Hence these two cancel out the effect of weach other as far as creating magnetic field at the concerned point `P` is considered . The point `(a,0,a) ` is in the ` X - Z` plane.
The magnetic field due to current in ` ABCDA` will be in ` +ve Z - direction` .
NOTE: Due to symmetry the ` y - components` and x- components` will cancel out each other .
Similarly the magnetic due to current in ` ADEFA` will be in ` x- direction .
:. The resultant magnetic field will be
` vec(B) = (1)/ ( sqrt(2))( hat (i) + hat( k))` .