Physics - Magnetism Question with Solution | TestHub

An infinite horizontal square grid of side '$a$' made of uniform material has current $i$ fed into it at node A, and this current leaves at B. The semi-infinite wires used for the entry and exit of current into this grid are shown in Figure 2, which is an enlarged version (not drawn to scale) of the grid in the locality of AB when observed along its plane. The feeding wires are parallel to AB and just above the plane of the grid by a small distance $\delta (\delta \to 0)$. These feeding wires are thus not visible in Figure 1. Consider an Amperean circular loop with AB as its axis, the midpoint of AB as its center, and R as its radius, as shown in Figure 2. C is the midpoint of the square formed by AB as its side, shown in Figure 1. Take $R=a/2$.

The magnitude of circulation $\oint \vec{B}\cdot d\vec{\ell }$ around the shown Amperean loop, if $\overrightarrow{B_1}$ is taken as due to the entire grid excluding the feeding wires, is $a$. If $\overrightarrow{B_1}=\frac{{\mu }_{0}i}{X}\left(\frac{1}{a}-\frac{1}{Y}\right)$, find $X+Y$.