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 the 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 $δ (δ \to 0)$ . These feeding wires are thus not visible in figure-1. Consider an Amperian circular loop with AB as axis, midpoint of AB as its centre and R as its radius as shown in figure 2. C is the midpoint of square formed by AB as its side shown in figure -1. Take $R = \frac{a}{2}$

Magnitude of circulation $\int \vec{B} . d \vec{l}$ around the shown Amperian loop if $\vec{B}$ is taken due to grid and feeding wires is x.
Magnitude of circulation $\int \vec{B} . d \vec{l}$ around the shown Amperian loop if $\vec{B}$ is taken due to entire grid excluding the feeding wires is y.
Magnitude of $\vec{B}$ at point C is z. Then

Question

Infinity Learn · Accepted Answer

$y = μ_{0} i (\frac{1}{\sqrt{2}} - \frac{1}{2})$

,

$y = μ_{0} i (\frac{3}{2} - \sqrt{2})$

Detailed Solution