\(B _{ ab }=\frac{\mu_0}{4 \pi} \frac{ I }{ r } \text { (out of the plane) }\)

\(B _{ bcd }=\frac{\mu_0}{4 \pi} \frac{ I }{ r }(2 \pi) \text { (in the plane) }\)

\(B _{ de }=\frac{\mu_0}{4 \pi} \frac{ I }{ r } \text { (out of the plane) }\)

Hence magnetic field at \(O\) is

\(B _0=-\frac{\mu_0}{4 \pi} \frac{ I }{ r }+\frac{\mu_0}{4 \pi} \frac{ I }{ r }(2 \pi)-\frac{\mu_0}{4 \pi} \frac{ I }{ r }\)

\(B _0=\frac{\mu_0}{2 \pi} \frac{ I }{ r }(\pi-1) \ldots \ldots . . \text { (III) }\)

(B)

\(B _{ ab }=\frac{\mu_0}{4 \pi} \frac{ I }{ r } \text { (out of the plane) }\)

\(B _{ bcd }=\frac{\mu_0}{4 \pi} \frac{ I }{ r }(\pi) \text { (out of the plane) }\)

\(B _{ de }=\frac{\mu_0}{4 \pi} \frac{ I }{ r } \text { (out of the plane) }\)

Hence magnetic field at \(O\) is

\(B _0=\frac{\mu_0}{4 \pi} \frac{ I }{ r }+\frac{\mu_0}{4 \pi} \frac{ I }{ r }(\pi)+\frac{\mu_0}{4 \pi} \frac{ I }{ r }\)

\(B _0=\frac{\mu_0}{4 \pi} \frac{ I }{ r }(\pi+2) \ldots .( I )\)

(C)

\(B _{ ab }=\frac{\mu_0}{4 \pi} \frac{ I }{ r } \text { (in the plane) }\)

\(B _{ bcd }=\frac{\mu_0}{4 \pi} \frac{ I }{ r }(\pi) \text { (in the plane) }\)

\(B _{ de }=0 \text { (at the axis) }\)

Hence magnetic field at \(O\) is

\(B _0=\frac{\mu_0}{4 \pi} \frac{ I }{ r }(1+\pi) \ldots(IV)\)

\(B _{ ab }=0 \text { (at the axis) }\)

\(B _{ bcd }=\frac{\mu_0}{4 \pi} \frac{ I }{ r }(\pi) \text { (out of the plane) }\)

\(B _{ de }=0 \text { (at the axis) }\)

Hence magnetic field at \(O\) is

\(B _0=\frac{\mu_0 I }{4 r } \ldots \text { (II) }\)