Sol. (A) $$ \begin{array}{ll} \mathrm{B}_{a b}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}} & \text { (out of the plane) } \\ \mathrm{B}_{c c}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}}(2 \pi) & \text { (in the plane) } \\ \mathrm{B}_{c c}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}} & \text { (out of the plane) } \end{array} $$ Hence magnetic field at O is $$ \begin{aligned} & B_0=-\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}}+\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}}(2 \pi)-\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}} \\ & B_0=-\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}}(\pi-1) \end{aligned} $$ B) $$ \begin{array}{ll} B_{a b}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}} & \text { (out of the plane) } \\ B_{b c d}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}}(\pi) & \text { (out of the plane) } \\ B_{c o}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}} & \text { (out of the plane) } \end{array} $$ Hence magnetic field at O is $$ \begin{aligned} & B_0=-\frac{\mu_0}{4 \pi} \frac{\mathrm{l}}{\mathrm{r}}+\frac{\mu_0}{4 \pi} \frac{\mathrm{l}}{\mathrm{r}}(\pi)-\frac{\mu_0}{4 \pi} \frac{\mathrm{l}}{\mathrm{r}} \\ & B_0=\frac{\mu_0}{4 \pi} \frac{\mathrm{l}}{\mathrm{r}}(\pi+2) \end{aligned} $$ (B) $$ \begin{array}{ll} \mathrm{B}_{a b}=\frac{\mu_0}{4 \pi} \frac{\mathrm{l}}{\mathrm{r}} & \text { (out of the plane) } \\ \mathrm{B}_{b c \sigma}=\frac{\mu_0}{4 \pi} \frac{\mathrm{l}}{\mathrm{r}}(\pi) & \text { (out of the plane) } \\ \mathrm{B}_{c o}=\frac{\mu_0}{4 \pi} \frac{\mathrm{l}}{\mathrm{r}} & \text { (out of the plane) } \end{array} $$ Hence magnetic field at $O$ is $$ \begin{aligned} & B_0=-\frac{\mu_0}{4 \pi} \frac{1}{r}+\frac{\mu_0}{4 \pi} \frac{1}{r}(\pi)-\frac{\mu_0}{4 \pi} \frac{1}{r} \\ & B_0=\frac{\mu_0}{4 \pi} \frac{1}{r}(\pi+2) \end{aligned} $$ (C) $$ \begin{array}{ll} \mathrm{B}_{a b}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}} & \text { (in the plane) } \\ \mathrm{B}_{b c d}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}}(\pi) & \text { (in the plane) } \\ \mathrm{B}_{\infty}=0 & \text { (at the axis) } \end{array} $$ Hence magnetic field at O is $$ \mathrm{B}_0=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}}(\pi+2) $$ (D) $$ \begin{array}{ll} \mathrm{B}_{a b}=0 & \text { (at the axis) } \\ \mathrm{B}_{b c d}=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}}(\pi) & \text { (out of the plane) } \\ \mathrm{B}_{\infty}=0 & \text { (at the axis) } \end{array} $$ Hence magnetic field at $O$ is $$ \mathrm{B}_0=\frac{\mu_0}{4 \pi} \frac{\mathrm{I}}{\mathrm{r}} $$