Question

Sol. [Key Concept: Magnetic field of a current-carrying loop]

$$\begin{matrix} & I_{1}R_{ABC} = I_{2}R_{ADC} \\ & I_{1} + I_{2} = I_{0} \\ & I_{1}\frac{R}{2} = I_{2}R \\ & \frac{I_{1}}{2} = I_{2} \end{matrix}$$

$$\begin{matrix} & l_{1} + l_{2} = l_{0} \\ & 2l_{2} + l_{2} = l_{0} \end{matrix}$$

$$\begin{matrix} & I_{2} = \frac{I_{0}}{3} \\ & I_{1} = \frac{2I_{0}}{3} \end{matrix}$$

$$\begin{matrix} & {\overrightarrow{B}}_{0} = {\overrightarrow{B}}_{1} + {\overrightarrow{B}}_{2} \\ & \ = \frac{\mu_{0}I_{1}( \cdot )}{4r} + \frac{\mu_{0}I_{2}( \times )}{4r} \\ & \ = \frac{\mu_{0}2I_{0}}{4r \times 3} - \frac{\mu_{0}I_{0}}{4r \times 3} = \frac{\mu_{0}I_{0}}{12r} \end{matrix}$$