A loop ABCDA, carrying current $$I = 12$$ A, is placed in a plane, consists of two semi-circular segments of radius $$R_1 = 6\pi$$ m and $$R_2 = 4\pi$$ m. The magnitude of the resultant magnetic field at center O is $$k \times 10^{-7}$$ T. The value of k is __________.
(Given $$\mu_0 = 4\pi \times 10^{-7}$$ Tm A$$^{-1}$$)

Straight segments $$AB$$ and $$CD$$: Their lines of extension pass through center $$O$$. $$\implies B_{AB} = 0 \quad \text{and} \quad B_{CD} = 0$$

Inner semicircular arc $$BC$$ ($$Radius = R_2$$): The current flows clockwise. According to the Right-Hand Rule, its magnetic field ($$B_2$$) points into the page ($$\otimes$$).

Outer semicircular arc $$DA$$ ($$Radius = R_1$$): The current flows counter-clockwise. According to the Right-Hand Rule, its magnetic field ($$B_1$$) points out of the page ($$\odot$$).

$$B_1 = \frac{\mu_0 I}{4 R_1} = \frac{(4\pi \times 10^{-7}) \times 12}{4 \times 6\pi} = \frac{48\pi \times 10^{-7}}{24\pi} = 2 \times 10^{-7}\text{ T} \quad (\odot)$$

$$B_2 = \frac{\mu_0 I}{4 R_2} = \frac{(4\pi \times 10^{-7}) \times 12}{4 \times 4\pi} = \frac{48\pi \times 10^{-7}}{16\pi} = 3 \times 10^{-7}\text{ T} \quad (\otimes)$$

$$B_{\text{net}} = B_2 - B_1 = (3 \times 10^{-7}) - (2 \times 10^{-7}) = 1 \times 10^{-7}\text{ T} \quad (\otimes)$$

$$k = 1$$