Magnetic Field from Semi-Circular Segments:

The magnetic field at the center due to a semi-circular loop is given by:

$ B = \frac{\mu_0 I}{4R} $

Calculating $B_{R_1}$ and $B_{R_2}$:

For the semi-circle with radius $R_1$:

$ B_{R_1} = \frac{\mu_0 I}{4R_1} = \frac{4\pi \times 10^{-7} \times 12}{4 \times 6\pi} $

For the semi-circle with radius $R_2$:

$ B_{R_2} = \frac{\mu_0 I}{4R_2} = \frac{4\pi \times 10^{-7} \times 12}{4 \times 4\pi} $

Net Magnetic Field at the Center (O):

The total magnetic field $B_0$ at the center is the difference between these two fields (since they are in opposite directions):

$ B_0 = |B_{R_1} - B_{R_2}| $

Calculate:

$ B_0 = \frac{4\pi \times 10^{-7} \times 12}{4} \left(\frac{1}{4\pi} - \frac{1}{6\pi}\right) $

$ B_0 = 12\pi \times 10^{-7} \left(\frac{1}{12\pi}\right) $

$ B_0 = 1 \times 10^{-7} \text{ T} $

Value of $k$:

Given that the magnitude of the resultant magnetic field is $k \times 10^{-7} \text{ T}$, we find $k = 1$.

Thus, the value of $k$ is 1.