(Solved): 1. A semicircular loop of radius \( a \) in free space carries a current of \( I \). Determine the ...

1. A semicircular loop of radius \( a \) in free space carries a current of \( I \). Determine the magnetic flux density at the center of the loop. 2. In free space the magnetic flux density is \( \overrightarrow{\mathbf{B}}=y^{2} \hat{\mathbf{a}}_{x}+z^{2} \hat{\mathbf{a}}_{y}+x^{2} \hat{\mathbf{a}}_{z}\left[\mathrm{~Wb} / \mathrm{m}^{2}\right] \). (a) Show that \( \overrightarrow{\mathbf{B}} \) is a valid magnetic flux density. (b) Find the magnetic flux through the surface \( x=1,0 \leq y \leq 1,1 \leq z \leq 4 \). (c) Find \( \overrightarrow{\mathbf{J}} \). (d) Determine the total magnetic flux through the surface of a cube defined by \( 0 \leq x=2,0 \leq y \leq 2,0 \leq z \leq 2 \). 3. Given that \( \overrightarrow{\mathbf{A}}=\frac{10}{r} \sin \theta \hat{\mathbf{a}}_{\phi}[\mathrm{Wb} / \mathrm{m}] \) in spherical coordinates, find \( \overrightarrow{\mathbf{H}} \) at the point \( (r, \theta, \phi)=\left(4 \mathrm{~m}, 60^{\circ}, 30^{\circ}\right) \).