Question $1[15$ points$]$:

The sinusoidal flux $$ in the magnetic core shown below links a winding of $500$ turns. The core also exhibits relative permeability of ${}_r=700,$ the core and gap cross$-$sectional areas of $A_c=10c{m}^2,A_g=11\mathrm{.}5c{m}^2,$ air gap length of $g=0\mathrm{.}050cm,$ and the mean core length of $l_c=50cm.$

Find the magnetic stored energy, W$,$ if the magnetic flux density $B_c=1\mathrm{.}75T$ and the inductance is $0\mathrm{.}3H_{.Do\text{n}\text{o}\text{t}\text{n}\text{e}\text{g}\text{l}\text{e}\text{c}\text{t}\text{c}\text{o}\text{r}\text{e}reluctancein\text{c}\text{o}\text{m}\text{p}\text{a}\text{r}\text{i}\text{s}\text{o}\text{n}\text{w}\text{i}\text{t}\text{h}\text{g}\text{a}\text{p}reluctance.}$ HINT: $R_c=\frac{l_c}{{}_{}{}_0{A}_c},R_g=\frac{g}{{}_0{A}_g},W=\frac{1}{2}L{i}^2.$

$[7$ points$]$