Problem $1.$ The cross$-$section of a coaxial cable is shown in the figure. The total length of the coaxial cable is $L.$ The radius of the inner conductor is $a.$ For the outer conductor, the radius of its inner surface is $b.$ Between the two conductors is an insulator with permittivity $={}_r{}_0,$ and permeability $={}_r{}_0.$ For the inner conductor, you may assume that charges and current are uniformly distributed on the surface $=a.$ For the outer conductor, you may assume charges and current are uniformly distributed on the inner surface $=b.$ Furthermore, you may assume that electric $/$ magnetic field can only exist within the insulator, i$.$e know ${}_r=3,{}_r=2,a=0\mathrm{.}5mm,b=2\mathrm{.}5mm,L=10m.$

$(a)(20pts)If$ the total charge $on$ the inner conductor $is1C,$ calculate the $absolute$ value $of$ voltage difference between the inner and outer conductor.

$(b)(20pts)$ Under the same condition $in(a),$ calculate the total energy stored $in$ the form $of$ electric field within the coaxial cable based $on$ integrating electric field energy density.

$(c)(20pts)$ Now suppose the inner conductor carries a current $of10A,$ calculate the total energy stored $in$ the form $of$ magnetic field within the cable based $on$ integrating magnetic field energy density. Explicitly confirms that the total magnetic energy equals $to\frac{1}{2}{L}_H{I}^2,$ where $L_{H}is$ the inductance $of$ the coaxial cable.