A wire of diameter $D=2mm$ and uniform temperature $T$ has an electrical resistance of $0\mathrm{.}01\frac{}{m}$ and a current flow of $16A.$

$($a$)$ What is the rate at which heat is dissipated per unit length of wire, in W$/$m$?$ What is the heat dissipation per unit volume within the wire, in $\frac{W}{m^3}?$

$($b$)$ If the wire is not insulated and is in ambient air and large surroundings for which $T_{}=T_{\text{s}\text{u}\text{r}}=20C,$ what is the temperature $T$ of the wire, in $C?$ The wire has an emissivity of $0\mathrm{.}3,$ and the coefficient associated with heat transfer by natural convection may be approximated by an expression of the form, $h=C[\frac{T-T_{}}{D}{]}^{\frac{1}{4}},$ where $C=1\mathrm{.}25\frac{W}{m^{\frac{7}{4}}}*K^{\frac{5}{4}}.$

$($c$)$ If the wire is coated with plastic insulation of $2-$mm thickness and a thermal conductivity of $0\mathrm{.}25\frac{W}{m}*K,$ what are the inner and outer surface temperatures of the insulation, both in $C?$ The insulation has an emissivity of $0\mathrm{.}9,$ and the convection coefficient is given by the expression of part $($b$).$

Part A

What is the rate at which heat is dissipated per unit length of wire, W$/$m$?$ What is the heat dissipation per unit volume within the wire, in $\frac{W}{m^3}?$

$E_g^{\text{'}}=,\frac{W}{m}$

$q^{}=,\frac{W}{m^3}$