= Solution
\b[Capacity convention.] The printed equation requires $c$ to be <volumetric heat capacity>, in $\mathrm{J\,m^{-3}\,K^{-1}}$. If “<specific heat capacity>” means per unit <mass> instead, the storage term is $\rho_m c\,\partial_tT$ and both denominators in the printed equation must contain $\rho_m c$. This distinction is fixed by dimensions.
For a fixed material region $V$, conservation of thermal <energy> gives the <heat balance with volumetric heat generation>,
$$
\frac d{dt}\int_VcT\,dV
=-\oint_{\partial V}\mathbf J\cdot\mathbf n\,dS+\int_VH\,dV.
$$
Use the <divergence theorem>, $\mathbf J=-\kappa\nabla T$ and constant coefficients. Since the region is arbitrary,
$$
\boxed{c\partial_tT=\kappa\nabla^2T+H,\qquad
\partial_tT=\frac\kappa c\nabla^2T+\frac Hc.}
$$
The <heat> source is power per volume, while $\kappa$ is <thermal conductivity> rather than <thermal diffusivity>.
For uniform <electrical conductivity> and imposed current, current <density> is $j=I/(\pi a^2)$ and <Joule heating> is $H=j^2/\sigma$. A long, steady, cylindrically symmetric wire therefore satisfies
$$
\frac1s\frac d{ds}\left(s\frac{dT}{ds}\right)=-\frac H\kappa.
$$
Integration gives $T'=-Hs/(2\kappa)+C/s$. Regularity at $s=0$ makes $C=0$, and the surface condition fixes the remaining constant. Thus the <Joule-heated cylindrical wire> has
$$
\boxed{T(s)=T_0+\frac{I^2}{4\pi^2a^4\kappa\sigma}(a^2-s^2).}
$$
Here $s$ denotes the printed radial coordinate $\rho$ and avoids confusing it with <mass> <density>. The centre is hotter by $I^2/(4\pi^2a^2\kappa\sigma)$.
For the air boundary, the internal flux at $s=a$ is $-\kappa T'(a)=Ha/2$. Equate this to $\alpha[T(a)-T_{\rm air}]$. The <Newton cooling of a Joule-heated wire> gives
$$
\boxed{T(a)=T_{\rm air}+\frac{Ha}{2\alpha}
=T_{\rm air}+\frac{I^2}{2\pi^2a^3\sigma\alpha}.}
$$
Multiplication by circumference $2\pi a$ verifies that surface loss per length equals $I^2/(\pi a^2\sigma)$, the electrical power per length. The full air-cooled profile uses this surface <temperature> plus the same parabolic interior excess, assuming constant material properties and negligible end losses.
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